In a diverse range of application areas--including astronomical imaging, microscopy, wireless communications, radar, sonar, and many others--one must confront the problem of estimating or extracting information from a signal that is governed by a small number of unknown (continuous-valued) parameters. Customized to each of these applications, unique methods for handling these unknown parameters have been proposed in the past. This research involves the development of a unified framework for addressing problems in all of these areas systematically, efficiently, with awareness of the fundamental limits, and with provable accuracy.
There are two main components of this framework. The first component involves the development of models where the signals of interest lie within or close to a subspace or a union of subspaces. In the case where the underlying parameters for the signal are continuous-valued, this is a challenging task. Rather than attempt to simply discredit these problems (a program potentially fraught with difficulty), this research involves the development of finite approximations for a locally infinite range of subspaces using efficient local subspace fits whose dimensions match the effective number of local degrees of freedom. The second component of this framework consists of techniques for identifying the range of subspaces responsible for generating a signal from a set of discrete observations, confronting difficulties posed by the coherence, by the continuous nature of the parameter space, and by the fact that some signals may be synthesized from a narrow (but continuously-indexed and therefore infinite) range of subspaces. These components combine to enable the efficient acquisition, processing, and estimation of signals arising from the parameterized subspaces model.
