Dequantizing compressed sensing with non-gaussian constraints

Laurent Jacques (1, 2) , David K. Hammond (1) , Jalal M. Fadili (3)

1 Electrical Engineering Institute - EPFL
2 Communications and Remote Sensing Laboratory [Louvain]
3 Equipe Image - Laboratoire GREYC - UMR6072

In this paper, following the Compressed Sensing 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 (BPDQp), 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 lp-norm, for 2 < p < infini. We show that in oversampled situations the performance of the BPDQp decoders are significantly better than that of BPDN, with reconstruction error due to quantization divided by racine(p 1). This reduction relies on a modified Restricted Isometry Property of the sensing matrix expressed in the lp-norm (RIPp); a property satisfied by Gaussian random matrices with high probability. We conclude with numerical experiments comparing BPDQp and BPDN for signal and image reconstruction problems.

Mots clés: Basis Pursuit

Mots clés: Compressed Sensing; Quantization; Sampling; Uniform noise; Convex Optimization; Basis Pursuit.

Domaines: Traitement des images [eess.IV]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Laurent Jacques, David K. Hammond, Jalal M. Fadili. Dequantizing compressed sensing with non-gaussian constraints. IEEE ICIP, Nov 2009, Cairo, Egypt. pp.1465 - 1468, ⟨10.1109/ICIP.2009.5414551⟩. ⟨hal-00813717⟩