Journal on Computing and Cultural Heritage (JOCCH)
Unified Computation of Strict Maximum Likelihood for Geometric Fitting
Journal of Mathematical Imaging and Vision
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This thesis is devoted to the problem of fitting parameterized curves to noisy data. In the first chapter we analyze the orthogonal least squares fit (OLSF) of simple curves such as lines and circles. We investigates theoretical issues such as the existence and the uniqueness of LSF and present various parameterization schemes. We evaluate several popular algorithms and propose a new one that surpasses the existing methods in reliability. Finally we discuss and compare direct (algebraic) circle fits. The second chapter is devoted to the statistical efficiency of curve fitting algorithms. Under certain assumptions (known as Cartesian and radial functional models), we derive asymptotic expressions for the bias and the covariance matrix of the parameter estimates. We extend Kanatani's version of the Cramer-Rao lower bound to more general estimates that include many popular algorithms (most notably, the orthogonal least squares and algebraic fits). We also show that the gradient-weighted algebraic fit (GRAF) is statistically efficient and describe all other statistically efficient algebraic fits. In the third chapter we show that, sometimes, the GRAF admits a substantial reduction of complexity and we determine the precise conditions under which this is possible. It turns out that this is, indeed, possible when one fits circles but not ellipses or hyperbolas.