James Gregson's Website

Dec 13, 2020

Essential Matrix

The essential matrix relates normalized coordinates in one view with those in another view of the same scene and is widely used in stereo reconstruction algorithms. It is a 3x3 matrix with rank 2, having only 8 degrees of freedom.

Given \(p\) as a normalized point in image P and \(q\) as the corresponding normalied point in another image Q of the same scene, the essential matrix provides the constraint:

\begin{equation*} q^T E p = 0 \end{equation*}

It's important that \(p\) and \(q\) be normalized homogeneous points, i.e. the following where \([u_p,v_p]\) are the pixel coordinates of point p:

\begin{equation*} p = K^{-1} \begin{bmatrix} u_p \\ v_p \\ 1 \end{bmatrix} \end{equation*}

By fixing the extrinsics of image P as \(R_P = I\) and \(t_P = [0,0,0]^T\) the essential matrix relating P and Q can defined by the relationship:

\begin{equation*} E = t_\times R \end{equation*}

where \(R\) and \(t\) are the relative transforms of camera Q with respect to camera P and \(t_\times\) is the matrix representation of the the cross-product:

\begin{equation*} t_\times = \begin{bmatrix} 0 & -t_z & t_y \\ t_z & 0 & -t_x \\ -t_y & t_x & 0 \end{bmatrix} \end{equation*}