Signal recovery from random projections

Can we recover a signal f∈R^N from a small number of linear measurements? A series of recent papers developed a collection of results showing that it is surprisingly possible to reconstruct certain types of signals accurately from limited measurements. In a nutshell, suppose that f is compressible in the sense that it is well-approximated by a linear combination of M vectors taken from a known basis Ψ. Then not knowing anything in advance about the signal, f can (very nearly) be recovered from about M log N generic nonadaptive measurements only. The recovery procedure is concrete and consists in solving a simple convex optimization program. In this paper, we show that these ideas are of practical significance. Inspired by theoretical developments, we propose a series of practical recovery procedures and test them on a series of signals and images which are known to be well approximated in wavelet bases. We demonstrate that it is empirically possible to recover an object from about 3M-5M projections onto generically chosen vectors with an accuracy which is as good as that obtained by the ideal M-term wavelet approximation. We briefly discuss possible implications in the areas of data compression and medical imaging

Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

Suppose we are given a vector $f$ in $\R^N$. How many linear measurements do we need to make about $f$ to be able to recover $f$ to within precision $\epsilon$ in the Euclidean ($\ell_2$) metric? Or more exactly, suppose we are interested in a class ${\cal F}$ of such objects--discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy $\epsilon$? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal $f \in {\cal F}$ decay like a power-law (or if the coefficient sequence of $f$ in a fixed basis decays like a power-law), then it is possible to reconstruct $f$ to within very high accuracy from a small number of random measurements.Comment: 39 pages; no figures; to appear. Bernoulli ensemble proof has been corrected; other expository and bibliographical changes made, incorporating referee's suggestion

Signal recovery from random projections

Can we recover a signal f∈R^N from a small number of linear measurements? A series of recent papers developed a collection of results showing that it is surprisingly possible to reconstruct certain types of signals accurately from limited measurements. In a nutshell, suppose that f is compressible in the sense that it is well-approximated by a linear combination of M vectors taken from a known basis Ψ. Then not knowing anything in advance about the signal, f can (very nearly) be recovered from about M log N generic nonadaptive measurements only. The recovery procedure is concrete and consists in solving a simple convex optimization program. In this paper, we show that these ideas are of practical significance. Inspired by theoretical developments, we propose a series of practical recovery procedures and test them on a series of signals and images which are known to be well approximated in wavelet bases. We demonstrate that it is empirically possible to recover an object from about 3M-5M projections onto generically chosen vectors with an accuracy which is as good as that obtained by the ideal M-term wavelet approximation. We briefly discuss possible implications in the areas of data compression and medical imaging

Counting faces of randomly-projected polytopes when the projection radically lowers dimension

This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing - with potential impacts in practical subjects like medical imaging and digital communications. Three such implications concern: convex hulls of Gaussian point clouds, signal recovery from random projections, and how many gross errors can be efficiently corrected from Gaussian error correcting codes.Comment: 56 page

Phase recovery from a Bayesian point of view: the variational approach

In this paper, we consider the phase recovery problem, where a complex signal vector has to be estimated from the knowledge of the modulus of its linear projections, from a naive variational Bayesian point of view. In particular, we derive an iterative algorithm following the minimization of the Kullback-Leibler divergence under the mean-field assumption, and show on synthetic data with random projections that this approach leads to an efficient and robust procedure, with a good computational cost.Comment: To appear in the proceedings of IEEE Int'l Conference on Acoustics, Speech and Signal Processing (ICASSP

Performance Bounds for Sparsity Pattern Recovery with Quantized Noisy Random Projections

In this paper, we study the performance limits of recovering the support of a sparse signal based on quantized noisy random projections. Although the problem of support recovery of sparse signals with real valued noisy projections with different types of projection matrices has been addressed by several authors in the recent literature, very few attempts have been made for the same problem with quantized compressive measurements. In this paper, we derive performance limits of support recovery of sparse signals when the quantized noisy corrupted compressive measurements are sent to the decoder over additive white Gaussian noise channels. The sufficient conditions which ensure the perfect recovery of sparsity pattern of a sparse signal from coarsely quantized noisy random projections are derived when the maximum likelihood decoder is used. More specifically, we find the relationships among the parameters, namely the signal dimension N, the sparsity index K, the number of noisy projections M, the number of quantization levels L, and measurement signal-to-noise ratio which ensure the asymptotic reliable recovery of the support of sparse signals when the entries of the measurement matrix are drawn from a Gaussian ensemble