CS: Q/A, Espace Vide, Recovering Signals from Lowpass Data, Sampling with Quasicrystals, a workshop

Here is what I received in my mailbox yesterday:

Dear Dr. Carron,

I have been closely following your webpages on Compressed Sensing....

I have tried the L1-magic toolkit for CS reconstruction of images using Total Variation Quadratic Constraints(TVQC) as this reconstruction is typically better compared other techniques. The toolbox uses Scrambled Fourier matrices for measurements and it is non-trivial to replace the measurement matrices with something else like wavelet etc. I would like to know if you could recommend me of a toolbox other than L1-magic that has solvers like TVQC and could easily be extended to include different measurement matrices.

Also, I would like to know if you could share some code that could be used to extract joint sparsity between two correlated signals measured using Distributed Compressed Sensing framework.

Here is what I responded:

I am glad you find the pages and the blog useful. With regards to TV techniques, there are indeed other solvers using TV and most of them show up when doing a simple search on the blog

http://nuit-blanche.blogspot.com/search?q=TV

with regards to your difficulty in getting to use a different measurement matrix, I think you ought to look at the SPARCO Toolbox and specifically use the very handy measurement matrix framework used there. In particular, you may want to reuse the scripts of the problems 501 and above:

http://www.cs.ubc.ca/labs/scl/sparco/index.php/Main/Problems

All the codes I am listing are on the blog or in the attendant more static pages (the Big Picture in Compressive Sensing or the Local Codes).

While SPARCO is good ground for benchmarks, it also is a good guide for newcomers to see how some of the measurement matrices are set up without having to reinvent the wheel,


Eric Tramel just created a blog called Espace Vide. Entries related to Compressive Sensing are here at:

http://espacevide.net/articles/tag/compressed-sensing

The title translates from French into void space but it already has two entries on CS:

• Optimied Dictionaries
• The starting point

It does not look void to me :-) Welcome to the blogosphere Eric!

Laurent Duval pointed out to me the following two papers of interest:

Recovering Signals from Lowpass Data by Yonina Eldar and Pohl Volker. The abstract reads:

The problem of recovering a signal from its low frequency components occurs often in practical applications due to the lowpass behavior of many physical systems. Here we study in detail conditions under which a signal can be determined from its low-frequency content. We focus on signals in shift-invariant spaces generated by multiple generators. For these signals, we derive necessary conditions on the cutoff frequency of the lowpass filter as well as necessary and sufficient conditions on the generators such that signal recovery is possible. When the lowpass content is not sufficient to determine the signal, we propose appropriate pre-processing that can improve the reconstruction ability. In particular, we show that modulating the signal with one or more mixing functions prior to lowpass filtering, can ensure the recovery of the signal in many cases, and reduces the necessary bandwidth of the filter.

Image Sampling with Quasicrystals by Mark Grundland, Jiri Patera, Zuzana Masakova, Neil A. Dodgson. The abstract reads:

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.

We talked about quasicrystals before. Howver in this paper the quasicrystral sampling is performed in the image space whereas in Yves Meyer's work ( Compressed sensing and transference and A variant of compressed sensing ) the sampling is performed in the Fourier space and the reconstruction is performed using a Linear Programming step.


Ron DeVore, Massimo Fornasier, Holger Rauhut are organizing a workshop entitled Workshop Sparsity and Computation at the Hausdorff Center for Mathematics Bonn on June 7-11, 2010. The website:

http://www.hausdorff-center.uni-bonn.de/event/2010/sparsity_and_computation/