Abstract

Low-rank matrix factorization is an essential problem in many areas including computer vision, with applications in e.g. affifine structure-from-motion, photometric stereo, and non-rigid structure from motion. However, very little attention has been drawn to minimal cases for this problem or to using the minimal confifiguration of observations to fifind the solution. Minimal problems are useful when either outliers are present or the observation matrix is sparse. In this paper, we fifirst give some theoretical insights on how to generate all the minimal problems of a given size using Laman graph theory. We then propose a new parametrization and a building-block scheme to solve these minimal problems by extending the solution from a small sized minimal problem. We test our solvers on synthetic data as well as real data with outliers or a large portion of missing data and show that our method can handle the cases when other iterative methods, based on convex relaxation, fail