All ETDs from UAB

Advisory Committee Chair

Nikolai Chernov

Advisory Committee Members

Boris Kunin

Charles Katholi

Ian Knowles

Wei-Shen Hsia

Document Type

Dissertation

Date of Award

2012

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

Fitting quadratic curves and surfaces to observed images is one of the basic tasks in pattern recognition and computer vision. The most accurate and robust fit is obtained by minimizing geometric (orthogonal) distances, but this problem has no closed form solution. All known algorithms are heuristic and either computationally costly or have drawbacks in accuracy and convergence. We develop a mathematically rigorous approach to the study of the geometric fitting problem. We begin with a thorough investigation of relevant theoretical aspects of the problem and then move on to its practical solution. We focus on image processing applications, where data points come from a picture, photograph, map, etc. Therefore we adopt standard statistical assumptions that are appropriate for these applications. We investigate the existence of the best fit, describe various parameterization schemes and analyze the behavior of the objective function on the parameter space. Our goal is to provide a robust, efficient projection method and to develop new fitting schemes, indicating how to combine them to achieve the best performance. Eberly discovered a remarkably fast and totally reliable projection algorithm for ellipses which we generalize to all the other quadratic curves and surfaces and provide proofs of convergence. Ahn has classified various approaches to the fitting problem. We develop our implicit fitting algorithm based on one of his approaches and demonstrate that it is the most efficient one by comparison to other known algorithms. By combining projection and minimization steps together, we give a complete, reliable and efficient geometric fitting scheme for fitting quadratic curves and surfaces of all kinds.

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