Statistical Optimization for Geometric Computation

Introduction - The aims of this book; the features of this book; organization and background the analytical mind: strengh and weakness. Fundamentals of linear algebra - Vector and matrix calculus; Eigenvalue problem; linear systems and optimization; matrix and tensor algebra. Probabilities and statistical estimation - probability distributions; manifolds and local distributions; gaussian distributions and X2 distributions; statistical estimation for gaussian models; general statistical estimation; maximum likelihood estimation; Akaike information criterion. Representation of geometric objects - image points and image lines; space points and space lines; space planes; conics; space conics and quadrics; coordinate transformation and projection. Geometric correction - general theory; correction of image points and image lines; correction of space points and space lines; correction of space planes; orthogonality correction; conic incidence correction. 3-D computation by stereo vision - epipolar constraint; optimal correction of correspondence; 3-D reconstruction of points ; 3-D reconstruction of lines; optimal back projection onto a space plane; scenes infinitely far away; camera calibration errors. Parametric fitting - general theory; optimal fitting for image points; optimal fitting for image lines; optimal fitting for space points; optimal fitting for space lines; optimal fitting for space planes. Optimal filter - general theory; iterative estimation scheme; effective gradient approximation; reduction from the klaman filter; estimation from linear hypotheses. Renormalization - eigenvector fit; unbiased eigenvector; generalized eigenvalue fit; renormalization; lincarization; second order renormalization. Applications of geometric estimation - image line fitting; conic fitting; space plane fitting by range sensing; space plane fitting by stereo vision. 3-D motion analysis - general theory; lincarization and renormalization; optimal correction and decomposition; reliability of 3-D reconstruction; critical surfaces; 3-D reconstruction from planar surface motion; camera rotation and information. 3-D interpretation of optical flow - optical flow detection; theoretical basis of 3-D interpretation; optimal estimation of motion parameters. (Part contents).