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IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO PO ISS ISSON AND ON AND ON AND ON AND ON AND ON AND ON AND SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONS A ThesisA ThesisA ThesisA Thesis A ThesisA ThesisA Thesis by ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to the Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studies the Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studiesthe Office of Graduate Studies Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University – Commerce CommerceCommerceCommerce Commerce In partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements for the degree of for the degree of for the degree of for the degree offor the degree of for the degree offor the degree of MASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCE MASTER OF SCIENCEMASTER OF SCIENCE MASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCE AugustAugustAugustAugustAugustAugust 2014 IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO POISPOISPOIS SON AND SON AND SON AND SON AND SON AND SON AND SON AND SON AND SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONS A ThesisA ThesisA ThesisA Thesis A ThesisA ThesisA Thesis by ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN Approved by:Approved by: Approved by:Approved by:Approved by: Approved by: Advisor:Advisor:Advisor:Advisor:Advisor:Advisor: Advisor: Nikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev Sirakov Committee: Committee:Committee:Committee:Committee:Committee:Committee:Committee: Tingxiu WangTingxiu WangTingxiu Wang Tingxiu Wang Tingxiu Wang Tingxiu WangTingxiu WangTingxiu Wang Yelin OuYelin OuYelin OuYelin OuYelin Ou Yelin Ou Head of Department:Head of Department:Head of Department:Head of Department: Head of Department:Head of Department: Head of Department: Head of Department:Head of Department:Head of Department: Head of Department: Tingxiu WangTingxiu WangTingxiu Wang Tingxiu Wang Tingxiu Wang Tingxiu WangTingxiu WangTingxiu Wang Dean of College:Dean of College:Dean of College:Dean of College: Dean of College:Dean of College:Dean of College:Dean of College:Dean of College: Daniel EdDaniel EdDaniel Ed Daniel EdDaniel EdDaniel Ed Daniel Edel manmanman Dean oDean oDean oDean o f Graduate Studies: f Graduate Studies: f Graduate Studies: f Graduate Studies:f Graduate Studies:f Graduate Studies: f Graduate Studies:f Graduate Studies: f Graduate Studies:f Graduate Studies:f Graduate Studies: Arlene HorneArlene Horne Arlene HorneArlene Horne Arlene HorneArlene HorneArlene Horneiii Copyright Copyright Copyright Copyright Copyright Copyright © 2014 Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden iv ABSTRACTABSTRACT ABSTRACTABSTRACTABSTRACTABSTRACT IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONS Adam Adam Adam Adam Bowden, MSBowden, MS Bowden, MS Bowden, MS Bowden, MS Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University – Commerce, 2014 Commerce, 2014Commerce, 2014Commerce, 2014 Commerce, 2014Commerce, 2014 Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is central tocentral tocentral to central tocentral tocentral tocentral to central to the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methods available ods available ods available ods available ods available ods available ods available ods available ods available ods available in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are the active contour modelthe active contour model the active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour model the active contour model the active contour model the active contour model the active contour modelthe active contour models. Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these models odels odels odels fall fallfall into into into threethree three categories based oncategories based oncategories based oncategories based oncategories based oncategories based on categories based oncategories based oncategories based on categories based oncategories based oncategories based on parametri parametriparametriparametriparametriparametriparametri c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets, c curves, level sets,c curves, level sets,c curves, level sets, c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets, c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets, or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this study, research is conducted intostudy, research is conducted intostudy, research is conducted into study, research is conducted intostudy, research is conducted into study, research is conducted intostudy, research is conducted intostudy, research is conducted intostudy, research is conducted into study, research is conducted into study, research is conducted intostudy, research is conducted into study, research is conducted into study, research is conducted intostudy, research is conducted intostudy, research is conducted into study, research is conducted intostudy, research is conducted into study, research is conducted into new and current active contour new and current active contournew and current active contournew and current active contournew and current active contour new and current active contournew and current active contour new and current active contour new and current active contour new and current active contournew and current active contournew and current active contournew and current active contournew and current active contournew and current active contour new and current active contour new and current active contour methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential equationequation equationequationequation s and the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energy and the minimization of energy and the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energy functionals. functionals. functionals. functionals. functionals. functionals. functionals. Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. Then aThen a Then a n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical method method method method is shown is shown is shown is shown is shown which which which which evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by solvsolv solvinging the Eulerthe Euler the Euler the Euler the Eulerthe EulerLagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial diff diffdifferential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. Ne xt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s equationequation equationequationequation s over image d over image dover image d over image dover image dover image dover image dover image d ata is ata is ata is ata is ata is ata is ata is presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with theirtheir theirtheir application inapplication in application inapplication inapplication inapplication inapplication inapplication in application in defining external energy indefining external energy indefining external energy in defining external energy in defining external energy in defining external energy in defining external energy in defining external energy indefining external energy in defining external energy indefining external energy indefining external energy in defining external energy indefining external energy indefining external energy indefining external energy in defining external energy in defining external energy in active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. Finally, Finally, Finally, Finally, Finally, Finally, Finally, experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. v ACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTS I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me made de this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and insightinsight insightinsightinsight . His . His . His . His support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu Wang, for their time, support,Wang, for their time, support,Wang, for their time, support, Wang, for their time, support, Wang, for their time, support, Wang, for their time, support,Wang, for their time, support, Wang, for their time, support,Wang, for their time, support,Wang, for their time, support,Wang, for their time, support, Wang, for their time, support, Wang, for their time, support, andand inputinput , Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and finally finally finallyfinallyfinally, my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants who havwho hav who have been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, andand encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to expressexpress expressexpress my gratitude tomy gratitude tomy gratitude tomy gratitude tomy gratitude to my gratitude tomy gratitude tomy gratitude tomy gratitude to my gratitude to my gratitude to the rest of the rest of the rest of the rest of the rest of the rest of the rest of the rest of the rest of great individualsgreat individuals great individualsgreat individualsgreat individuals great individuals great individualsgreat individualsgreat individuals great individualsgreat individuals in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M University University University University University University University University University – Comm Commerce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in which to work.which to work. which to work.which to work. which to work.which to work. which to work. vi TABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTS TABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTS LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURESLIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURES................................ ................................ ................................ ................................ VIIVII CHAPTER CHAPTERCHAPTERCHAPTERCHAPTERCHAPTER ................................ ................................ ................................ ................................ ............... 1 1. INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION ................................ ................................ ................................ .................... 1 2. THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND ................................ ................................ .......................... 3 EulerEuler EulerEulerLagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange Equation Lagrange EquationLagrange EquationLagrange Equation ................................ ................................ ................................ .. 3 Active ContoursActive ContoursActive ContoursActive ContoursActive ContoursActive ContoursActive Contours Active Contours ................................ ................................ ................................ ................ 4 Poisson EquationPoisson Equation Poisson EquationPoisson EquationPoisson Equation Poisson Equation Poisson EquationPoisson EquationPoisson Equation ................................ ................................ ................................ ............... 7 Schrödinger's EquationSchrödinger's EquationSchrödinger's Equation Schrödinger's Equation Schrödinger's EquationSchrödinger's EquationSchrödinger's EquationSchrödinger's EquationSchrödinger's Equation Schrödinger's Equation Schrödinger's EquationSchrödinger's EquationSchrödinger's Equation ................................ ................................ ................................ ..... 8 3. METHODSMETHODSMETHODSMETHODSMETHODSMETHODSMETHODS ................................ ................................ ................................ ............................... 9 Numerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerLagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange Equation Lagrange EquationLagrange EquationLagrange Equation ................................ .......... 9 Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy Definition ................................ ................................ ................ 12 Schrödinger External Energy DefinitionSchrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy DefinitionSchrödinger External Energy Definition ................................ ................................ ......... 14 4. FINDING AND RESULTSFINDING AND RESULTS FINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTS FINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTS ................................ ................................ ................................ ..... 16 Poisson Poisson Poisson Poisson Poisson External EnergyExternal Energy External EnergyExternal Energy External EnergyExternal Energy External EnergyExternal EnergyExternal EnergyExternal Energy ................................ ................................ ................................ . 16 Schrödinger External EnergySchrödinger External EnergySchrödinger External Energy Schrödinger External Energy Schrödinger External EnergySchrödinger External Energy Schrödinger External Energy Schrödinger External EnergySchrödinger External Energy Schrödinger External EnergySchrödinger External Energy Schrödinger External Energy Schrödinger External EnergySchrödinger External Energy ................................ ................................ .......................... 18 5. CONCLUSION CONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSION ................................ ................................ ................................ ....................... 22 REFERENCES REFERENCESREFERENCESREFERENCES REFERENCESREFERENCES REFERENCES................................ ................................ ................................ ................................ ........ 24 VITAVITAVITAVITA ................................ ................................ ................................ ................................ ....................... 27 vii LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURESLIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURES FIGUREFIGUREFIGUREFIGUREFIGURE 1. Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this imageimageimageimageimage ................................ ................................ ................................ ................................ ........... 13 2. 𝛻𝐸𝑖𝑚𝑔 vector field asvector field asvector field asvector field asvector field as vector field asvector field asvector field as vector field asvector field as received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. ........................... 14 3. Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o n this image, and the n this image, and the n this image, and then this image, and the n this image, and then this image, and then this image, and then this image, and then this image, and the n this image, and then this image, and the n this image, and the 𝛻𝐸𝑖𝑚𝑔 vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image. vector field over this filtered image. vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image................................. ................................ ................ 15 4. Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α = 0.5 and = 0.5 and β = 0.2 β = 0.2 ................................ ................................ ................................ ................................ ......... 16 5. Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α = 0.1 and = 0.1 and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ......... 16 6. Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α = 300 and = 300 and = 300 and β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. filter of kernel size 10. filter of kernel size 10.filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. ................................ ................................ .. 17 7. Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α = 1 and = 1 and β = 1 β = 1 ................................ ................................ ................................ ................................ ............ 17 8. Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α = 3750 and = 3750 and β = 1 β = 1 ................................ ................................ ................................ ................................ ............ 17 9. Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, α = 0.4 and = 0.4 and β = 0.2 β = 0.2 ................................ ................................ ................................ ...................... 18 10. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. 100 contour points 100 contour points100 contour points 100 contour points 100 contour points100 contour points100 contour points, ε = 0.01, α ε = 0.01, α = 0.1 and and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ... 19 11. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α = 0.5 and and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ... 19 viii 12. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α = 300 and = 300 and β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. filter of kernel size 10. filter of kernel size 10.filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. ................................ ................................ .. 20 13. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 , α = 1 and = 1 and β = 1 β = 1 ................................ ................................ ................................ ................................ ............ 20 14. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, α = 3750 and β = 1 β = 1 ................................ ................................ ................................ ................................ ...... 20 15. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, α = 0.1 and and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ... 21 16. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation attempt with 140 contour points,empt with 140 contour points,empt with 140 contour points, empt with 140 contour points, empt with 140 contour points,empt with 140 contour points,empt with 140 contour points, empt with 140 contour points,empt with 140 contour points, empt with 140 contour points, empt with 140 contour points,empt with 140 contour points,empt with 140 contour points,empt with 140 contour points, ε = 0.01, ε = 0.01, α = 0.4 and = 0.4 and β = 0.2 β = 0.2 ................................ ................................ ................................ ...................... 21 1 Chapter 1 Chapter 1 Chapter 1Chapter 1 INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medic
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Title  Implementation of a Splitting Finite Difference Scheme to Segment Images with the Solution to the Poisson and Schr?dinger Equations 
Author  Bowden, Adam 
Subject  Mathematics; Computer science; Information technology 
Abstract  IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO PO ISS ISSON AND ON AND ON AND ON AND ON AND ON AND ON AND SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER SCHRÖDINGER EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONS A ThesisA ThesisA ThesisA Thesis A ThesisA ThesisA Thesis by ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to Submitted to the Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studies the Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studies the Office of Graduate Studiesthe Office of Graduate Studiesthe Office of Graduate Studies Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University – Commerce CommerceCommerceCommerce Commerce In partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirementsIn partial fulfillment of the requirementsIn partial fulfillment of the requirements In partial fulfillment of the requirements for the degree of for the degree of for the degree of for the degree offor the degree of for the degree offor the degree of MASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCE MASTER OF SCIENCEMASTER OF SCIENCE MASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCEMASTER OF SCIENCE AugustAugustAugustAugustAugustAugust 2014 IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO POISPOISPOIS SON AND SON AND SON AND SON AND SON AND SON AND SON AND SON AND SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONS A ThesisA ThesisA ThesisA Thesis A ThesisA ThesisA Thesis by ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN ADAM BOWDENADAM BOWDENADAM BOWDENADAM BOWDEN Approved by:Approved by: Approved by:Approved by:Approved by: Approved by: Advisor:Advisor:Advisor:Advisor:Advisor:Advisor: Advisor: Nikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev SirakovNikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev SirakovNikolay Metodiev Sirakov Nikolay Metodiev Sirakov Committee: Committee:Committee:Committee:Committee:Committee:Committee:Committee: Tingxiu WangTingxiu WangTingxiu Wang Tingxiu Wang Tingxiu Wang Tingxiu WangTingxiu WangTingxiu Wang Yelin OuYelin OuYelin OuYelin OuYelin Ou Yelin Ou Head of Department:Head of Department:Head of Department:Head of Department: Head of Department:Head of Department: Head of Department: Head of Department:Head of Department:Head of Department: Head of Department: Tingxiu WangTingxiu WangTingxiu Wang Tingxiu Wang Tingxiu Wang Tingxiu WangTingxiu WangTingxiu Wang Dean of College:Dean of College:Dean of College:Dean of College: Dean of College:Dean of College:Dean of College:Dean of College:Dean of College: Daniel EdDaniel EdDaniel Ed Daniel EdDaniel EdDaniel Ed Daniel Edel manmanman Dean oDean oDean oDean o f Graduate Studies: f Graduate Studies: f Graduate Studies: f Graduate Studies:f Graduate Studies:f Graduate Studies: f Graduate Studies:f Graduate Studies: f Graduate Studies:f Graduate Studies:f Graduate Studies: Arlene HorneArlene Horne Arlene HorneArlene Horne Arlene HorneArlene HorneArlene Horneiii Copyright Copyright Copyright Copyright Copyright Copyright © 2014 Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden Adam Bowden iv ABSTRACTABSTRACT ABSTRACTABSTRACTABSTRACTABSTRACT IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES IMPLEMENTATION OF A SPLITTING FINITE DIFFERENCE SCHEME TO SEGMENT IMAGES WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO WITH THE SOLUTION TO POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND POISSON AND SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER SCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGERSCHRÖDINGER EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONS Adam Adam Adam Adam Bowden, MSBowden, MS Bowden, MS Bowden, MS Bowden, MS Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University Texas A&M University – Commerce, 2014 Commerce, 2014Commerce, 2014Commerce, 2014 Commerce, 2014Commerce, 2014 Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhD Advisor: Nikolay Metodiev Sirakov, PhDAdvisor: Nikolay Metodiev Sirakov, PhD Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is Image segmentation is central tocentral tocentral to central tocentral tocentral tocentral to central to the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one the theory and practice of image analysis is one fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of meth fundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methfundamental concepts in the applications of computer vision and recognition. Of methods available ods available ods available ods available ods available ods available ods available ods available ods available ods available in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are in this field, some of the most often applied arein this field, some of the most often applied arein this field, some of the most often applied are in this field, some of the most often applied are the active contour modelthe active contour model the active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour modelthe active contour model the active contour model the active contour model the active contour model the active contour modelthe active contour models. Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these . Broadly, these models odels odels odels fall fallfall into into into threethree three categories based oncategories based oncategories based oncategories based oncategories based oncategories based on categories based oncategories based oncategories based on categories based oncategories based oncategories based on parametri parametriparametriparametriparametriparametriparametri c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets, c curves, level sets,c curves, level sets,c curves, level sets, c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets, c curves, level sets,c curves, level sets,c curves, level sets,c curves, level sets, or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this or partial differential equations. In this study, research is conducted intostudy, research is conducted intostudy, research is conducted into study, research is conducted intostudy, research is conducted into study, research is conducted intostudy, research is conducted intostudy, research is conducted intostudy, research is conducted into study, research is conducted into study, research is conducted intostudy, research is conducted into study, research is conducted into study, research is conducted intostudy, research is conducted intostudy, research is conducted into study, research is conducted intostudy, research is conducted into study, research is conducted into new and current active contour new and current active contournew and current active contournew and current active contournew and current active contour new and current active contournew and current active contour new and current active contour new and current active contour new and current active contournew and current active contournew and current active contournew and current active contournew and current active contournew and current active contour new and current active contour new and current active contour methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential methods using partial differential equationequation equationequationequation s and the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energy and the minimization of energy and the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energy and the minimization of energyand the minimization of energy functionals. functionals. functionals. functionals. functionals. functionals. functionals. Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods Furthermore, new methods implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and implemented and coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. coded in MatLab are demonstrated. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. First, relevant theory is discussed. Then aThen a Then a n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical n original numerical method method method method is shown is shown is shown is shown is shown which which which which evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by evolves a contour to object boundaries by solvsolv solvinging the Eulerthe Euler the Euler the Euler the Eulerthe EulerLagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial Lagrange partial diff diffdifferential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. erential equation. Ne xt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson'sxt, the solution to Poisson's xt, the solution to Poisson'sxt, the solution to Poisson's and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s and Schrödinger’s equationequation equationequationequation s over image d over image dover image d over image dover image dover image dover image dover image d ata is ata is ata is ata is ata is ata is ata is presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with presented along with theirtheir theirtheir application inapplication in application inapplication inapplication inapplication inapplication inapplication in application in defining external energy indefining external energy indefining external energy in defining external energy in defining external energy in defining external energy in defining external energy in defining external energy indefining external energy in defining external energy indefining external energy indefining external energy in defining external energy indefining external energy indefining external energy indefining external energy in defining external energy in defining external energy in active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. active contour methods. Finally, Finally, Finally, Finally, Finally, Finally, Finally, experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded experimental results are given to validate the theory and implementation by applying coded methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. methods to a set of synthetic and skin lesion images. v ACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTSACKNOWLEDGMENTS I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me maI would like to express my gratitude all who have supported and encouraged me ma I would like to express my gratitude all who have supported and encouraged me made de this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. this work possible. In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I In particular, I would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank would like to thank my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, my advisor Dr. Nikolay Sirakov for his time, knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and knowledge, patience, and insightinsight insightinsightinsight . His . His . His . His support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity support was indispensable, and I am grateful for the opportunity to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my to work with him. I would also like thank my committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu committee members, Dr. Yelin Ou and Tingxiu Wang, for their time, support,Wang, for their time, support,Wang, for their time, support, Wang, for their time, support, Wang, for their time, support, Wang, for their time, support,Wang, for their time, support, Wang, for their time, support,Wang, for their time, support,Wang, for their time, support,Wang, for their time, support, Wang, for their time, support, Wang, for their time, support, andand inputinput , Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support Dr. Pamela Webster and KaSai Un for their support encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, and encouragement throughout my time as a graduate assistant, andencouragement throughout my time as a graduate assistant, and finally finally finallyfinallyfinally, my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants my fellow graduate assistants who havwho hav who have been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support,e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, e been an ever present source of friendship, support, andand encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to encouragement. I would like to expressexpress expressexpress my gratitude tomy gratitude tomy gratitude tomy gratitude tomy gratitude to my gratitude tomy gratitude tomy gratitude tomy gratitude to my gratitude to my gratitude to the rest of the rest of the rest of the rest of the rest of the rest of the rest of the rest of the rest of great individualsgreat individuals great individualsgreat individualsgreat individuals great individuals great individualsgreat individualsgreat individuals great individualsgreat individuals in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M in the Mathematics Department at Texas A&M University University University University University University University University University – Comm Commerce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in erce, quite possibly the best department ever in which to work.which to work. which to work.which to work. which to work.which to work. which to work. vi TABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTS TABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTSTABLE OF CONTENTS LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURESLIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURES................................ ................................ ................................ ................................ VIIVII CHAPTER CHAPTERCHAPTERCHAPTERCHAPTERCHAPTER ................................ ................................ ................................ ................................ ............... 1 1. INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION ................................ ................................ ................................ .................... 1 2. THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND THEORETICAL BACKGROUNDTHEORETICAL BACKGROUNDTHEORETICAL BACKGROUND ................................ ................................ .......................... 3 EulerEuler EulerEulerLagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange Equation Lagrange EquationLagrange EquationLagrange Equation ................................ ................................ ................................ .. 3 Active ContoursActive ContoursActive ContoursActive ContoursActive ContoursActive ContoursActive Contours Active Contours ................................ ................................ ................................ ................ 4 Poisson EquationPoisson Equation Poisson EquationPoisson EquationPoisson Equation Poisson Equation Poisson EquationPoisson EquationPoisson Equation ................................ ................................ ................................ ............... 7 Schrödinger's EquationSchrödinger's EquationSchrödinger's Equation Schrödinger's Equation Schrödinger's EquationSchrödinger's EquationSchrödinger's EquationSchrödinger's EquationSchrödinger's Equation Schrödinger's Equation Schrödinger's EquationSchrödinger's EquationSchrödinger's Equation ................................ ................................ ................................ ..... 8 3. METHODSMETHODSMETHODSMETHODSMETHODSMETHODSMETHODS ................................ ................................ ................................ ............................... 9 Numerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerNumerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the Euler Numerical Scheme for Solving the EulerNumerical Scheme for Solving the EulerLagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange EquationLagrange EquationLagrange EquationLagrange Equation Lagrange Equation Lagrange EquationLagrange EquationLagrange Equation ................................ .......... 9 Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy Definition Poisson External Energy Definition Poisson External Energy DefinitionPoisson External Energy DefinitionPoisson External Energy Definition ................................ ................................ ................ 12 Schrödinger External Energy DefinitionSchrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy Definition Schrödinger External Energy Definition Schrödinger External Energy DefinitionSchrödinger External Energy DefinitionSchrödinger External Energy Definition ................................ ................................ ......... 14 4. FINDING AND RESULTSFINDING AND RESULTS FINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTS FINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTSFINDING AND RESULTS ................................ ................................ ................................ ..... 16 Poisson Poisson Poisson Poisson Poisson External EnergyExternal Energy External EnergyExternal Energy External EnergyExternal Energy External EnergyExternal EnergyExternal EnergyExternal Energy ................................ ................................ ................................ . 16 Schrödinger External EnergySchrödinger External EnergySchrödinger External Energy Schrödinger External Energy Schrödinger External EnergySchrödinger External Energy Schrödinger External Energy Schrödinger External EnergySchrödinger External Energy Schrödinger External EnergySchrödinger External Energy Schrödinger External Energy Schrödinger External EnergySchrödinger External Energy ................................ ................................ .......................... 18 5. CONCLUSION CONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSIONCONCLUSION ................................ ................................ ................................ ....................... 22 REFERENCES REFERENCESREFERENCESREFERENCES REFERENCESREFERENCES REFERENCES................................ ................................ ................................ ................................ ........ 24 VITAVITAVITAVITA ................................ ................................ ................................ ................................ ....................... 27 vii LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURESLIST OF FIGURESLIST OF FIGURES LIST OF FIGURESLIST OF FIGURES LIST OF FIGURES FIGUREFIGUREFIGUREFIGUREFIGURE 1. Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this Simple image and the plot of solution values to Poisson’s equation solved over this imageimageimageimageimage ................................ ................................ ................................ ................................ ........... 13 2. 𝛻𝐸𝑖𝑚𝑔 vector field asvector field asvector field asvector field asvector field as vector field asvector field asvector field as vector field asvector field as received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition. received from the Poisson potential energy definition.received from the Poisson potential energy definition.received from the Poisson potential energy definition. ........................... 14 3. Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o Simple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) oSimple image, the result of Schrödinger filter (kernel size 3x3) o n this image, and the n this image, and the n this image, and then this image, and the n this image, and then this image, and then this image, and then this image, and then this image, and the n this image, and then this image, and the n this image, and the 𝛻𝐸𝑖𝑚𝑔 vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image. vector field over this filtered image. vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image. vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image.vector field over this filtered image................................. ................................ ................ 15 4. Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α Poisson external energy. One object result with 100 contour points, ε = 0.01, α = 0.5 and = 0.5 and β = 0.2 β = 0.2 ................................ ................................ ................................ ................................ ......... 16 5. Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α Poisson external energy. Multiple object results with 512 contour points, ε = 0.01, α = 0.1 and = 0.1 and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ......... 16 6. Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α Poisson external energy. Skin lesion results with 256 contour points, ε = 0.1, α = 300 and = 300 and = 300 and β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. filter of kernel size 10. filter of kernel size 10.filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. ................................ ................................ .. 17 7. Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α = 1 and = 1 and β = 1 β = 1 ................................ ................................ ................................ ................................ ............ 17 8. Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α Poisson external energy. Boat image result with 100 contour points, ε = 0.02, α = 3750 and = 3750 and β = 1 β = 1 ................................ ................................ ................................ ................................ ............ 17 9. Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, Poisson external energy. Internal segmentation attempt with 140 contour points, ε = 0.01, α = 0.4 and = 0.4 and β = 0.2 β = 0.2 ................................ ................................ ................................ ...................... 18 10. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. Schrödinger external energy. Incomplete segmentation. 100 contour points 100 contour points100 contour points 100 contour points 100 contour points100 contour points100 contour points, ε = 0.01, α ε = 0.01, α = 0.1 and and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ... 19 11. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. Schrödinger external energy. Complete segmentation. 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α 100 contour points, ε = 0.01, α = 0.5 and and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ... 19 viii 12. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α external energy. Skin lesion results with 256 contour points, ε = 0.1, α = 300 and = 300 and β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre β = 10, and Schrödinger pre filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. filter of kernel size 10. filter of kernel size 10.filter of kernel size 10. filter of kernel size 10.filter of kernel size 10.filter of kernel size 10.filter of kernel size 10. ................................ ................................ .. 20 13. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 external energy. Boat image result with 100 contour points, ε = 0.02 , α = 1 and = 1 and β = 1 β = 1 ................................ ................................ ................................ ................................ ............ 20 14. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1 external energy. Boat image result with 1external energy. Boat image result with 1external energy. Boat image result with 1 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, 00 contour points, ε = 0.02, α = 3750 and β = 1 β = 1 ................................ ................................ ................................ ................................ ...... 20 15. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with external energy. Multiple object results with 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, 512 contour points, ε = 0.01, α = 0.1 and and β = 0.1 β = 0.1 ................................ ................................ ................................ ................................ ... 21 16. Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation attexternal energy. Internal segmentation att external energy. Internal segmentation attexternal energy. Internal segmentation attempt with 140 contour points,empt with 140 contour points,empt with 140 contour points, empt with 140 contour points, empt with 140 contour points,empt with 140 contour points,empt with 140 contour points, empt with 140 contour points,empt with 140 contour points, empt with 140 contour points, empt with 140 contour points,empt with 140 contour points,empt with 140 contour points,empt with 140 contour points, ε = 0.01, ε = 0.01, α = 0.4 and = 0.4 and β = 0.2 β = 0.2 ................................ ................................ ................................ ...................... 21 1 Chapter 1 Chapter 1 Chapter 1Chapter 1 INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION INTRODUCTIONINTRODUCTIONINTRODUCTION Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary Of particular interest in the field of image analysis is segmentation via boundary detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medical detection and extraction. This task has applications in many important, disparate fields such as medic 
Date  2014 
Faculty Advisor  Sirakov, Nikolay M 
Committee Members 
Ou, Yelin Wang, Tingxiu 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSMathematics 
Degree Awarded  M.S. 
Pages  35 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



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