A Novel Approach in Computer Vision and Photogrammetry to Recover the Relative Position and Orientation of Cameras in Stereo Images Using the SVD Decomposition of the Essential Matrix

The relative position and orientation between two cameras in a stereo pair are included within the Essential matrix, E. The decomposition of this matrix into a rotation matrix, R, and a skew-symmetric matrix, S, is an efficient tool for retrieving the relative position and orientation of the cameras. In this paper, a new method is proposed to recover the relative position and orientation of the cameras in a stereo pair using the singular value decomposition (SVD) of the Essential matrix. First, the existing formulas in the decomposition of the Essential matrix into a rotation matrix and a skew-symmetric matrix using the SVD decomposition are directly proved using the SVD properties. Then, based on these results, a new method in the decomposition of the Essential matrix using SVD will be presented. The Essential matrix decomposition in this method is accomplished by extracting the base vector of the left null space of the Essential matrix and then by SVD decomposition of the skew-symmetric matrix corresponding to this base vector. In this method, the initial mapping of the Essential matrix, recovered from the erroneous coordinates of the corresponding image points in two images, into the space of Essential matrices does not require. This mapping is performed by determining the skew-symmetric matrix, S. The proposed numerical analysis shows that the results of the new presented method are correct and identical with the results of the existing formulas.