Fundamental matrix (computer vision)

In computer vision, the fundamental matrix $mathbf\{F\}$ is a $3\; imes\; 3$ matrix of rank 2 which relates corresponding points in stereo images. In epipolar geometry, with homogeneous image coordinates $mathbf\{y\_1\}$ and $mathbf\{y\_2\}$ of corresponding points in a stereo image pair, $mathbf\{F\; y\_1\}$ describes a line (an epipolar line) on which the corresponding point $y\_2$ on the other image must lie. That means, for all pairs of corresponding points holds

: $mathbf\{\; y\_2^T\; F\; y\_1\}\; =\; 0.$

Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone.

The term "fundamental matrix" was coined by Luong in his influential PhD thesis. It is sometimes also referred to as the "bifocal tensor".

The above relation which defines the fundamental matrix was published in 1992 by both Faugeras and Hartley. Although Longuet-Higgins' essential matrix satisfies a similar relationship, the essential matrix is a metric object pertaining to calibrated cameras, while the fundamental matrix describes the correspondence in more general and fundamental terms of projective geometry.

Introduction

The fundamental matrix is a relationship between any two images of a same scene that constrains where the projection of points from the scene can occur in both images. Given the projection of a scene point into one of the images the corresponding point in the other image is constrained to a line, helping the search, and allowing for the detection of wrong correspondences. The relation between corresponding image points which the fundamental matrix represents is referred to as "epipolar constraint", "matching constraint", "discrete matching constraint", or "incidence relation".

Solving the correspondence problem, one can do 3d scene reconstruction.

References

* cite conference
title=What can be seen in three dimensions with an uncalibrated stereo rig?
author=Olivier D. Faugeras
booktitle=Proceedings of European Conference on Computer Vision
year=1992

* cite conference
title=Camera self-calibration: Theory and experiments
author=Olivier D. Faugeras
coauthors=Quang-Tuan Luong and Steven Maybank
booktitle=Proceedings of European Conference on Computer Vision
year=1992

*cite book
author=Olivier Faugeras and Q. T. Luong
title=The Geometry of Multiple Images
publisher=MIT Press
year=2001
id=ISBN 0-262-06220-8

* cite conference
title=Estimation of relative camera positions for uncalibrated cameras
author=Richard I. Hartley
booktitle=Proceedings of European Conference on Computer Vision
year=1992

*cite book
author=Richard Hartley and Andrew Zisserman
title=Multiple View Geometry in computer vision
publisher=Cambridge University Press
year=2003
id=ISBN 0-521-54051-8

* cite book
title=Fundamental matrix and self-calibration
author=Q. T. Luong
series=PhD Thesis, University of Paris, Orsay
year=1992

*cite book
author=Yi Ma
coauthors=Stefano Soatto, Jana Košecká and S. Shankar Sastry
title=An Invitation to 3-D Vision
publisher=Springer
year=2004

*cite book
author=Gang Xu and Zhengyou Zhang
title=Epipolar geometry in Stereo, Motion and Object Recognition
publisher=Kluwer Academic Publishers
year=1996
id=ISBN 0-7923-4199-6

*cite journal
author=Richard I. Hartley
title=In Defense of the Eight-Point Algorithm
journal=IEEE Transactions on Pattern Analysis and Machine Intelligence
volume=19(6)
pages=580–593
year=1997

*cite journal
author=Zhengyou Zhang
title=Determining the epipolar geometry and its uncertainty: A review
journal=International Journal of Computer Vision
volume=27(2)
pages=161–195
year=1998
doi=10.1023/A:1007941100561

Toolboxes

* [http://eia.udg.es/%7Eqsalvi/recerca.html Fundamental Matrix Estimation Toolbox (Joaquim Salvi)]
* [http://egt.dii.unisi.it/ The Epipolar Geometry Toolbox Toolbox (EGT)]

External links

* [http://citeseer.ist.psu.edu/article/zhang96determining.html Determining the epipolar geometry and its uncertainty: A review (Zhengyou Zhang)]