Fundamental Matrix

Related Concepts

Epipolar Geometry; Essential Matrix

Definition

Fundamental matrix is a special 3 ×3 matrix which captures the geometric relationship between two cameras or between two locations of a single moving camera.

Background

See entry Epipolar Geometry for details.

Theory

If two points $\boldsymbol{m}$ and $\boldsymbol{m}\prime$, expressed in pixel image coordinates in the first and second camera, are in correspondence, they must satisfy the following equation

$$\widetilde{\boldsymbol{{m}}}^{T}\mathbf{F}\widetilde{\boldsymbol{m}}\prime = 0\;,$$

(1)

where

$$\displaystyle\begin{array}{rcl} \mathbf{F}& =&{ \mathbf{A}}^{-T}\mathbf{E}\mathbf{{A\prime}^{-1}}\;,\end{array}$$

(2)

$$\displaystyle\begin{array}{rcl} \mathbf{E}& =& [\boldsymbol{t}]_{\times }\mathbf{R}\;,\end{array}$$

(3)

A and A′ are respectively the intrinsic matrix of the first and second camera, and $(\mathbf{R},\boldsymbol{t})$is the rigid transformation between the first and second camera. This is a...