While the recent theory of compressed sensing provides an opportunity toovercome the Nyquist limit in recovering sparse signals, a solution approachusually takes a form of inverse problem of the unknown signal, which iscrucially dependent on specific signal representation. In this paper, wepropose a drastically different two-step Fourier compressive sampling frameworkin continuous domain that can be implemented as a measurement domaininterpolation, after which a signal reconstruction can be done using classicalanalytic reconstruction methods. The main idea is originated from thefundamental duality between the sparsity in the primary space and thelow-rankness of a structured matrix in the spectral domain, which shows that alow-rank interpolator in the spectral domain can enjoy all the benefit ofsparse recovery with performance guarantees. Most notably, the proposedlow-rank interpolation approach can be regarded as a generalization of recentspectral compressed sensing to recover large class of finite rate ofinnovations (FRI) signals at near optimal sampling rate. Moreover, for the caseof cardinal representation, we can show that the proposed low-rankinterpolation will benefit from inherent regularization and the optimalincoherence parameter. Using the powerful dual certificates and golfing scheme,we show that the new framework still achieves the near-optimal sampling ratefor general class of FRI signal recovery, and the sampling rate can be furtherreduced for the class of cardinal splines. Numerical results using various typeof FRI signals confirmed that the proposed low-rank interpolation approach hassignificant better phase transition than the conventional CS approaches.
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