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This paper designs and extensively evaluates an online algorithm, called practical recursive projected compressive sensing (Prac-ReProCS), for recovering a time sequence of sparse vectors <formula formulatype="inline"><tex Notation="TeX">$S_{t}$</tex></formula> and a time sequence of dense vectors <formula formulatype="inline"><tex Notation="TeX">$L_{t}$</tex></formula> from their sum, <formula formulatype="inline"> <tex Notation="TeX">$M_{t}:=S_{t}+L_{t}$</tex></formula>, when the <formula formulatype="inline"><tex Notation="TeX">$L_{t}$</tex> </formula>'s lie in a slowly changing low-dimensional subspace of the full space. A key application where this problem occurs is in real-time 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 on-the-fly. Prac-ReProCS is a practical modification of its theoretical counterpart which was analyzed in our recent work. Extension to the undersampled case is also developed. Extensive experimental comparisons demonstrating the advantage of the approach for both simulated and real videos, over existing batch and recursive methods, are shown.

An Online Algorithm for Separating Sparse and Low-Dimensional Signal Sequences From Their Sum