Subspace Methods and Equilibration in Computer Vision

Many computer vision problems (e.g. the estimation of the fundamental matrix, the homography matrix, or camera calibration as well as the factorization method) belong to the class of subspace problems which are well-known in signal processing. They essentially consist in dividing a vector space into a lower dimensional data space and an orthogonal error space, based on the information available from a set of measurement vectors. This can be formulated as a rank reduction problem on a perturbed measurement matrix. In this paper, a powerful technique for adjusting the error structure of this measurement matrix is introduced. Doing this, the rank reduction works in a statistically optimized way and much better estimates of the sought entities (lower variance, reduced or eliminated bias) can be obtained. It is well known that a proper weighting of the measurement matrix changes the error metrics of the optimization problem belonging to the rank reduction task. But very little is known how these weighting matrices have to be chosen in a statistically optimal way. In this paper, we will describe four applications of our method (fundamental matrix estimation, camera calibration, factorization method, and homography matrix estimation).