All ETDs from UAB

Advisory Committee Chair

Nikolai Chernov

Advisory Committee Members

Wei-Shen Hsia

Charles R Katholi

Ian Knowles

Boris Kunin

Document Type

Dissertation

Date of Award

2011

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

This dissertation is devoted to the problem of fitting geometric curves such as lines, circles, and ellipses to a set of experimental observations whose both coordinates are contaminated with noisy errors. This kind of regression is called Errors-in-Variables models (EIV), which is quite different and much more difficult to solve than the classical regression. This research study is motivated by the wide range of EIV applications in computer vision and image processing. We adopted statistical assumptions suitable for these applications and we studied the statistical properties of two kinds of fits; geometric fit and algebraic fit for line, circle and ellipse fittings. The main contribution of the dissertation is proposing several fits for both circle and ellipse fitting problems. These proposed fits were discovered after we developed our unconventional statistical analysis that allowed us to effectively assess EIV parameter estimates. This approach was validated through a series of numerical tests. We theoretically compared the most popular fits for circles and ellipses to each other and we showed why, and by how much, each fit differs from others. Our theoretical comparison leads to new unbeatable fits with superior characteristics that surpass all existing fits theoretically and experimentally. Another contribution is discussing some statistical issues in circle fitting. We proved that the most popular and accurate fits have infinite absolute first moment while the one with finite first moment is, paradoxically, the least accurate and has the heaviest bias. Also, we proved that the geometric fit returns absolutely continuous estimator.

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