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.
Highlights
We concisely recall some basics of compressed sensing, emphasizing on the role of mutual coherence between the sparsity and sensing bases
5 Conclusion We have presented a compressed sensing strategy that consists of a wide bandwidth pre-modulation of the signal of interest before projection onto randomly selected vectors of an orthonormal basis
In a digital setting with a random pre-modulation, the technique was proved to be universal for sensing bases such as the Fourier or Hadamard bases, where it may be implemented efficiently
Summary
We concisely recall some basics of compressed sensing, emphasizing on the role of mutual coherence between the sparsity and sensing bases. 1.1 Compressed sensing basics Compressed sensing is a recent theory aiming at merging data acquisition and compression [1,2,3,4,5,6,7] It predicts that sparse or compressible signals can be recovered from a small number of linear and non-adaptative measurements. In this context, Gaussian and Bernouilli random matrices, respectively with independent standard normal and ± 1 entries, have encountered a particular interest as they provide optimal conditions in terms of the number of measurements needed to recover sparse signals [3,4,5]. The measurement vector y Î Cm reads as (1)
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