The authors of this paper clearly define their problem: to minimize the functional ∫_&OHgr;f(u, v)dx+&bgr; ∫_&OHgr;|▿u|dx, where v is an image on &OHgr;, f is a convex function of u (the restored image) like f (u, v)=| u (x)- v (x)|^p, p=1,2, and &bgr; is a positive constant.

The authors refer to related works, including one presenting an exact solution in one dimension and fast algorithms for approximation; compare their algorithm with the others from various points of view (amount of memory needed, numerical results, complexity); and summarize their contributions--a fast algorithm that computes an exact minimizer of the above functional by reformulating the problem into an independent binary Markov random field (MRF) attached to each level set of the image. Exact minimization is performed thanks to a minimum cost cut algorithm. They also prove that minimizing the model L1 + total variation yields a contrast invariant and self dual filter. Section 5 is devoted to numerical results. In section 6, the authors mention what the next part of their work will contain: an extension of the proposed approach of energy decomposition on the level sets to a more general class of energies, and proof that the case of the total variation is indeed a particular case.

This is indeed a very nice applied mathematical work. It would have been accessible to a wider audience if the facts described by formulas had been presented in a more intuitive way.