Recently, a new signal processing paradigm based on "sparsity" has emerged. This new paradigm exploits an empirical observation: many types of signals, e.g., audio, natural images, and video, can be well represented by a sparse expansion with respect to a suitable basis or frame. Compressed sensing, which has gained vast interest in applied mathematics as well as in engineering in recent years, exploits this fact in order to recover signals from a very limited amount of linear measurements.
It is now well established that compressed sensing can be used for dimension reduction. In this talk, we focus on another important problem: How can one quantize compressed sensing measurements in an efficient way? Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candes, Romberg and Tao, and of Donoho. In particular, if the measurement operator satisfies certain generic conditions, e.g., the restricted isometry property, these results guarantee that an approximation with accuracy of order $\delta$ can be obtained if each measurement of a sparse signal is quantized within an accuracy of $\delta$. In this talk, we propose a novel approach to quantization of compressed sensing measurements that utilizes "noise-shaping". It turns out that our previous work on "Sobolev duals" and frame quantization is the key for constructing superior quantizers in the compressed sensing setting. Our results are based on an interplay between frame theory techniques, sigma-delta quantization methods, and techniques from probability theory.