All ETDs from UAB

Advisory Committee Chair

Nikolai Chernov

Advisory Committee Members

Wei-Shen Hsia

Ian Knowles

Boris Kunin

Charles Katholi

Document Type

Dissertation

Date of Award

2012

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

This dissertation is devoted to the study of a popular regression model: Error-In-Variable Model, which has been commonly recognized as one of the key components of computer vision research. In EIV model, a set of data points whose (x,y) coordinates are subject to random errors is fitted by some geometric shapes such as lines, circles and ellipses. The geometric fitting which minimizes the sum of orthogonal distances from points to geometric shapes is universally recognized as the most desirable solution of the fitting problem. However, there is no explicit form of the solution for nonlinear models (circles, ellipses etc). The problem of fitting circles has been investigated intensively over past a few decades and all major issue appeared to be resolved. Our analysis will focus on a more sophisticated model - fitting ellipses to a set of points. We will address the issues of existence and uniqueness of the best fitting solution, study the parameter space of all quadratic curves and properties of the objective function and show some peculiar feature of the estimates of geometric parameters for the best fitting ellipse: they have no finite moments. Our results promote understanding of why computer algorithms keep diverging, return nonsense or crash altogether and help development of more robust, efficient fitting schemes.

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