| Model-based signal processing is concerned with measuring, understanding, and communicating data under the assumption that the (potentially high-dimensional) data in hand has in fact few degrees of freedom and can be accurately represented with a concise signal model.;For instance, when the signal model is the class of sparse signals (i.e., signals with a concise representation in some basis), compressive sensing has proved effective in combining the sensing and compression stages, thereby allowing for more efficient sensors and powerful signal processing algorithms. A canonical result in compressive sensing states that sparse signals can be accurately recovered from a small number of generic linear measurements by solving a tractable convex optimization problem.;More generally, however, manifold-based signal processing focuses on (possibly high-dimensional) data which lives on (or near) low-dimensional surfaces (or manifolds, to use the precise term). Typically, this is a stronger notion of conciseness compared to sparsity and results in a more efficient representation of data, one that often leads to better signal processing algorithms for manifold-modeled data.;With this introduction, the central question in this dissertation can be phrased as follows: What can we learn about data that lives on (or near) a manifold from limited (or compressive) measurements?;In adding to the collective knowledge of this question, we study the quality of signal reconstruction from compressive measurements. Moreover, we make strides in answering the question above for certain specific manifolds that often arise in applications. For instance, we estimate (from limited measurements) the unknown parameters of signals that live on the shift-manifold (which is formed from translated copies of a template signal). Lastly, we focus on a framework for time-series analysis that relies heavily on manifold models. |
