Recovering Compressively Sampled Signals Using Partial Support Information

Abstract

We study recovery conditions of weighted 1 minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50% of the (partial) support information is accurate, then weighted 1 minimization is stable and robust under weaker sufficient conditions than the analogous conditions for standard 1 minimization. Moreover, weighted 1 minimization provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals. Index TermsCompressed sensing, weighted 1 minimization, adaptive recovery. I. INTRODUCTIONCompressed sensing (see, e.g., [1]-[3]) is a paradigm for effective acquisition of signals that admit sparse (or approximately sparse) representations in some transform domain. The approach can be used to reliably recover such signals from significantly fewer linear measurements than their ambient dimension.Because a wide range of natural and man-made signals-e.g., audio, natural and seismic images, video, and wideband radio frequency signals-are sparse or approximately sparse in appropriate transform domains, the potential applications of compressed sensing can be immense.

Compressed sensing of data with a known distribution

“…For example, the paper [MDR14] analyzes the case where a signal similar to the one to be recovered is known beforehand. Also, [VL10,FMSY12] assume information about the support of the signal x 0 . Specifically, the first paper assumes to know part of the support entries, and the second one assumes to have prior knowledge about the support location.…”

Recursive Reconstruction of Sparse Signal Sequences

“…The work of [46] obtains exact recovery thresholds for weighted 1 , similar to those in [48], for the case when a probabilistic prior on the signal support is available. Some later work motivated by modified-CS includes modified OMP [49], modified CoSaMP [50], modified block CS [51], error bounds on modified BPDN [52], [22], [53], [20], better conditions for modified-CS based exact recovery [54], and exact support recovery conditions for multiple measurement vectors (MMV) based recursive recovery [33].…”

Adaptive compressed sensing for video acquisition

“…Our work is motivated by recent results on adaptive recovery from standard CS measurements using weighted 1 minimization [7]. Given a support estimate set T , the weighted 1 minimization problem is defined as…”

“…It was shown in [7] that if α > 0.5 and the matrix A satisfies RIP with constant δ (a+1)k < a−γ a+γ for γ = γ(α, ω) < 1 for some a > 1, then…”

“…Moreover, empirical studies in [7] have shown that a value ω ≈ 0.5 results in the most reliable recovery especially when the accuracy of the support estimate is not known. The recovery from compressed sensing measurements does not only depend on the sparsity of the signal but also on the relative decay of the transform coefficients.…”