We consider compressed sensing (CS) using partially coherent sensing matrices. Most of CS theory to date is focused on incoherent sensing, that is, columns from the sensing matrix are highly uncorrelated. However, sensing systems with naturally occurring correlations arise in many applications, such as signal detection, motion detection and radar. Moreover, in these applications it is often not necessary to know the support of the signal exactly, but instead small errors in the support and signal are tolerable. In this paper, we focus on d-tolerant recovery, in which support set reconstructions are considered accurate when their locations match the true locations within d indices. Despite the abundance of work utilizing incoherent sensing matrices, for d-tolerant recovery we suggest that coherence is actually beneficial. This is especially true for situations with only a few and very noisy measurements as we demonstrate via numerical simulations. As a first step towards the theory of tolerant coherent sensing we introduce the notions of d-coherence and d-tolerant recovery. We then provide some theoretical arguments for a greedy algorithm applicable to d-tolerant recovery of signals with sufficiently spread support.
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