Factorization with Uncertainty and Missing Data: Exploiting Temporal Coherence

The problem of "Structure From Motion" is a central problem in vision: given the 2D locations of certain points we wish to recover the camera motion and the 3D coordinates of the points. Under simplified camera models, the problem reduces to factorizing a measurement matrix into the product of two low rank matrices. Each element of the measurement matrix contains the position of a point in a particular image. When all elements are observed, the problem can be solved trivially using SVD, but in any realistic situation many elements of the matrix are missing and the ones that are observed have a different directional uncertainty. Under these conditions, most existing factorization algorithms fail while human perception is relatively unchanged. In this paper we use the well known EM algorithm for factor analysis to perform factorization. This allows us to easily handle missing data and measurement uncertainty and more importantly allows us to place a prior on the temporal trajectory of the latent variables (the camera position). We show that incorporating this prior gives a significant improvement in performance in challenging image sequences.