Tracking sparse signal sequences from nonlinear/non-Gaussian measurements and applications in illumination-motion tracking

Abstract

In this work, we develop algorithms for tracking time sequences of sparse spatial signals with slowly changing sparsity patterns, and other unknown states, from a sequence of nonlinear observations corrupted by (possibly) non-Gaussian noise. A key example of the above problem occurs in tracking moving objects across spatially varying illumination changes, where motion is the small dimensional state while the illumination image is the sparse spatial signal satisfying the slow-sparsity-pattern-change property.Index Terms-particle filtering, compressed sensing, tracking INTRODUCTIONWe study the problem of tracking (causally estimating) a time sequence of sparse spatial signals with slowly changing sparsity patterns, as well as other unknown states, from a sequence of nonlinear observations corrupted by (possibly) non-Gaussian noise. In many practical applications, the unknown state can be split into a small dimensional part and a spatial signal (large dimensional part). The spatial signal is often well modeled as being sparse in some domain. For a long sequence, its sparsity pattern can change over time, although the changes are usually slow. A key example of the above problem occurs in tracking moving objects across spatially varying illumination changes, e.g. persons walking under a tree (different lighting falling on different parts of the face due to the leaves blocking or not blocking the sunlight and this pattern changes with time as the leaves move) or in indoor sequences with variable lighting. In all these cases, one needs to explicitly track the motion (small dimensional part) as well as the illumination "image" (illumination at each pixel in the image), which is the spatial signal satisfying the slow-sparsity-pattern-change property [see Sec 4].Related Work. In recent years, starting with the seminal papers of Candes, Romberg, Tao and of Donoho [1,2] there has been a large amount of work on sparse signal recovery / compressive sensing (CS). The problem of recursively recovering a time sequence of sparse signals, with slowly changing sparsity patterns and signal values, from linear measurements has also been extensively studied [3,4,5,6,7,8,9,10,11,12,13,14,15].For tracking problems that need to causally estimate a time sequence of hidden states, Xt, from nonlinear and possibly nonGaussian measurements, Yt, the most common and efficient solution is to use a particle filter (PF). The PF uses sequential importance sampling [16] along with a resampling step [17] to obtain a sequential Monte Carlo estimate of the posterior distribution, f X t |Y 1:t (xt|y1:t), of the state Xt conditioned on all observations up to the current time, Y1:t. In our problem, part of the state vector is a discrete spatial signal and hence very high dimensional. As a result, in this case, the original PF [17] will require too many particles for accurate tracking and hence becomes impractical to use. As

Recursive Recovery of Sparse Signal Sequences From Compressive Measurements: A Review

“…On one end, one can try to use sequential Monte Carlo techniques (particle filtering) to approximate the MMSE estimate. Some attempts to do this are described in [84], [85]. But these can get computationally expensive for high dimensional problems and it is never clear what number of particles is sufficient to get an accurate enough estimate.…”

Jointly Optimized Target Detection and Tracking Using Compressive Samples

“…The existing CS tracking algorithms mainly focus on how to track time-varying sparse signals whose sparse nonzero elements change slowly. Different types of filters based on CS are exploited to reduce the workload when sensing dynamic sparse signals, such as the CS-Kalman, CS-MUSIC, CS-Bayesian, and CS-Particle filters [15]- [18]. However, these filters are not suitable for radar target tracking when the target signals are changing rapidly.…”

Dictionary Learning and Sparse Representation for Image Analysis with Application to Segmentation, Classification and Event Detection

“…Sparse representation based dictionary learning has been an active area of research in the field of image processing and computer vision with applications ranging from, image retrieval [10,11], classification [9,[12][13][14][15][16], segmentation [17,18] to video tracking [19,20], scene change detection [21]. Sparse coding techniques provide compact representations but do not incorporate discrimination.…”

“…The literature is vast and spans the problems of image denoising, registration, segmentation, classification, quality assessment and extends to analysis of video for object tracking, event detection, activity recognition, etc. [1,6,8,14,16,19,20,[31][32][33][34][35][36][37].…”

“…In [19], the authors proposed a work for visual tracking under varying illumination. Here the intensity change of the target was represented as a sparse combination of Legendre polynomials, which are specifically designed for illumination change prediction on the template.…”