Chenlu Qiu

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

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 St and a time sequence of dense vectors Lt from their sum, Mt: = St + Lt, when the Lts 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.

Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise

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 Lts. 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.

Real-time dynamic MR image reconstruction using Kalman Filtered Compressed Sensing

In recent work, Kalman Filtered Compressed Sensing (KF-CS) was proposed to causally reconstruct time sequences of sparse signals, from a limited number of “incoherent” 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 “incoherent” 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.

Recursive sparse recovery in large but correlated noise

In this work, we focus on the problem of recursively recovering a time sequence of sparse signals, with time-varying sparsity patterns, from highly undersampled measurements corrupted by very large but correlated noise. It is assumed that the noise is correlated enough to have an approximately low rank covariance matrix that is either constant, or changes slowly, with time. We show how our recently introduced Recursive Projected CS (ReProCS) and modified-ReProCS ideas can be used to solve this problem very effectively. To the best of our knowledge, except for the recent work of dense error correction via ℓ1 minimization, which can handle another kind of large but “structured” noise (the noise needs to be sparse), none of the other works in sparse recovery have studied the case of any other kind of large noise.

Recursive sparse recovery in large but structured noise — Part 2

We study the problem of recursively recovering a time sequence of sparse vectors, S_t, from measurements M_t := S_t + L_t that are corrupted by structured noise L_t which is dense and can have large magnitude. The structure that we require is that L_t should 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 S_t at all times; and the reconstruction errors of both S_t and L_t are upper bounded by a time-invariant and small value.

Practical ReProCS for separating sparse and low-dimensional signal sequences from their sum — Part 1

This paper designs and evaluates a practical algorithm, called Prac-ReProCS, for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt := St + Lt, when any subsequence of the Lts 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.

Performance guarantees for ReProCS - Correlated low-rank matrix entries case

Online or recursive robust PCA can be posed as a problem of recovering a sparse vector, St, and a dense vector, Lt, which lies in a slowly changing low-dimensional subspace, from Mt := St+Lt on-the-fly as new data comes in. For initialization, it is assumed that an accurate knowledge of the subspace in which L0 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 als result. Their work required that the Lts be mutually independent over time. However this is not a practical assumption, e.g., in the video application, Lt is the background image sequence and one would expect it to be correlated over time. In this work we relax this and allow the Lts 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 St at all times; the reconstruction errors of both St and Lt 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.

Performance guarantees for undersampled recursive sparse recovery in large but structured noise

We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt +BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, and has support which is correlated over time. We introduce a solution which we call Recursive Projected Modified Compressed Sensing (ReProMoCS), which exploits the correlated support change of St. We show that, under weaker assumptions than previous work, with high probability, ReProMoCS will exactly recover the support set of St and the reconstruction error of St is upper bounded by a small time-invariant value. A motivating application where the above problem occurs is in functional MRI imaging of the brain to detect regions that are “activated” in response to stimuli. In this case both measurement matrices are the same (i.e. A = B). The active region image constitutes the sparse vector St and this region changes slowly over time. The background brain image changes are global but the amount of change is very little and hence it can be well modeled as lying in a slowly changing low dimensional subspace, i.e. this constitutes Lt.

Practical ReProCS for separating sparse and low-dimensional signal sequences from their sum — Part 2

In this work, we experimentally evaluate and verify model assumptions for our recently proposed algorithm (practical ReProCS) for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt := St + Lt, when Lt 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. Practical-ReProCS is the practical analog of its theoretical counterpart that was studied in our recent work.