학술논문

Message-Passing De-Quantization With Applications to Compressed Sensing
Document Type
Periodical
Source
IEEE Transactions on Signal Processing IEEE Trans. Signal Process. Signal Processing, IEEE Transactions on. 60(12):6270-6281 Dec, 2012
Subject
Signal Processing and Analysis
Communication, Networking and Broadcast Technologies
Computing and Processing
Quantization
Estimation
Linear systems
Transforms
Compressed sensing
Approximation algorithms
AC-DC converters
Analog-to-digital conversion
approximate message passing
belief propagation
compressed sensing
frames
non-regular quantizers
overcomplete representations
quantization
Slepian-Wolf coding
Wyner-Ziv coding
Language
ISSN
1053-587X
1941-0476
Abstract
Estimation of a vector from quantized linear measurements is a common problem for which simple linear techniques are suboptimal—sometimes greatly so. This paper develops message-passing de-quantization (MPDQ) algorithms for minimum mean-squared error estimation of a random vector from quantized linear measurements, notably allowing the linear expansion to be overcomplete or undercomplete and the scalar quantization to be regular or non-regular. The algorithm is based on generalized approximate message passing (GAMP), a recently-developed Gaussian approximation of loopy belief propagation for estimation with linear transforms and nonlinear componentwise-separable output channels. For MPDQ, scalar quantization of measurements is incorporated into the output channel formalism, leading to the first tractable and effective method for high-dimensional estimation problems involving non-regular scalar quantization. The algorithm is computationally simple and can incorporate arbitrary separable priors on the input vector including sparsity-inducing priors that arise in the context of compressed sensing. Moreover, under the assumption of a Gaussian measurement matrix with i.i.d. entries, the asymptotic error performance of MPDQ can be accurately predicted and tracked through a simple set of scalar state evolution equations. We additionally use state evolution to design MSE-optimal scalar quantizers for MPDQ signal reconstruction and empirically demonstrate the superior error performance of the resulting quantizers. In particular, our results show that non-regular quantization can greatly improve rate-distortion performance in some problems with oversampling or with undersampling combined with a sparsity-inducing prior.