An Online Algorithm for Separating Sparse and Low-dimensional Signal Sequences from their Sum

This paper designs and evaluates a practical algorithm, called practical recursive projected compressive sensing (Prac-ReProCS), for recovering a time sequence of sparse vectors $S_t$ and a time sequence of dense vectors $L_t$ from their sum, $M_t:= S_t + L_t$, when any subsequence of the $L_t$'s lies in a slowly changing low-dimensional subspace. A key application where this problem occurs is in video layering where the goal is to separate a video sequence into a slowly changing background sequence and a sparse foreground sequence that consists of one or more moving regions/objects. Prac-ReProCS is a practical modification of its theoretical counterpart which was analyzed in our recent work. Experimental comparisons demonstrating the advantage of the approach for both simulated and real videos are shown. Extension to the undersampled case is also developed.