Universal and efficient compressed sensing by spread spectrum and application to realistic Fourier imaging techniques

TL;DR

This paper introduces a universal compressed sensing method using spread spectrum modulation that improves measurement efficiency and is applicable to Fourier imaging, confirmed by numerical and practical experiments.

Contribution

The paper presents a universal compressed sensing approach with spread spectrum modulation, reducing coherence and enabling efficient, basis-independent signal recovery in digital and analog Fourier imaging.

Findings

Universal measurement efficiency independent of sparsity basis

Effective in digital and analog Fourier imaging scenarios

Validated by numerical phase transition analysis

Abstract

We advocate a compressed sensing strategy that consists of multiplying the signal of interest by a wide bandwidth modulation before projection onto randomly selected vectors of an orthonormal basis. Firstly, in a digital setting with random modulation, considering a whole class of sensing bases including the Fourier basis, we prove that the technique is universal in the sense that the required number of measurements for accurate recovery is optimal and independent of the sparsity basis. This universality stems from a drastic decrease of coherence between the sparsity and the sensing bases, which for a Fourier sensing basis relates to a spread of the original signal spectrum by the modulation (hence the name "spread spectrum"). The approach is also efficient as sensing matrices with fast matrix multiplication algorithms can be used, in particular in the case of Fourier measurements. Secondly, these results are confirmed by a numerical analysis of the phase transition of the l1- minimization problem. Finally, we show that the spread spectrum technique remains effective in an analog setting with chirp modulation for application to realistic Fourier imaging. We illustrate these findings in the context of radio interferometry and magnetic resonance imaging.