Low-Rank Matrix Fitting Based on Subspace Perturbation Analysis with Applications to Structure from Motion

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 31; no. 5; pp. 841 - 854
• Main Authors: Hongjun Jia, Martinez, A.M.
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.05.2009; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive noise; Algorithms; Applied sciences; Artificial Intelligence; Bioinformatics; Computer graphics; Computer science; control theory; systems; Computer vision; Criteria; Data engineering; Exact sciences and technology; Fittings; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Imaging, Three-Dimensional - methods; Intelligence; Low-rank matrix; Mathematical analysis; Matrices; Matrix methods; matrix perturbation; missing data; Motion; Motion analysis; Movement; Noise; Optical noise; Pattern analysis; Pattern recognition; Pattern Recognition, Automated - methods; Pattern recognition. Digital image processing. Computational geometry; Photography - methods; random matrix; Reproducibility of Results; Sensitivity and Specificity; structure from motion; Studies; subspace analysis; Upper bound; noise; matrix perturbation; subspace analysis; structure from motion; Additive noise; Occlusion; Validation; Computer vision; Random matrix; Low-rank matrix; Pattern recognition; Modeling; Missing data; Upper bound; Perturbation method; Vector space; Incomplete information; Pattern analysis; Artificial intelligence; Occultation
• DOI: 10.1109/TPAMI.2008.122
• ISSN: 0162-8828

The task of finding a low-rank (r) matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering. The problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements. The former problem can be solved by concatenating a set of r-column matrices that share a common single r-dimensional solution space. Unfortunately, the number of possible submatrices is generally very large and, hence, the results obtained with one set of r-column matrices will generally be different from that captured by a different set. Ideally, we would like to find that solution that is least affected by noise. This requires that we determine which of the r-column matrices (i.e., which of the original feature points) are less influenced by the unknown noise term. This paper presents a criterion to successfully carry out such a selection. Our key result is to formally prove that the more distinct the r vectors of the r-column matrices are, the less they are swayed by noise. This key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r-column matrices. It is shown how this criterion can be effectively used to recover the noise-free matrix of rank r. Finally, we derive the affine and projective structure-from-motion (SFM) algorithms using the proposed criterion. Extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art.