Linear fitting with missing data: applications to structure-from-motion and to characterizing intensity images

Several vision problems can be reduced to the problem of fitting a linear surface of low dimension to data, including the problems of structure-from-affine-motion, and of characterizing the intensity images of a Lambertian scene by constructing the intensity manifold. For these problems, one must deal with a data matrix with some missing elements. In structure-from-motion, missing elements will occur if some point features are not visible in some frames. To construct the intensity manifold missing matrix elements will arise when the surface normals of some scene points do not face the light source in some images. We propose a novel method for fitting a low rank matrix to a matrix with missing elements. We show experimentally that our method produces good results in the presence of noise. These results can be either used directly, or can serve as an excellent starting point for an iterative method.