A Probabilistic Approach to Linear Subspace Fitting for Computer Vision Problems

Several computer vision problems, such as some of photometric problems and the problem of affine structure from motion, are formulated as fitting linear subspace(s) to point data in a multi-dimensional space. In ideal cases the linear subspaces can easily be computed by PCA/SVD algorithms. Unfortunately this will not apply to real cases, since there are outliers and missing components in real data. Furthermore it is sometimes necessary to fit multiple different subspaces to a set of point data in a situation where each point belongs to one of the subspaces but it is unknown which subspace each point belongs to. One straightforward solution to these advanced cases is to adopt the expectation maximization framework based on Bayesian inference. However, this solution does not seem to have been well considered in computer vision community, as far as the above problems of linear subspace fitting are concerned. This paper presents expectation maximization algorithms and its extension, variational Bayes-based algorithm, for several cases of linear subspace fitting and applies them to computer vision problems.