Signal Recovery with Unknown Clustered Sparsity and Quantization

实数信号通常是稀疏的，因为它们通过使用适当的基而具有紧凑的表示。值得注意的例子是数字图像，可以通过各种压缩技术来压缩和压缩。稀疏性利用已被证明是一个强大的工具，用于采集，传输和存储高维信号。值得注意的是，它允许从相对较少的测量中恢复整个信号。诸如图像、音频和视频信号的许多稀疏信号也表现出某种稀疏模式（例如，群集或块稀疏系数），其允许更有效的信号恢复。然而，当稀疏模式是不规则的或未知的，如何有效地恢复信号从一些测量仍然是一个未知的领域。这是该项目的第一个研究方向。同时，对于信号采集，实际系统通常采用量化，将每个测量值转换为几个二进制比特，以便于处理和存储。然而，大多数经典的稀疏信号恢复方法忽略了量化的影响。直到最近，稀疏信号重建/估计，明确占量化的效果开始受到关注，虽然大多数这样的作品只是考虑启发式量化器。该项目的第二个研究重点是研究与量化相关的两个问题：（1）开发具有量化测量的有效信号恢复算法;以及（2）用于稀疏信号恢复的最佳量化器设计，特别是对于许多具有带宽/功率约束的传感系统所需的低速率量化的情况。该项目还有一个重要的教育组成部分，旨在为本科生和研究生提供综合研究经验和培训。该项目采取了几种方法来解决上述问题。首先，在聚类稀疏信号恢复方面，PI提出了一种新的贝叶斯学习框架，用于未知块稀疏结构的稀疏信号恢复。所提出的研究建立在一种新的贝叶斯分层模型，涉及耦合超参数，以促进块稀疏性，而不施加刚性块结构。基于所提出的模型，一系列的贝叶斯推理算法的开发，考虑到计算效率和鲁棒性噪声。其次，在低速率量化器设计上，PI提出了一种自适应量化方法，以提高量化测量的稀疏信号恢复算法的重建性能。所提出的方法包括顺序量化每个测量，使用过去的量化位来预测下一个要量化的样本，最后在预测值阈值。第三，在使用量化测量的重建算法的开发上，PI提出采用S形函数来在重建信号和量化测量之间施加一致性约束，这被示出为提供比现有方法更低的计算复杂度和显著的重建精度提高的益处。

Real-word signals are often sparse in that they have compact representation by using proper bases. Notable examples are digital images which can be compactly represented and compressed via various compression techniques. Sparsity exploitation has proven a powerful tool for the acquisition, transmission, and storage of high-dimensional signals. Remarkably, it allows recovering the entire signal from relatively few measurements. Many sparse signals such as images, audio and video signals also exhibit some sparsity pattern (e.g., clustered or block sparse coefficients) that admits more efficient signal recovery. However, when the sparsity pattern is irregular or unknown, how to efficiently recover the signal from a few measurements is still largely an uncharted territory. This is the first research thrust pursued in this project. Meanwhile, for signal acquisition, practical systems usually employ quantization which converts each measurement into a few binary bits to facilitate processing and storage. The effect of quantization, however, was neglected by most classical sparse signal recovery methods. Only recently, sparse signal reconstruction/estimation which explicitly accounts for the effect of quantization started to receive attention, although most such works just considered heuristic quantizers. A second research thrust of this project examines two problems related to quantization: (1) development of efficient signal recovery algorithms with quantized measurements; and (2) optimum quantizer design for sparse signal recovery, in particular for the case of low-rate quantization which is required in many sensing systems that have bandwidth/power constraints. This project also has a significant educational component aimed to provide integrated research experience and training for undergraduate and graduate students. This project takes several approaches to address the above problems. First, on clustered sparse signal recovery, the PI proposes a new Bayesian learning framework for sparse signal recovery with unknown block sparsity structure. The proposed research builds on a novel Bayesian hierarchical model involving coupled hyperparameters to promote block sparsity without imposing rigid block structures. Based on the proposed models, a range of Bayesian inference algorithms are to be developed, taking into account of computational efficiency and robustness to noise. Second, on low-rate quantizer design, the PI proposes an adaptive quantization approach to enhance the reconstruction performance of sparse signal recovery algorithms taking quantized measurements. The proposed approach involves sequentially quantizing each measurement, using past quantized bits to predict the next sample to be quantized, and finally thresholding at the predicted value. Third, on the development of reconstruction algorithms using quantized measurements, the PI proposes to employ a sigmoid function to impose a consistency constraint between the reconstructed signal and the quantized measurements, which is shown to offer the benefit of computational complexity reduction and significant reconstruction accuracy improvement over existing methods.