Fundamental matrix (computer vision)

From Wikipedia, the free encyclopedia

In computer vision, the fundamental matrix is a matrix of rank 2 which relates corresponding points in stereo images. In epipolar geometry, describes a line of which the corresponding point y₂ on the other image must lie. That means, for all pairs of corresponding points with homogeneous image coordinates and holds

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.

The above relation which defines the fundamental matrix was published in 1992 by both Faugeras and Hartley. Although Longuett-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.

[edit] Introduction

The existence of the fundamental matrix means that it is possible to test if image points from each image in a stereo pair correspond to the same point in the scene, thereby solving the correspondence problem. It should be noted that the above condition is necessary but not sufficient for correspondence. This means that hypothetical correspondences can be rejected but not definitely confirmed. This application of the fundamental matrix makes it a useful tool for scene reconstruction.

The relation between corresponding image points which the fundamental matrix represent is referred to as epipolar constraint, matching constraint, or incidence relation.

[edit] References

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

• 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. 

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

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

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

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

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