Compressed sensing and related areas: bridging theory with practice

从20世纪80年代中期开始，应用和计算谐波分析社区构建了构建模块（例如，小波和曲波），其允许许多感兴趣的信号的有效分解（例如，自然图像和音频）。这一领域的活动导致了工业的根本性变化，促进了数字革命。小波技术已经渗透到许多相关领域，包括图像压缩、模式识别、雷达、医学成像和生物物理学。构建这些构建模块的主要动机之一是获得实际感兴趣的信号的稀疏近似。这项研究开启了一门新技术：压缩传感（CS）。 CS的中心观察可以简单地说：具有（近似）稀疏表示的信号可以通过易于处理的算法从很少的测量中恢复。挑战在于稀疏信号的非线性和组合性质。这个想法很快被推广到其他非线性结构，包括低秩矩阵和流形。结构化信号无处不在，CS的思想有望彻底改变几个不同的领域和应用。事实上，在过去的十年里，已经看到了CS理论的全面基础的建设。然而，CS的实践并没有像小波那样渗透到工业中。 我的研究计划的主题是桥接CS理论与真实的应用程序，有各种严格的限制。这些经常导致困难，实际相关的数学问题。以下是我打算（继续）探索的一些具体方向： (1)模数转换：为了将CS建立为可行的信号采集方案，它需要能够产生对所采集信号的高精度估计。在数字世界中，这必须通过量化测量来实现。虽然在这一领域开展了一些工作，但仍有几个基本问题有待回答。我们将在以前的工作基础上回答这些问题，并为压缩感知实现其最初的目标之一做出贡献：将感知与压缩合并。 (2)基于模型和数据的恢复：CS通常不假设除稀疏性之外的关于目标信号的任何先验知识。然而，在许多应用中，存在可以并入测量和/或重建方案的先验信息。例如，在视频帧序列中，连续的帧通常是相似的。我们希望探索各种方法，将先验信息的重建方法，同时保持感测非自适应。 (3)基于模型和数据的采集：我们将专注于基于模型的采集和CS定制恢复方法的联合设计。我们还将通过显式建模和通过深度学习的隐式建模来解决各种“模型不匹配”问题。

Starting in the mid 1980s, the applied and computational harmonic analysis community constructed building blocks (e.g., wavelets and curvelets) that allow for efficient decomposition of many signals of interest (e.g., natural images and audio). The activity in this area led to fundamental changes in industry that have contributed to the digital revolution. Wavelet techniques have penetrated many related areas, including image compression, pattern recognition, radar, medical imaging, and geophysics. One of the main motivations when constructing these building blocks was to obtain sparse approximations for signals of practical interest. This research opened the door to a new technology: compressed sensing (CS). The central observation of CS can be stated simply: signals that have (approximately) sparse representations can be recovered from few measurements via tractable algorithms. The challenge lies in the non-linear and combinatorial nature of sparse signals. This idea was soon generalized to allow for other non-linear structures including low-rank matrices and manifolds. Structured signals are ubiquitous, and the ideas of CS promised to revolutionize several different fields and applications. Indeed, the last decade has seen the construction of a comprehensive foundation for the theory of CS. Yet, the practice of CS has not penetrated industry to the same degree that wavelets have. The main theme of my research programme is bridging the CS theory with real applications that have various rigid constraints. These often result in difficult, practically relevant mathematical problems. Below are some specific directions I intend to (continue to) explore: (1) Analogue-to-digital conversion: For CS to be established as a viable signal acquisition scheme, it needs to be able to produce high accuracy estimates of the acquired signal. In a digital world, this must be achieved from quantized measurements. While there is some work in this area, there are several fundamental questions left to answer. We will build on our previous work to answer these and contribute to have compressed sensing fulfill one of its original goals: merging sensing with compression. (2) Model and data based recovery: CS typically does not assume any prior knowledge about the target signal other than sparsity. However, in many applications there is prior information that can incorporated to the measurement and/or the reconstruction schemes. For example, in a video frame sequence, consecutive frames are usually similar. We wish to explore various approaches for incorporating prior information into the reconstruction method, while keeping sensing non-adaptive. (3) Model and data based acquisition: We will focus on the joint design of model based acquisition and tailored recovery methods for CS. We will also address various "model mismatch" issues both by explicit modeling and by implicit modeling via deep learning.