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• W2128804753 abstract "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 <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</sub> ), 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 <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</sub> -norm of the residual error for 2 ≤ p ≤ ∞. 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 Iρ norm, and (ii), for Gaussian random matrices and uniformly quantized measurements, BPDQ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</sub> performance exceeds that of BPDN by dividing the reconstruction error due to quantization by √(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 <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> . To demonstrate the theoretical power of BPDQ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</sub> , we report numerical simulations on signal and image reconstruction problems." @default.
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• W2128804753 date "2011-01-01" @default.
• W2128804753 modified "2023-10-07" @default.
• W2128804753 title "Dequantizing Compressed Sensing: When Oversampling and Non-Gaussian Constraints Combine" @default.
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• W2128804753 doi "https://doi.org/10.1109/tit.2010.2093310" @default.
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