Dequantizing Compressed Sensing: When Oversampling and Non-Gaussian Constraints Combine

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 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 p -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 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 √ 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.

One‐Bit Compressed Sensing by Linear Programming

“…One may endeavor to circumvent this problem by considering quantization errors as a source of noise, thereby reducing the quantized compressed sensing problem to the noisy classical compressed sensing problem. Further, in some cases the theory and algorithms of noisy compressed sensing may be adapted to this problem as in [28,11,17,25]; the method of quantization may be specialized in order to minimize the recovery error. As noted in [19] if the range of the signal is unspecified, then such a noise source is unbounded, and so the classical theory does not apply.…”

Joint Source-Channel Vector Quantization for Compressed Sensing

“…The optimality is addressed by minimizing the MSE performance measure E[ X − X stress that we use the VQ in its generic form. This is different from the design methods using uniform quantization [9] or 1-bit quantization of CS measurements [12]- [15]. Our contributions include…”

“…Examples are [6]- [15]. To elaborate, let us consider [9] where CS measurements are uniformly quantized and a convex optimization-based CS reconstruction algorithm, called basis pursuit dequantizing (BPDQ), is developed to suit the effect of uniform quantization. Further, the design of CS reconstruction algorithms and their performance bounds for reconstructing a sparse source from 1-bit quantized measurements have been investigated in [12]- [15].…”

“…minimizing end-to-end distortion, quantization distortion or ℓ 1 -norm of reconstruction vector. Some examples of system models following Figure 3 include [3], [5], [9], [11] (assuming a noiseless channel) and the conventional nearestneighbor coding of CS measurements. In general, according to this system model, quantizer decoder D outputs the vector Y ∈ R M after receiving channel output.…”

“…• Basis Pursuit DeQuantizing (BPDQ) [9]: Using this method, the encoder uniformly scalar-quantizes CS measurements, and the BPDQ algorithm [9] reconstructs the sparse source (from the quantized measurements) by solving the following convex optimization problem…”