Abstract:

The Rudin-Osher-Fatemi (ROF) denoising model, a cornerstone of signal and image processing, consists in minimizing the total variation seminorm bal­anced by the squared L² distance to the datum. Its l¹-anisotropic version is a natural choice for images with an underlying rectilinear geometry. Moreover, it can be particularly well approximated by piecewise polynomials on recti­linear grids. The first result we will examine more closely in this talk states that the orthogonal projection onto the space of piecewise constant functions maps subgradients of the anisotropic total variation to subgradients. This inconspicuous fact has some remarkable implications. First, it implies that the minimizer of the anisotropic ROF model is piecewise constant on a rectilinear grid, whenever the datum is. Second, the orthogonal projection is total varia­tion diminishing, from which one obtains a convergence rate of order up to 1/2 for the piecewise constant approximations. Third, a certain property of the approximate solutions which we call universal minimality (meaning that they simultaneously minimize a large class of convex functionals, including all p-norms, over a certain neighbourhood of the datum) carries over to the contin­uous setting.