Partial Sum Minimization of Singular Values in RPCA for Low-Level Vision

Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise/outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values. The proposed objective function implicitly encourages the target rank constraint in rank minimization. Our experimental analyses show that our approach performs better than conventional rank minimization when the number of samples is deficient, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g. high dynamic range imaging, photometric stereo and image alignment, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method.