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It is known that the decomposition in low-rank and sparse matrices(\textbf{L+S} for short) can be achieved by several Robust PCA techniques.Besides the low rankness, the local smoothness (\textbf{LSS}) is a vitallyessential prior for many real-world matrix data such as hyperspectral imagesand surveillance videos, which makes such matrices have low-rankness and localsmoothness properties at the same time. This poses an interesting question: Canwe make a matrix decomposition in terms of \textbf{L\&LSS +S } form exactly? Toaddress this issue, we propose in this paper a new RPCA model based onthree-dimensional correlated total variation regularization (3DCTV-RPCA forshort) by fully exploiting and encoding the prior expression underlying suchjoint low-rank and local smoothness matrices. Specifically, using amodification of Golfing scheme, we prove that under some mild assumptions, theproposed 3DCTV-RPCA model can decompose both components exactly, which shouldbe the first theoretical guarantee among all such related methods combining lowrankness and local smoothness. In addition, by utilizing Fast Fourier Transform(FFT), we propose an efficient ADMM algorithm with a solid convergenceguarantee for solving the resulting optimization problem. Finally, a series ofexperiments on both simulations and real applications are carried out todemonstrate the general validity of the proposed 3DCTV-RPCA model.

Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices