Compressed sensing with corrupted Fourier measurements

TL;DR

This paper demonstrates that in compressed sensing with corrupted Fourier measurements, the true signal and corruption can be exactly recovered using reweighted l1 minimization under certain conditions, without requiring prior assumptions on the signal.

Contribution

It proves exact recovery of sparse signals and corruptions from Fourier measurements with minimal assumptions, extending previous results by removing restrictive conditions.

Findings

Exact recovery with high probability when m > O(card(x0) log(n) log(n))

No extra assumptions needed on the signal besides sparsity

Applicable when n is prime

Abstract

This paper studies a data recovery problem in compressed sensing (CS), given a measurement vector b with corruptions: b=Ax0+f0, can we recover x0 and f0 via the reweighted l1 minimization: minimize |x| + lambda*|f| subject to Ax+f=b? Here the m by n measurement matrix A is a partial Fourier matrix, x0 denotes the n dimensional ground true signal vector, f0 denotes the m-dimensional corrupted noise vector, it is assumed that a positive fraction of entries in the measurement vector b are corrupted by the non-zero entries of f0. This problem had been studied in literatures [1-3], unfortunately, certain random assumptions (which are often hard to meet in practice) are required for the signal x0 in these papers. In this paper, we show that x0 and f0 can be recovered exactly by the solution of the above reweighted l1 minimization with high probability provided that m>O(card(x0)log(n)log(n)) and n is prime, here card(x0) denotes the cardinality (number of non-zero entries) of x0. Except the sparsity, no extra assumption is needed for x0.