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We develop a theory for image restoration with a learned regularizer that isanalogous to that of Meyer's characterization of solutions of the classicalvariational method of Rudin-Osher-Fatemi (ROF). The learned regularizer we useis a Kantorovich potential for an optimal transport problem of mapping adistribution of noisy images onto clean ones, as first proposed by Lunz,\"Oktem and Sch\"onlieb. We show that the effect of their restoration method onthe distribution of the images is an explicit Euler discretization of agradient flow on probability space, while our variational problem, dubbedWasserstein ROF (WROF), is the corresponding implicit discretization. We obtainour geometric characterisation of the solution in this setting by first provinga more general convex analysis theorem for variational problems with solutionscharacterised by projections. We then use optimal transport arguments to obtainour WROF theorem from this general result, as well as a decomposition of atransport map into large scale "features" and small scale "details", wherescale refers to the magnitude of the transport distance. Further, we leverageour theory to analyze two algorithms which iterate WROF. We refer to these asiterative regularization and multiscale transport. For the former we proveconvergence to the clean data. For the latter we produce successiveapproximations to the target distribution that match it up to finer and finerscales. These algorithms are in complete analogy to well-known effectivemethods based on ROF for iterative denoising, respectively hierarchical imagedecomposition. We also obtain an analogue of the Tadmor Nezzar Vese energyidentity which decomposes the Wasserstein 2 distance between two measures intoa sum of non-negative terms that correspond to transport costs at differentscales.

An optimal transport analogue of the Rudin Osher Fatemi model and its corresponding multiscale theory