Fundamental matrix (computer vision)

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In computer vision, the fundamental matrix $\mathbf{F}$ is a $3 \times 3$ 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.

1 Introduction
2 References
3 Toolboxes
4 External links

[edit] 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.

[edit] References

Olivier D. Faugeras (1992). "What can be seen in three dimensions with an uncalibrated stereo rig?". Proceedings of European Conference on Computer Vision.

Olivier D. Faugeras; Quang-Tuan Luong and Steven Maybank (1992). "Camera self-calibration: Theory and experiments". Proceedings of European Conference on Computer Vision.

Olivier Faugeras and Q. T. Luong (2001). The Geometry of Multiple Images. MIT Press. ISBN 0-262-06220-8.

Richard I. Hartley (1992). "Estimation of relative camera positions for uncalibrated cameras". Proceedings of European Conference on Computer Vision.

Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press. ISBN 0-521-54051-8.

Q. T. Luong (1992). Fundamental matrix and self-calibration, PhD Thesis, University of Paris, Orsay.

Yi Ma; Stefano Soatto, Jana Košecká and S. Shankar Sastry (2004). An Invitation to 3-D Vision. Springer.

Gang Xu and Zhengyou Zhang (1996). Epipolar geometry in Stereo, Motion and Object Recognition. Kluwer Academic Publishers. ISBN 0-7923-4199-6.

Richard I. Hartley (1997). "In Defense of the Eight-Point Algorithm". IEEE Transactions on Pattern Analysis and Machine Intelligence 19(6): 580–593.

Zhengyou Zhang (1998). "Determining the epipolar geometry and its uncertainty: A review". International Journal of Computer Vision 27(2): 161–195. doi:10.1023/A:1007941100561.