期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 57, 期 1, 页码 559-571出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2010.2093310
关键词
Compressed sensing; convex optimization; denoising; optimality; oversampling; quantization; sparsity
In this paper, 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 subject to a data-fidelity constraint expressed in the l(p)-norm of the residual error for 2 <= p <= infinity. We show theoretically that, (i) the reconstruction error of these new decoders is bounded if the sensing matrix satisfies an extended Restricted Isometry Property involving the l(p) norm, and (ii), for Gaussian random matrices and uniformly quantized measurements, BPDQ(p) performance exceeds that of BPDN by dividing the reconstruction error due to quantization by root p+1. This last effect happens with high probability when the number of measurements exceeds a value growing with p, i.e., in an oversampled situation compared to what is commonly required by BPDN = BPDQ(2). To demonstrate the theoretical power of BPDQ(p), we report numerical simulations on signal and image reconstruction problems.
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