[Paper Review] One-Bit Quantization Design and Adaptive Methods for Compressed Sensing

Jun Fang, Yanning Shen+1 more|arXiv (Cornell University)|Apr 7, 2013

Sparse and Compressive Sensing Techniques20 references3 citations

TL;DR

This paper proposes an adaptive one-bit quantization method for compressed sensing that iteratively refines quantization thresholds based on estimated signal values, enabling arbitrarily small reconstruction error. By dynamically adjusting thresholds toward unquantized measurements (which are unknown), the method resolves magnitude ambiguity and significantly improves signal recovery accuracy over fixed-threshold approaches.

ABSTRACT

There have been a number of studies on sparse signal recovery from one-bit quantized measurements. Nevertheless, little attention has been paid to the choice of the quantization thresholds and its impact on the signal recovery performance. This paper examines the problem of one-bit quantizer design for sparse signal recovery. Our analysis shows that the magnitude ambiguity that ever plagues conventional one-bit compressed sensing methods can be resolved, and an arbitrarily small reconstruction error can be achieved by setting the quantization thresholds close enough to the original data samples without being quantized. Note that unquantized data samples are unaccessible in practice. To overcome this difficulty, we propose an adaptive quantization method that adaptively adjusts the quantization thresholds in a way such that the thresholds converges to the optimal thresholds. Numerical results are illustrated to collaborate our theoretical results and the effectiveness of the proposed algorithm.

Motivation & Objective

• Address the lack of systematic design for one-bit quantization thresholds in compressed sensing.
• Overcome the magnitude ambiguity in conventional one-bit CS, where only signs are retained.
• Enable arbitrarily small reconstruction error by designing thresholds close to original unquantized measurements.
• Develop a practical method to approximate optimal thresholds when unquantized data are unavailable.
• Demonstrate improved performance through adaptive threshold refinement using iterative signal estimates.

Proposed method

• Proposes an adaptive quantization scheme that updates thresholds iteratively using signal estimates.
• Uses the estimate of the reconstructed signal to refine thresholds as $ \boldsymbol{\tau}^{(t)} = \boldsymbol{\hat{y}}^{(t)} + \xi^{(t)}\boldsymbol{\delta}^{(t)} $, where $ \boldsymbol{\hat{y}}^{(t)} $ is the estimated measurement vector.
• Employs a diminishing deviation parameter $ \xi^{(t)} $, updated as $ \xi^{(t+1)} = \xi^{(t)}/10 $, to gradually reduce threshold perturbation.
• Initial thresholds are set as $ \boldsymbol{\tau}^{(0)} = \boldsymbol{y} + \xi^{(0)}\boldsymbol{\delta}^{(0)} $, with $ \xi^{(0)} $ proportional to the norm of the measurements.
• Applies $ \ell_1 $-minimization or log-sum minimization for signal reconstruction at each iteration.
• Employs convergence criteria based on $ \|\boldsymbol{\hat{x}}^{(t)} - \boldsymbol{\hat{x}}^{(t-1)}\|_2 < 0.01 $ to stop iteration.

Experimental results

Research questions

• RQ1 How does the choice of quantization threshold affect the reconstruction accuracy in one-bit compressed sensing?
• RQ2 Can arbitrarily small reconstruction error be achieved with one-bit quantization if thresholds are optimally chosen?
• RQ3 What strategies can be used to approximate optimal thresholds when unquantized measurements are unavailable?
• RQ4 How does adaptive threshold refinement improve signal recovery compared to fixed-threshold schemes?
• RQ5 What is the convergence behavior and performance gain of iterative threshold adaptation in one-bit CS?

Key findings

• Arbitrarily small reconstruction error is theoretically achievable by setting quantization thresholds close to the original unquantized measurements.
• The proposed adaptive method significantly improves reconstruction performance over non-adaptive schemes using fixed thresholds.
• The algorithm converges within ten iterations on average, with consistent performance improvement across iterations.
• Numerical results show that the adaptive scheme outperforms non-adaptive one-bit methods in terms of normalized mean-squared error (NMSE) across varying numbers of measurements.
• Image recovery experiments demonstrate that the adaptive scheme produces visibly sharper and more accurate reconstructions than the non-adaptive baseline.
• The method effectively resolves magnitude ambiguity, enabling better signal reconstruction despite one-bit quantization.