Rank 1 Weighted Factorization for 3D Structure Recovery: Algorithms and Performance Analysis

Thepaper describes the rank 1weighted factorization solution to the structure frommotionproblem. Thismethod recovers the 3Dstructure from the factorization of a datamatrix that is rank 1 rather than rank 3. Thismatrix collects the estimates of the 2Dmotions of a set of feature points of the rigid object. These estimates are weighted by the inverse of the estimates error standard deviation so that the 2Dmotionestimates for “sharper” features,whichare usuallywell-estimated, are givenmoreweight,while the noisiermotionestimates for “smoother” features are weighted less. We analyze the performance of the rank 1 weighted factorization algorithm to determine what are the most suitable 3D shapes or the best 3D motions to recover the 3D structure of a rigid object from the 2D motions of the features. Our approach is developed for the orthographic cameramodel. It avoids expensive singular value decompositions by using the powermethod and is suitable to handledensesets of featurepointsand longvideosequences.Experimental studieswith synthetic and real data illustrate the good performance of our approach.