Binary Compressive Sensing and Super-Resolution With Unknown Threshold

Subhadip Mukherjee, Anjany Kumar Sekuboyina, Chandra Sekhar Seelamantula

Abstract: We consider the problem of binary compressive sensing (CS), where random linear projections of a sparse signal are encoded using threshold-crossing information. The threshold used by the binary encoder for acquisition is unknown to the decoder and is estimated jointly with the signal. We cast the problem of signal reconstruction and threshold estimation as one of learning a hyperplane that separates the sampling vectors corresponding to the +1 and -1 measurements, and develop a reconstruction algorithm that entails iterative minimization of reweighted ℓ 1 -norm subject to a set of linear constraints that enforce measurement separability. The proposed algorithm leads to a reconstruction performance comparable with that obtained using a popular binary CS algorithm, namely binary iterative hard-thresholding, which assumes that the threshold is set to zero. We consider binary super-resolution as an application, where a signal consisting of point sources needs to be estimated from sign measurements of its blurred version. The proposed algorithm successfully recovers the locations and amplitudes of the point sources, even in the presence of significant blurring.