Direct Estimation of Homogeneous Vectors: An Ill-Solved Problem in Computer Vision

Computer Vision theory is firmly rooted in Projective Geometry, whereby geometric objects can be effectively modeled by homogeneous vectors. We begin from Gauss's 200 year old theorem of least squares to derive a generic algorithm for the direct estimation of homogeneous vectors. We uncover the common link of previous methods, showing that direct estimation is not an ill-conditioned problem as is the popular belief, but has merely been an ill-solved problem. Results show improvements in goodness-of-fit and numerical stability, and demonstrate that “data normalization” is unnecessary for a well-founded algorithm.

[1]  W. Gander.  Least squares with a quadratic constraint , 1980 .

[2]  Wolfgang Förstner,et al.  Uncertainty and Projective Geometry , 2005 .

[3]  Gene H. Golub,et al.  The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate , 1972, Milestones in Matrix Computation.

[4]  G. Stewart,et al.  A generalization of the Eckart-Young-Mirsky matrix approximation theorem , 1987 .

[5]  Richard I. Hartley,et al.  In Defense of the Eight-Point Algorithm , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[6]  PAUL D. SAMPSON,et al.  Fitting conic sections to "very scattered" data: An iterative refinement of the bookstein algorithm , 1982, Comput. Graph. Image Process..

[7]  Yves Nievergelt,et al.  Hyperspheres and hyperplanes fitted seamlessly by algebraic constrained total least-squares , 2001 .

[8]  Andrew Zisserman,et al.  Multiple View Geometry in Computer Vision (2nd ed) , 2003 .

[9]  Gabriel Taubin,et al.  Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  Carl Friedrich Gauss,et al.  Méthode des moindres carrés : mémoires sur la combinaison des observations , 1981 .

[12]  Bernhard P. Wrobel,et al.  Multiple View Geometry in Computer Vision , 2001 .

[13]  F. Bookstein.  Fitting conic sections to scattered data , 1979 .

[14]  Matthew Harker,et al.  Direct type-specific conic fitting and eigenvalue bias correction , 2008, Image Vis. Comput..

[15]  Gabriel Taubin,et al.  An improved algorithm for algebraic curve and surface fitting , 1993, 1993 (4th) International Conference on Computer Vision.

[16]  Carl Friedrich Gauss.  Méthode des moindres carrés , .

[17]  Andrew W. Fitzgibbon,et al.  Invariant Fitting of Two View Geometry , 2003, BMVC.