The Optimization Model For Robust Principal Component Analysis And Its Applications In Computer Vision

Recovering the low-rank structure of observation data from outliers、gross corrup-tions and missing values has a wide range of applications in computer vision. Robust prin-cipal component analysis is recently attracting much attention as extension of the classicalprincipal component analysis due to the development of theory and algorithm. In this pa-per, we study the problems of robust bilinear factorization and tensor recovery, both can beclassified as the variants of robust principal component analysis.Learning low rank model by bilinear factorization is a popular method for the rep-resentation of shape、appearance or motion. However, traditional approaches sufer fromlocal minima or poor convergence performance under the circumstances of serious outliersand missing data. This paper proposes a robust bilinear factorization model for consideringboth outliers and missing values. This model is a constrained optimization problem and anequivalent reformulation is introduced. We then utilize the augmented Lagrangian alternat-ing direction method based algorithm, named RBF-ALADM, to tackle the equivalent formof robust bilinear factorization problem. The experimental results on synthetic data showthat the proposed method in general has a much faster speed than the state-of-the-art meth-ods, and the performances on Structure from Motion and Photometric Stereo also indicatethe good recoverability by our method.The rapid advance in modern engineering technology has given rise to the wide pres-ence of multidimensional data (tensor data). Traditional robust principal component analy-sis is inherently two-dimensional, which limits its usefulness in recovering low dimensionalstructure from a multidimensional perspective. In this paper, we exploit the higher-ordergeneralization of robust principal component analysis, named higher-order principal com-ponent pursuit (HOPCP). Unlike the convexification (nuclear norm) for matrix rank func-tion, the tensorial nuclear norm is still an open problem. While existing preliminary workson the tensor completion field provide a viable way to indicate the low complexity esti-mate of tensor, therefore, we focus on low multi-linear rank tensor and adopt its convexrelaxation to formulate the convex optimization model of HOPCP. We further propose twomethods for HOPCP based on alternative minimization scheme: the augmented Lagrangianalternating direction method (ALADM) and its truncated higher-order singular value de-composition (ALADM-THOSVD) version. The former can obtain high accuracy solutionwhile the latter is more efcient to handle the computationally intractable problems. Ex-perimental results on both synthetic data and real MRI data show the applicability of ourmethods in high-dimensional tensor data processing.