 
Summary: Factorization as a Rank 1 Problem
Pedro M. Q. Aguiar
Jos´e M. F. Moura
Instituto de Sistemas e Rob´otica Carnegie Mellon University
IST, Lisboa, Portugal Pittsburgh, PA, USA
aguiar@isr.ist.utl.pt moura@ece.cmu.edu
Abstract
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 pa
per 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.
