## 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:

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*:

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:

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: