  Recently, it has been shown that incoherence is an unrealistic assumption for
compressed sensing when applied to many inverse problems. Instead, the key
property that permits efficient recovery in such problems is so-called local
incoherence. Similarly, the standard notion of sparsity is also inadequate for
many real world problems. In particular, in many applications, the optimal
sampling strategy depends on asymptotic incoherence and the signal sparsity
structure. The purpose of this paper is to study asymptotic incoherence and its
implications towards the design of optimal sampling strategies and efficient
sparsity bases. It is determined how fast asymptotic incoherence can decay in
general for isometries. Furthermore it is shown that Fourier sampling and
wavelet sparsity, whilst globally coherent, yield optimal asymptotic
incoherence as a power law up to a constant factor. Sharp bounds on the
asymptotic incoherence for Fourier sampling with polynomial bases are also
provided. A numerical experiment is also presented to demonstrate the role of
asymptotic incoherence in finding good subsampling strategies.
