Multibody factorization with uncertainty and missing data using the EM algorithm

Multibody factorization algorithms [2, 1, 16] give an elegant and simple solution to the problem of structure from motion even for scenes containing multiple independent motions. Despite this elegance, it is still quite diffcult to apply these algorithms to arbitrary scenes. First, their performance deteriorates rapidly with increasing noise. Second, they cannot be applied unless all the points can be tracked in all the frames (as will rarely happen in real scenes). Third, they cannot incorporate prior knowledge on the structure or the motion of the objects. In this paper we present a multibody factorization algorithm that can handle arbitrary noise covariance for each feature as well as missing data. We show how to formulate the problem as one of factor analysis and derive an expectation-maximization based maximum-likelihood algorithm. One of the advantages of our formulation is that we can easily incorporate prior knowledge, including the assumption of temporal coherence. We show that this assumption greatly enhances the robustness of our algorithm and present results on challenging sequences.