Namrata Vaswani

Modified-CS: Modifying Compressive Sensing for Problems With Partially Known Support

We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known” part of the support, denoted T, may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known” part. The idea of our proposed solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of T. We obtain sufficient conditions for exact reconstruction using modified-CS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called regularized modified-CS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation comparisons for both sparse and compressible signals are shown.

Tracking Deforming Objects Using Particle Filtering for Geometric Active Contours

Tracking deforming objects involves estimating the global motion of the object and its local deformations as a function of time. Tracking algorithms using Kalman filters or particle filters have been proposed for finite dimensional representations of shape, but these are dependent on the chosen parametrization and cannot handle changes in curve topology. Geometric active contours provide a framework which is parametrization independent and allow for changes in topology, in the present work, we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects. To the best of our knowledge, this is the first attempt to implement an approximate particle filtering algorithm for tracking on a (theoretically) infinite dimensional state space.

Recognition of dynamic hand gestures

This paper is concerned with the problem of recognition of dynamic hand gestures. We have considered gestures which are sequences of distinct hand poses. In these gestures hand poses can undergo motion and discrete changes. However, continuous deformations of the hand shapes are not permitted. We have developed a recognition engine which can reliably recognize these gestures despite individual variations. The engine also has the ability to detect start and end of gesture sequences in an automated fashion. The recognition strategy uses a combination of static shape recognition (performed using contour discriminant analysis), Kalman filter based hand tracking and a HMM based temporal characterization scheme. The system is fairly robust to background clutter and uses skin color for static shape recognition and tracking. A real time implementation on standard hardware is developed. Experimental results establish the effectiveness of the approach.

LS-CS-Residual (LS-CS): Compressive Sensing on Least Squares Residual

We consider the problem of recursively and causally reconstructing time sequences of sparse signals (with unknown and time-varying sparsity patterns) from a limited number of noisy linear measurements. The sparsity pattern is assumed to change slowly with time. The key idea of our proposed solution, LS-CS-residual (LS-CS), is to replace compressed sensing (CS) on the observation by CS on the least squares (LS) residual computed using the previous estimate of the support. We bound CS-residual error and show that when the number of available measurements is small, the bound is much smaller than that on CS error if the sparsity pattern changes slowly enough. Most importantly, under fairly mild assumptions, we show “stability” of LS-CS over time for a signal model that allows support additions and removals, and that allows coefficients to gradually increase (decrease) until they reach a constant value (become zero). By “stability,” we mean that the number of misses and extras in the support estimate remain bounded by time-invariant values (in turn implying a time-invariant bound on LS-CS error). Numerical experiments, and a dynamic MRI example, backing our claims are shown.

Modified-CS: Modifying compressive sensing for problems with partially known support

We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known. This may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known” part of the support. The idea of our solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and whose support contains the smallest number of new additions to the known support. We obtain sufficient conditions for exact reconstruction using modified-CS. These turn out to be much weaker than those needed for CS, particularly when the known part of the support is large compared to the unknown part.

Particle filtering for geometric active contours with application to tracking moving and deforming objects

Geometric active contours are formulated in a manner which is parametrization independent. As such, they are amenable to representation as the zero level set of the graph of a higher dimensional function. This representation is able to deal with singularities and changes in topology of the contour. It has been used very successfully in static images for segmentation and registration problems where the contour (represented as an implicit curve) is evolved until it minimizes an image based energy functional. But tracking involves estimating the global motion of the object and its local deformations as a function of time. Some attempts have been made to use geometric active contours for tracking, but most of these minimize the energy at each frame and do not utilize the temporal coherency of the motion or the deformation. On the other hand, tracking algorithms using Kalman filters or particle filters have been proposed for finite dimensional representations of shape. But these are dependent on the chosen parametrization and cannot handle changes in curve topology. In the present work, we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects.

A Generic Framework for Tracking Using Particle Filter With Dynamic Shape Prior

Tracking deforming objects involves estimating the global motion of the object and its local deformations as functions of time. Tracking algorithms using Kalman filters or particle filters (PFs) have been proposed for tracking such objects, but these have limitations due to the lack of dynamic shape information. In this paper, we propose a novel method based on employing a locally linear embedding in order to incorporate dynamic shape information into the particle filtering framework for tracking highly deformable objects in the presence of noise and clutter. The PF also models image statistics such as mean and variance of the given data which can be useful in obtaining proper separation of object and background

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.

Regularized Modified BPDN for Noisy Sparse Reconstruction With Partial Erroneous Support and Signal Value Knowledge

We study the problem of sparse reconstruction from noisy undersampled measurements when the following knowledge is available. (1) We are given partial, and partly erroneous, knowledge of the signals support, denoted by T . (2) We are also given an erroneous estimate of the signal values on T, denoted by (μ̂)T . In practice, both of these may be available from prior knowledge. Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant as T and (μ̂)T. In this paper, we introduce regularized modified basis pursuit denoising (BPDN) (reg-mod-BPDN) to solve this problem and obtain computable bounds on its reconstruction error. Reg-mod-BPDN tries to find the signal that is sparsest outside the set T, while being “close enough” to (μ̂)T on T and while satisfying the data constraint. Corresponding results for modified-BPDN and BPDN follow as direct corollaries. A second key contribution is an approach to obtain computable error bounds that hold without any sufficient conditions. This makes it easy to compare the bounds for the various approaches. Empirical reconstruction error comparisons with many existing approaches are also provided.