Wednesday, December 24, 2014

How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT

Jakob, a map maker, just sent me the following:
Dear Igor,

Thank you for keeping the Nuit Blanche an important CS resource that I have been following for several years now.

I thought you and some readers might be interested in an update on our work on phase transitions in computed tomography that you recently posted about:
That paper has been published online (with open access) by Inverse Problems in Science and Engineering:

Furthermore, we have just put on arXiv our latest results:
How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT by Jakob S. Jørgensen and Emil Y. Sidky

where we compare the empirically observed phase transitions in CT with the theoretical results for Gaussian matrices i.e. the Donoho-Tanner phase transition as well as the results by Amelunxen, Lotz, McCoy and Tropp:
Surprisingly, at least to us, we find essentially the same phase-transition behavior in CT as in Gaussian case!

Further we study the use of randomized sampling strategies in CT and give preliminary results on how well the phase transitions can predict the sufficient number of samples to acquire in a large-scale CT setting depending on the sparsity.

Happy holidays and best wishes,

Jakob Sauer Jørgensen

Postdoc Scientific Computing
Technical University of Denmark Department of Applied Mathematics and Computer Science Matematiktorvet Building 303 B
2800 Kgs. Lyngby
Thanks Jakob ! Here is the paper: How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT by Jakob S. Jørgensen, Emil Y. Sidky

We introduce phase-diagram analysis, a standard tool in compressed sensing, to the X-ray CT community as a systematic method for determining how few projections suffice for accurate sparsity-regularized reconstruction. In compressed sensing a phase diagram is a convenient way to study and express certain theoretical relations between sparsity and sufficient sampling. We adapt phase-diagram analysis for empirical use in X-ray CT for which the same theoretical results do not hold. We demonstrate in three case studies the potential of phase-diagram analysis for providing quantitative answers to questions of undersampling: First we demonstrate that there are cases where X-ray CT empirically performs comparable with an optimal compressed sensing strategy, namely taking measurements with Gaussian sensing matrices. Second, we show that, in contrast to what might have been anticipated, taking randomized CT measurements does not lead to improved performance compared to standard structured sampling patterns. Finally, we show preliminary results of how well phase-diagram analysis can predict the sufficient number of projections for accurately reconstructing a large-scale image of a given sparsity by means of total-variation regularization.  
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