2009/02/13 by Laurent Jacques, Jacques, Laurent, David K. Hammond +3
Computer Science · Engineering · #Distributed Sensor Networks and Detection Algorithms #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Microwave Imaging and Scattering Analysis #Optimization and Control (math.OC) #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.0902.2367
openalex publication_date 2009/02/13 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
In this paper we study the problem of recovering sparse or compressible\nsignals from uniformly quantized measurements. We present a new class of convex\noptimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment\np (BPDQp), that model the quantization distortion more faithfully than the\ncommonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by\nminimizing the sparsity of the signal to be reconstructed subject to a\ndata-fidelity constraint expressed in the \ℓp-norm of the residual error\nfor 2\≤ p\≤ \∞.\n We show theoretically that, (i) the reconstruction error of these new\ndecoders is bounded if the sensing matrix satisfies an extended Restricted\nIsometry Property involving the \ℓp norm, and (ii), for Gaussian random\nmatrices and uniformly quantized measurements, BPDQp performance exceeds\nthat of BPDN by dividing the reconstruction error due to quantization by\n\√(p+1). This last effect happens with high probability when the number of\nmeasurements exceeds a value growing with p, i.e. in an oversampled situation\ncompared to what is commonly required by BPDN = BPDQ2. To demonstrate the\ntheoretical power of BPDQp, we report numerical simulations on signal and\nimage reconstruction problems.\n