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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*}