Universal and efficient compressed sensing by spread spectrum and application to realistic Fourier imaging techniques
Puy
Gilles
Vandergheynst
Pierre
Gribonval
RĂ©mi
Wiaux
Yves
2012
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 \emph{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 \emph{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 $\ell_1$-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.
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