Quantization, frames, and compressed sensing

数字计算机及其处理数据的效率是我们现代技术背后的主要驱动力之一。我的研究计划包括与获取固有模拟数据相关的数学问题（例如，音频和图像）、将它们转换成数字比特流、以及为了诸如去噪和压缩的若干目的而数字地处理数据。该研究计划在应用和计算谐波分析，信号处理和信息理论领域的三个主要研究方向：（i）冗余扩展量化的近似理论，（ii）压缩感知量化，以及（iii）稀疏近似和压缩感知领域的各种开放数学问题。（i）和（ii）之间的相互作用，即，稀疏信号的压缩采样测量的量化是本研究计划的重点。

Digital computers and their efficiency at processing data is one of the main driving forces behind much of our modern technology. My research program includes mathematical problems related to acquiring inherently analogue data (e.g., audio and images), converting them to digital bit-streams, and processing the data digitally for several purposes such as denoising and compression. This research proposal is on three main research lines in the areas of applied and computational harmonic analysis, signal processing, and information theory: (i) approximation theory of quantization of redundant expansions, (ii) quantization for compressed sensing, and (iii) various open mathematical problems in the area of sparse approximations and compressed sensing. The interactions between (i) and (ii), i.e., quantization of compressively sampled measurements of sparse signals, are in the focus of this research program.