Wednesday, October 09, 2013

Infinity Continues to Matter: Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum

When writing this entry on Compressive Sensing and Uncertainty Quantification a while ago, the following slide was telling me a different story than the one we should be expecting.
Yes, a CS approach to uncertainty quantification does provide some relief in allowing you to have access to the largest coefficients. However, this finding has to be tempered with the fact that lower level coefficients do have an impact (in effect the norm used to evaluate the error is not the real norm that which one uses to evaluate a technique) In fact, I was not overly surprised to see some of the slides in yesterday's entry. The post featured the slides of the second edition of the International Workshop on Compressed Sensing applied to Radar (CoSeRa 2013), where few presentations tell a story as to why Radar is not directly applicable to actual data. The reason the work of Yonina and collaborators is more successful is because of their focus on the analog based sensing. In MRI, the most advanced field using some of the concepts of compressive sensing, there is a focus on less than random sampling especially for low frequency items, there is not a well theoretically understood reason as to why  This is not exactly true, some of these reasons are featured in the following paper (previous coverage and Q&A with some of the authors is also listed below). On a personal level, my mind races through these arguments and wonder how we should change some of the phase transitions we have been accustomed to use and ultimately how sensors should be changed as a result.

Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum by Ben Adcock, Anders Hansen, Bogdan Roman, Gerd Teschke
The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinite-dimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via so-called generalized sampling (GS). Second, the extension of generalized sampling to inverse and ill-posed problems. And third, the combination of generalized sampling with sparse recovery techniques. This final contribution leads to a theory and set of methods for infinite-dimensional compressed sensing, or as we shall also refer to it, compressed sensing over the continuum.


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