Total Variation Regularized Tensor RPCA for Background Subtraction from Compressive Measurements

Background subtraction has been a fundamental and widely studied task in video analysis, with a wide range of applications in video surveillance, teleconferencing and 3D modeling. Recently, motivated by compressive imaging, background subtraction from compressive measurements (BSCM) is becoming an active research task in video surveillance. In this paper, we propose a novel tensor-based robust PCA (TenRPCA) approach for BSCM by decomposing video frames into backgrounds with spatial-temporal correlations and foregrounds with spatio-temporal continuity in a tensor framework. In this approach, we use 3D total variation (TV) to enhance the spatio-temporal continuity of foregrounds, and Tucker decomposition to model the spatio-temporal correlations of video background. Based on this idea, we design a basic tensor RPCA model over the video frames, dubbed as the holistic TenRPCA model (H-TenRPCA). To characterize the correlations among the groups of similar 3D patches of video background, we further design a patch-group-based tensor RPCA model (PG-TenRPCA) by joint tensor Tucker decompositions of 3D patch groups for modeling the video background. Efficient algorithms using alternating direction method of multipliers (ADMM) are developed to solve the proposed models. Extensive experiments on simulated and real-world videos demonstrate the superiority of the proposed approaches over the existing state-of-the-art approaches.Comment: To appear in IEEE TI

Asymptotic Analysis of ADMM for Compressed Sensing

In this paper, we analyze the asymptotic behavior of alternating direction method of multipliers (ADMM) for compressed sensing, where we reconstruct an unknown structured signal from its underdetermined linear measurements. The analytical tool used in this paper is recently developed convex Gaussian min-max theorem (CGMT), which can be applied to various convex optimization problems to obtain its asymptotic error performance. In our analysis of ADMM, we analyze the convex subproblem in the update of ADMM and characterize the asymptotic distribution of the tentative estimate obtained at each iteration. The result shows that the update equations in ADMM can be decoupled into a scalar-valued stochastic process in the asymptotic regime with the large system limit. From the asymptotic result, we can predict the evolution of the error (e.g. mean-square-error (MSE) and symbol error rate (SER)) in ADMM for large-scale compressed sensing problems. Simulation results show that the empirical performance of ADMM and its theoretical prediction are close to each other in sparse vector reconstruction and binary vector reconstruction.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl