Compressive Sampling using Annihilating Filter-based Low-Rank Interpolation by Jong Chul Ye, Jong Min Kim, Kyong Hwan Jin
While the recent theory of compressed sensing or compressive sampling (CS) provides an opportunity to overcome the Nyquist limit in recovering sparse signals, a recovery algorithm usually takes the form of penalized least squares or constraint optimization framework that is different from classical signal sampling theory. In this paper, we provide a drastically different compressive sampling framework that can exploit all the benefits of the CS, but can be still implemented in a classical sampling framework using a digital correction filter. The main idea is originated from the fundamental duality between the sparsity in the primary space and the low-rankness of a structured matrix in the reciprocal spaces, which demonstrates that the low-rank interpolator as a digital correction filter can enjoy all the optimality of the standard CS. We show that the idea can be generalised to recover signals in large class of signals such as piece-wise polynomial, and spline representations. Moreover, by restricting signal class as cardinal splines, the proposed low-rank interpolation approach can achieve inherent regularization to improve the noise robustness. Using the powerful dual certificates and golfing scheme by Gross, we show that the new framework still achieves the near-optimal sampling rate for signal recovery. Numerical results using various type of signals confirmed that the proposed scheme has significant better phase transition than the conventional CS approaches.
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