CS: Literature Review on Sparse Optimization

After having made available a detailed explanation of Antonin Chambolle's algorithm for the resolution of compressed sensing with TV regularization, Gabriel Peyre has written a much welcomed literature review on some of the iterative algorithms used with L1/TV regularization in a paper entitled: Literature Review on Sparse Optimization. The abstract reads:

Some thoughts about algorithms for minimizing l1 and TV norms for signal and image recovery. In particular, I put the emphasis on the connections between Lagrangian and l2 constrained optimizations, and between analysis and synthesis priors.

This a nice addition to guide some of us in the "new" techniques that are showing up (they were only three directly downloadable a year and half ago). One important fact is that this recent surge in new methods has not kept up with an actual quantification of the speed of convergence on known examples or benchmarks (see end of list for images) as featured in the Sparco easy-to-use-and-solver-agnostic framework. More importantly the connection between these solvers and specific CS measurement matrices is still ripe for further investigation. In Compressive Sensing (CS), we are not only interested in the convergence speed (which is of interest to numerical analysts looking at these schemes) but we are also interested in the minimum number of compressed measurements required to recover the original signal/image. Maybe some enterprising student would want to duplicate for images something like what David Mary did in this post ( A Comparison of the Reconstruction Capability of CoSaMP, OMP, Subspace Pursuit and Reweighted Lp) for 1-d signals and help the community out ? On top of being a low hanging fruit, I will make sure it gets advertized here and it should receive some attention.


Images Credit: NASA/JPL-Caltech/University of Arizona/Texas A&M, Image taken by the SSI instrument on the Phoenix lander on Sol 87.