利用标量压缩扩展器稳定非均匀量化压缩感知
Stabilizing Nonuniformly Quantized Compressed Sensing with Scalar Companders
摘要：
本文研究了从压缩感知测量数据中重构稀疏或可压缩信号的问题。基于高斯模型（有界l2范数）的量化失真的量化压缩感知（QCS）问题的先前方法产生的结果，虽然通常是可接受的，但可能不是完全一致的：重建信号的重新测量和量化不一定匹配的初始观察。量化失真反而更接近于异方差均匀噪声，方差取决于观察到的量化仓。推广我们以前的工作均匀量化，我们表明，非均匀量化器所描述的“压扩器”的形式主义，量化失真可能会更好地表征为有界加权Lp-范数（p >= 2），对于一个特定的权重。我们开发了一种新的重建方法，称为广义基追踪去噪（GBPDN），它最大限度地减少了这个加权Lp范数保真度约束下的重建信号的稀疏性。我们证明，对于B位每次测量和下过采样QCS的情况下（当测量的数量是大的信号稀疏相比），如果传感矩阵满足一个建议的广义限制等距属性，然后，GBPDN提供了稀疏信号的重建误差减少像O（2^{-B}/ \sqrt{p+1}）：减少了一个因子\sqrt{p+1}相对于使用l2-范数产生的。除了QCS的情况下，我们还表明，GBPDN直接适用于异方差广义高斯噪声与可证明的重建误差减少损坏的CS测量的相关情况。最后，我们描述了一个有效的数值计算过程，通过原始-对偶凸优化方案计算GBPDN，并通过模拟证明其有效性。
英文摘要：
This paper studies the problem of reconstructing sparse or compressible signals from compressed sensing measurements that have undergone nonuniform quantization. Previous approaches to this Quantized Compressed Sensing (QCS) problem based on Gaussian models (bounded l2-norm) for the quantization distortion yield results that, while often acceptable, may not be fully consistent: re-measurement and quantization of the reconstructed signal do not necessarily match the initial observations. Quantization distortion instead more closely resembles heteroscedastic uniform noise, with variance depending on the observed quantization bin. Generalizing our previous work on uniform quantization, we show that for nonuniform quantizers described by the "compander" formalism, quantization distortion may be better characterized as having bounded weighted lp-norm (p >= 2), for a particular weighting. We develop a new reconstruction approach, termed Generalized Basis Pursuit DeNoise (GBPDN), which minimizes the sparsity of the reconstructed signal under this weighted lp-norm fidelity constraint. We prove that for B bits per measurement and under the oversampled QCS scenario (when the number of measurements is large compared to the signal sparsity) if the sensing matrix satisfies a proposed generalized Restricted Isometry Property, then, GBPDN provides a reconstruction error of sparse signals which decreases like O(2^{-B}/\sqrt{p+1}): a reduction by a factor \sqrt{p+1} relative to that produced by using the l2-norm. Besides the QCS scenario, we also show that GBPDN applies straightforwardly to the related case of CS measurements corrupted by heteroscedastic Generalized Gaussian noise with provable reconstruction error reduction. Finally, we describe an efficient numerical procedure for computing GBPDN via a primal-dual convex optimization scheme, and demonstrate its effectiveness through simulations.
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