This paper designs and evaluates a practical algorithm, called Prac-ReProCS, for recovering a time sequence of sparse vectors Stand a time sequence of dense vectors Ltfrom their sum, Mt:= St+ Lt, when any subsequence of the Lt'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 the practical analog of its theoretical counterpart that was studied in our recent work.

In this work, we propose an online algorithm Recursive Projected Sparse Matrix Recovery (ReProSMR) for recovering a time sequence of sparse matrix Stand a time sequence of dense matrix Ltfrom their sum Mt= Lt+ Stwhen Lts' lies in a slowly changing low dimensional tensor subspace. A key application where this problem occurs is in video layer separation where the goal is to separate a video sequence into a slowly changing low dimensional background sequence and a sparse foreground sequence. Mathematically, a 2D image can be thought of as a second order tensor. ReProSMR is a modification of Recursive Projected Compressive Sensing (ReProCS) based on tensor PCA. Experimental comparisons demonstrating the advantages and computation gain of ReProSMR are shown for both simulated and real videos.

Online or recursive robust PCA can be posed as a problem of recovering a sparse vector, S t , and a dense vector, L t , which lies in a slowly changing low-dimensional subspace, from M t := S t +L t on-the-fly as new data comes in. For initialization, it is assumed that an accurate knowledge of the subspace in which L 0 lies is available. In recent works, Qiu et al proposed and analyzed a novel solution to this problem called recursive projected compressed sensing or ReProCS. In this work, we relax one limiting assumption of Qiu et al's result. Their work required that the L t 's be mutually independent over time. However this is not a practical assumption, e.g., in the video application, L t is the background image sequence and one would expect it to be correlated over time. In this work we relax this and allow the L t 's to follow an autoregressive model. We are able to show that under mild assumptions and under a denseness assumption on the unestimated part of the changed subspace, with high probability (w.h.p.), ReProCS can exactly recover the support set of S t at all times; the reconstruction errors of both S t and L t are upper bounded by a time invariant and small value; and the subspace recovery error decays to a small value within a finite delay of a subspace change time.

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above problem, in recent work, we introduced a novel solution called recursive projected CS (ReProCS). In this paper, we develop a simple modification of the original ReProCS idea and analyze it. This modification assumes knowledge of a subspace change model on the Lt's. Under mild assumptions and a denseness assumption on the unestimated part of the subspace of Lt at various times, we show that, with high probability, the proposed approach can exactly recover the support set of St at all times, and the reconstruction errors of both St and Lt are upper bounded by a time-invariant and small value. In simulation experiments, we observe that the last assumption holds as long as there is some support change of St every few frames.

We study the recursive robust principal components' analysis (PCA) problem. Here, “robust” refers to robustness to both independent and correlated sparse outliers. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt: the noise needs to lie in a “slowly changing” low dimensional subspace. We study a novel solution called Recursive Projected CS (ReProCS). Under mild assumptions, we show that, with high probability (w.h.p.), at all times, ReProCS can exactly recover the support set of St; and the reconstruction errors of both Stand Ltare upper bounded by a time-invariant and small value.

We study the problem of recursively recovering a time sequence of sparse vectors, St, from measurements Mt:= St+ Ltthat are corrupted by structured noise Ltwhich is dense and can have large magnitude. The structure that we require is that Ltshould lie in a low dimensional subspace that is either fixed or changes “slowly enough” and the eigenvalues of its covariance matrix are “clustered”. We do not assume any model on the sequence of sparse vectors. Their support sets and their nonzero element values may be either independent or correlated over time (usually in many applications they are correlated). The only thing required is that there be some support change every so often. We introduce a novel solution approach called Recursive Projected Compressive Sensing with cluster-PCA (ReProCS-cPCA) that addresses some of the limitations of earlier work. Under mild assumptions, we show that, with high probability, ReProCS-cPCA can exactly recover the support set of Stat all times; and the reconstruction errors of both Stand Ltare upper bounded by a time-invariant and small value.

This work proposes a causal and recursive algorithm for solving the “robust” principal components' analysis problem. We primarily focus on robustness to correlated outliers. In recent work, we proposed a new way to look at this problem and showed how a key part of its solution strategy involves solving a noisy compressive sensing (CS) problem. However, if the support size of the outliers becomes too large, for a given dimension of the current principal components' space, then the number of “measurements” available for CS may become too small. In this work, we show how to address this issue by utilizing the correlation of the outliers to predict their support at the current time; and using this as “partial support knowledge” for solving Modified-CS instead of CS.

In recent work, Kalman Filtered Compressed Sensing (KF-CS) was proposed to causally reconstruct time sequences of sparse signals, from a limited number of ldquoincoherentrdquo measurements. In this work, we develop the KF-CS idea for causal reconstruction of medical image sequences from MR data. This is the first real application of KF-CS and is considerably more difficult than simulation data for a number of reasons, for example, the measurement matrix for MR is not as ldquoincoherentrdquo and the images are only compressible (not sparse). Greatly improved reconstruction results (as compared to CS and its recent modifications) on reconstructing cardiac and brain image sequences from dynamic MR data are shown.