The problem of detecting the sparsity pattern of a k-sparse vector in Rn from m random noisy measurements is of interest in many areas such as system identification, denoising, pattern recognition, and compressed sensing. This paper addresses the scaling of the number of measurements m, with signal dimension n and sparsity-level nonzeros k, for asymptotically-reliable detection. We show a neces...

In this article, we review the literature on recursive algorithms for reconstructing a time sequence of sparse signals from a greatly reduced number of linear projection measurements. The signals are sparse in some transform domain referred to as the sparsity basis, and their sparsity pattern (support set of the sparsity basis coefficients’ vector) can change with time. We also summarize the th...

In this paper, we investigate the theoretical guarantees of penalized l1-minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with non-necessarily random noise, when the sensing operator belongs to the Gaussian ensemble (i.e. random design matrix with i.i.d. Gaussian entries). More precisely, we ...

In this paper we present a linear programming solution for support recovery. Support recovery involves the estimation of sign pattern of a sparse signal from a set of randomly projected noisy measurements. Our solution of the problem amounts to solving min ‖Z‖1 s.t. Y = GZ, and quantizing/thresholding the resulting solution Z. We show that this scheme is guaranteed to perfectly reconstruct a di...

The problem of consistently estimating the sparsity pattern of a vector β ∈ R based on observations contaminated by noise arises in various contexts, including signal denoising, sparse approximation, compressed sensing, and model selection. We analyze the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is...

This paper addresses the problem of sparsity pattern detection for unknown ksparse n-dimensional signals observed through m noisy, random linear measurements. Sparsity pattern recovery arises in a number of settings including statistical model selection, pattern detection, and image acquisition. The main results in this paper are necessary and sufficient conditions for asymptotically-reliable s...

In this paper, we propose a new image inpainting method based on the property that much of the image information in the transform domain is sparse. We add a redundancy to the original image by mapping the transform coefficients with small amplitudes to zero and the resultant sparsity pattern is used as the side information in the recovery stage. If the side information is not available, the rec...

The problem of consistently estimating the sparsity pattern of a vector β∗ ∈ R based on observations contaminated by noise arises in various contexts, including subset selection in regression, structure estimation in graphical models, sparse approximation, and signal denoising. We analyze the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering th...

Compressed sensing provided a new sampling paradigm for sparse signals. Remarkably, it has been shown that practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sample rate. In real-world applications, a signal of interest is typically sparse not in the canonical basis but in a certain transform domain, such as the wavelet or the finite differenc...

In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between...