Graduation Semester and Year

2009

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Guojun Liao

Abstract

The purpose of this dissertation is twofold: To present a new method of orthogonal grid generation and to investigate certain theoretical aspects of the optimal control approach to the image registration problem.In the first part of this dissertation, we will present a variational method of orthogonal grid generation which is based on the deformation method and solving Euler-Lagrange equations. Although the concept of grid generation has been studied extensively, the generation of orthogonal grids is still one of the most challenging problems of the grid generation methods.An orthogonal grid should offer significant advantages in the solution of systems of partial differential equations. Previous work requires a uniform grid to improve the orthogonality of grids.The grid deformation method provides size control via a monitor function but it has no control over gridline orthogonality.Our approach is to improve orthogonality while providing size distribution by the deformation method. In the second part, we replace the cost functional in the orthogonality problem with the sum of squared differences known as SSD to solve the Image registration problem. Image registration is a significant part of image processing. It is a process of finding an optimal geometric transformation between corresponding pixels that minimizes the SSD.In this optimal control approach, we use the grid deformation equations as constraints. In this dissertation, we prove the existence of optimal solutions using the direct method in the calculus of variations; discuss the non-uniqueness of solutions and the existence of Lagrange multipliers of the optimal control problem for image registration using an abstract theorem concerning the existence of Lagrange multipliers on Banach spaces.

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

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