CS: A talk, BCS of a Highly Impulsive Signal in Heavy-Tailed Noise, Subspace correction methods for TV and l1-minimization, Domain decomposition CS

Svetlana Avramov-Zamurovic has a new video introducing Compressive Sensing but it is in Serbian. The slides are here whereas the video is here (after 30 minutes into the video the discussion goes into CS). I'll add it to the video section soon. I may even start a page dedicated to introduction to Compressive Sensing in languages other than English if there is enough of them. Since I don't speak many languages I would really appreciate it, if you record such video or know of such recording to let me know where it is located. Thanks in advance.

here are three new papers:

Bayesian Compressed Sensing of a Highly Impulsive Signal in Heavy-Tailed Noise using a Multivariate Cauchy Prior, by George Tzagkarakis and Panagiotis Tsakalides. The abstract reads:

Recent studies reveal that if a signal is highly compressible in some orthonormal basis, then an accurate reconstruction can be obtained from random projections using a very small subset of the projection coefficients, and thus, reducing the complexity of the sensing system. A Bayesian framework was introduced recently with respect to the reconstruction of the original (noisy) signal, providing some advantages when compared with reconstruction methods, employing norm-based constrained minimization approaches. These Bayesian methods were designed by using mixtures of Gaussians to approximate the sparsity of the prior distribution of the projection coefficients. However, there are cases in which a signal exhibits a highly impulsive behavior, and thus, resulting in an even sparser coefficient vector. In this paper, we develop a Bayesian approach for estimating the original signal based on a set of compressed-sensing measurements corrupted by heavy-tailed noise. The prior belief that the vector of projection coefficients should be sparse is enforced by fitting its prior distribution by means of a heavy-tailed multivariate Cauchy distribution. The experimental results show that our proposed method achieves an improved reconstruction performance, in terms of a smaller reconstruction error, while increasing the sparsity using less basis functions, compared with the recently introduced Gaussian-based Bayesian implementation.

Subspace correction methods for total variation and l1-minimization by Carola-Bibiane Schoenlieb and Massimo Fornasier. The abstract reads:

This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via oblique thresholding, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for singular elliptic PDEs arising in total variation minimization and in accelerated sparse recovery algorithms based on ℓ1-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing

Matlab software and simulations are available here. A parallel implementation is also available.

Domain decomposition methods for compressed sensing by Massimo Fornasier , Andreas Langer, Carola-Bibiane Schoenlieb. The abstract reads:

We present several domain decomposition algorithms for sequential and parallel minimization of functionals formed by a discrepancy term with respect to data and total variation constraints. The convergence properties of the algorithms are analyzed. We provide several numerical experiments, showing the successful application of the algorithms for the restoration 1D and 2D signals in interpolation/inpainting problems respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.

Massimo Fornasier will also give a talk in University of Vienna today.



Image Credit: NASA/JPL/Space Science Institute, Saturn as taken on May 11, 2009.