Thursday, December 16, 2010

CS: Sinc approximations ?, RHESSI, Theory and Applications of Compressive Sensing and more.

As I was reading the latest NA-Digest newletter, I came across this announcement for a Handbook of Sinc Numerical Methods by Frank Stenger.. From the Summary of Sinc Numerical Methods by the same author,
Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers. This article summarizes results obtained to date, on Sinc numerical methods of computation.  Sinc methods provide procedures for function approximation over bounded or unbounded  regions, encompassing interpolation, approximation of derivatives, approximate definite and indefinite integration, solving initial value ordinary differential equation problems, approxi­mation and inversion of Fourier and Laplace transforms, approximation of Hilbert transforms, and approximation of indefinite convolutions, the approximate solution of partial differen­tial equations, and the approximate solution of integral equations, methods for constructing conformal maps, and methods for analytic continuation. Indeed, Sinc are ubiquitous for  approximating every operation of calculus.

Let me wonder aloud the following: isn't the approximation of a compactly supported function with a series of sinc, a little bit like projecting that function on a set of incoherent functions. At the very least, the approximation capabilities of sincs are exponential (in the right space), couldn't they be used in some dictionaries ?

I just found the following abstract of a presentation entitled: A compressive sensing approach to image reconstruction in X-ray solar astronomy by Silvia Allavena. The abstract reads:
The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) is a small satellite launched by NASA to produce images of the most explosive phenomena in the solar atmosphere named solar flares at hard X-ray energies. This instrument uses indirect collimator-based imaging techniques, the native output of which is in the form of visibilities, i.e. two dimensional spatial Fourier components of the incoming radiation. Since the sampling of the data in the frequency plane is very sparse, we apply an iterative algorithm (OMP, Orthogonal Matching Pursuit) belonging to the family of compressive sensing methods, in order to reconstruct X-ray images of solar flares. This presentation focuses on the application of this algorithm to both synthetic and real visibilities measured by RHESSI, with particular interest to the fidelity, accuracy and robustness of the method.
If you recall RHESSI is a coded aperture camera for X-ray, so it is a natural candidate for reconstruction using compressive sensing solvers as it is a compressive sensing system in the first place.. I look forward to the result of this study.

Finally, Atul Divekar's thesis entitled . Theory and Applications of Compressive Sensing is out. The abstarct reads:
This thesis develops algorithms and applications for compressive sensing, a topic in signal processing that allows reconstruction of a signal from a limited number of linear combinations of the signal. New algorithms are described for common remote sensing problems including superresolution and fusion of images. The algorithms show superior results in comparison with conventional methods. We describe a method that uses compressive sensing to reduce the size of image databases used for content based image retrieval. The thesis also describes an improved estimator that enhances the performance of Matching Pursuit type algorithms, several variants of which have been developed for compressive sensing recovery.
Congratulations Atul!

There was recently a course in Hong-Kong entitled Adaptive Data Analysis via Nonlinear Compressed Sensing taught by Prof Thomas Hou, Dr. Zuoqiang Shi.

Ialso found a project called Compressed Sensing Analog to Information Converter but I don;t have much information on it.

Credit: NASA.

1 comment:

Anonymous said...

An interesting point you mention about the sinc. Using traditional sampling that follows Shanon's approach it is proven that sinc interpolator allows a perfect reconsturction your signal. Can we link this to CS acquisition and incoherence terms. A framework establishing the link between the two approaches will be nice to find