Essential matrix

In computer vision, the essential matrix is a $3\times 3$ matrix $\mathbf {E}$ which relates corresponding points in stereo images. More specifically, if $\mathbf {\tilde {y}} _{1}$ and $\mathbf {\tilde {y}} _{2}$ are homogeneous normalized image coordinates in image 1 and 2, respectively, then

$\mathbf {\tilde {y}} _{1}^{T}\mathbf {E} \mathbf {\tilde {y}} _{2}=0$

if $\mathbf {\tilde {y}} _{1}$ and $\mathbf {\tilde {y}} _{2}$ correspond to the same 3D point in the scene.

The above relation which defines the essential matrix was published in 1981 by Longuet-Higgins.

Introduction

The essential matrix can be seen as a precursor to the fundamental matrix which has proven to be of more practical use. Both matrices can be used for establishing constraints between matching image points, but the essential matrix can only be used in relation to calibrated cameras since the inner camera parameters must be known in order to achieve the normalization.

References

• H. Christopher Longuet-Higgins (1981). "A computer algorithm for reconstructing a scene from two projections". Nature. 293: 133–135.
• Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press. ISBN 0-521-54051-8.
• Yi Ma (2004). An Invitation to 3-D Vision. Springer. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Gang Xu and Zhengyou Zhang (1996). Epipolar geometry in Stereo, Motion and Object Recognition. Kluwer Academic Publishers. ISBN 0-7923-4199-6.