Factorization as a Rank 1 Problem

Tomasi and Kanade [1] introduced the factorization method for recovering 3D structure from 2D video. In their formulation, the 3D shape and 3D motion are computed by using an SVD to approximate a matrix that is rank 3 in a noiseless situation. In this paper we reformulate the problem using the fact that the x and y coordinates of each feature are known from their projection onto the image plane in frame 1. We show how to compute the 3D shape, i.e., the relative depths z, and the 3D motion by a simple factorization of a matrix that is rank 1 in a noiseless situation. This allows the use of very fast algorithms even when using a large number of features and large number of frames. We also show how to accommodate confidence weights for the feature trajectories. This is done without additional computational cost by rewriting the problem as the factorization of a modified matrix.