Computing the Uncertainty of the 8 point Algorithm for Fundamental Matrix Estimation

Fundamental matrix estimation is difficult since it is often based on correspondences that are spoilt by noise and outliers. Outliers must be thrown out via robust statistics, and noise gives uncertainty. In this article we provide a closed-form formula for the uncertainty of the so-called 8 point algorithm, which is a basic tool for fundamental matrix estimation via robust methods. As an application, we modify a well established robust algorithm accordingly, leading to a new criterion to recover point correspondences under epipolar constraint, balanced by the uncertainty of the estimation.

[1]  H. Neudecker,et al.  The density of the Moore-Penrose inverse of a random matrix , 1996 .

[2]  Robert C. Bolles,et al.  Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography , 1981, CACM.

[3]  Philip H. S. Torr,et al.  Bayesian Model Estimation and Selection for Epipolar Geometry and Generic Manifold Fitting , 2002, International Journal of Computer Vision.

[4]  Jean-Michel Morel,et al.  From Gestalt Theory to Image Analysis: A Probabilistic Approach , 2007 .

[5]  H.F. Durrant-Whyte,et al.  A new approach for filtering nonlinear systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[6]  Serge J. Belongie,et al.  Covariance Propagation for Guided Matching , 2006 .

[7]  G LoweDavid,et al.  Distinctive Image Features from Scale-Invariant Keypoints , 2004 .

[8]  Charles T. Loop,et al.  Estimating the Fundamental Matrix by Transforming Image Points in Projective Space , 2001, Comput. Vis. Image Underst..

[9]  Olivier D. Faugeras,et al.  Characterizing the Uncertainty of the Fundamental Matrix , 1997, Comput. Vis. Image Underst..

[10]  O. Faugeras,et al.  The Geometry of Multiple Images , 1999 .

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

[12]  Zhengyou Zhang,et al.  Determining the Epipolar Geometry and its Uncertainty: A Review , 1998, International Journal of Computer Vision.

[13]  Andrew Zisserman,et al.  MLESAC: A New Robust Estimator with Application to Estimating Image Geometry , 2000, Comput. Vis. Image Underst..

[14]  Manolis I. A. Lourakis,et al.  Estimating the Jacobian of the Singular Value Decomposition: Theory and Applications , 2000, ECCV.

[15]  Lionel Moisan,et al.  A Probabilistic Criterion to Detect Rigid Point Matches Between Two Images and Estimate the Fundamental Matrix , 2004, International Journal of Computer Vision.

[16]  Jiri Matas,et al.  Matching with PROSAC - progressive sample consensus , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[17]  Sami S. Brandt.  On the Probabilistic Epipolar Geometry , 2004, BMVC.

[18]  Kenichi Kanatani,et al.  High Accuracy Computation of Rank-Constrained Fundamental Matrix , 2007, BMVC.

[19]  Jean-Michel Morel,et al.  A Theory of Shape Identification , 2008 .