DeQuantizing Compressed Sensing with non-Gaussian constraints

In this paper, following the Compressed Sensing (CS) paradigm, we study the problem of recovering sparse or compressible signals from uniformly quantized measurements. We present a new class of convex optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment p (BPDQ_p), that model the quantization distortion more faithfully than the commonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by minimizing the sparsity of the signal to be reconstructed while enforcing a data fidelity term of bounded ¿_p-norm, for 2 < p ¿ ¿. We show that in oversampled situations, i.e. when the number of measurements is higher than the minimal value required by CS, the performance of the BPDQ_p decoders outperforms that of BPDN, with reconstruction error due to quantization divided by. This reduction relies on a modified Restricted Isometry Property of the sensing matrix expressed in the ¿_p-norm (RIP_p); a property satisfied by Gaussian random matrices with high probability. We conclude with numerical experiments comparing BPDQ_p and BPDN for signal and image reconstruction problems.