On asymptotic incoherence and its implications for compressed sensing for inverse problems

Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to infinite-dimensional inverse problems. Instead, the key property that permits efficient recovery in such problems is so-called asymptotic incoherence. The purpose of this paper is to study this new concept, and its implications towards the design of optimal sampling strategies. In particular, we show that Fourier sampling and wavelet sparsity, whilst globally coherent, yield optimal asymptotic incoherence up to a constant factor. We also provide sharp bounds on the asymptotic incoherence in the case of Fourier sampling with polynomial bases. Finally, we analyse how the orderings of the sampling and sparsity bases affect the asymptotic incoherence in one or more dimensions, and the implication of this on sampling strategies.