Statistical Analysis Of Curve Fitting In Errors In-Variables Models

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

Statistical Analysis of Curve Fitting in Errors-In-Variables Models

Fitting geometric curves such as lines, circles, and ellipses to data is a fundamental task in computer vision, image processing, and pattern recognition. These problems are classified as Errors-in-Variables models (EIV), where both coordinates of observations are subject to noise, making them more challenging than classical regression. This talk explores statistical methods to assess EIV parameter estimators, comparing geometric and algebraic fits for circles and ellipses. We present new estimators with superior accuracy and bias reduction

Fitting Concentric Elliptical Shapes Under General Model

The problem of fitting concentric ellipses is a vital problem in image processing, pattern recognition, and astronomy. Several methods have been developed but all address very special cases. In this paper, this problem has been investigated under a more general setting, and two estimators for estimating the parameters have been proposed. Since both estimators are obtained iterative fashion, several numerical schemes are investigated and the best initial guess is determined. Furthermore, the constraint Cram\'{e} Rao lower bound for this problem is derived and it is compared with the variance of each estimator. Finally, our theory is assessed and validated by a series of numerical experiments on both real and synthetic data.Comment: it has a serious mistake that needs major revision and it could take several months to fi