Recursive robust PCA or recursive sparse recovery in large but structured noise

Abstract

Abstract-This work studies the recursive robust principal components analysis (PCA) problem. If the outlier is the signalof-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 work 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 (w.h.p.), 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.

Separating sparse and low-dimensional signal sequences from time-varying undersampled projections of their sums

“…As shown in Fig 1, the "slowly changing" low dimensional subspace assumption indeed holds for the background sequence in fMRI. It has also been verified for the background sequences of video data in earlier work [2]. Moreover, the assumption that a short sequence of measurements of only the low dimensional sequence (background sequence) are available is also easy to satisfy in either application.…”

“…Every α frames, these can then be used in a noisy matrix completion algorithm [7] recover the ℓt's. The noisy version is needed becauselt = Atℓt + Atet, where et := st −ŝt is the error in estimating st. PCA on the last 2α estimates oflt or projection PCA [2] can then be used to computePt. We summarize the complete algorithm in Algorithm 1.…”

Compressive Online Video Background-Foreground Separation Using Multiple Prior Information and Optical Flow

“…Some studies on recursive recovery from low-dimensional measurements have been proposed to leverage prior information [18,19,21,23]. The study in [23] provided a comprehensive overview of the domain, reviewing a class of recursive algorithms.…”

“…The problem of separating a sequence of time-varying frames using prior information brings significant improvements in the context of online RPCA [18,21,22]. Some studies on recursive recovery from low-dimensional measurements have been proposed to leverage prior information [18,19,21,23].…”

“…The studies in [18,21] used modified-CS [25] to leverage prior knowledge under the condition of slowly varying support and signal values. However, this method as well as the methods in [12,13,15] do not explore the correlations between the current frame and multiple previously separated frames.…”