Universal and efficient compressed sensing by spread spectrum and application to realistic Fourier imaging techniques
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RESEARCH
Open Access
Universal and efficient compressed sensing by spread spectrum and application to realistic Fourier imaging techniques Gilles Puy1,2*, Pierre Vandergheynst1, Rémi Gribonval3 and Yves Wiaux1,4,5
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. First, 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. Second, 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. 1 Introduction In this section we concisely recall some basics of compressed sensing, emphasizing on the role of mutual coherence between the sparsity and sensing bases. We discuss the interest of improving the standard acquisition strategy in the context of Fourier imaging techniques such as radio interferometry and magnetic resonance imaging (MRI). Finally we highlight the main contributions of our study advocating a universal and efficient compressed sensing strategy coined spread spectrum, and describe the organization of this article. 1.1 Compressed sensing basics
Compressed sensing is a recent theory aiming at merging data acquisition and compression [1-7]. 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 * Correspondence: [email protected] 1 Institute of Electrical Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Full list of author information is available at the end of the article
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 [3-5]. However, the use of these matrices for real-world applications is limited for several reasons: no fast matrix multiplication algorithm is available, huge memory requirements for large scale problems, difficult implementation on hardware, etc. Let us consider ssparse digital
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