Analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing animage into cartoon plus texture

We study the Osher-Solé-Vese model [11], which is the gradient flow of an energy consisting of the total variation functional plus an $H^{-1}$ fidelity term. A variational inequality weak formulation for this problem is proposed along the lines of that of Feng and Prohl [7] for the Rudin-Osher-Fatemi model [12]. A regularized energy is considered, and the minimization problems corresponding to both the original and regularized energies are shown to be well-posed. The Galerkin method of Lions [9] is used to prove the well-posedness of the weak problem corresponding to the regularized energy. By letting the regularization parameter $\epsilon$ tend to $0$, we recover the well-posedness of the weak problem corresponding to the original energy. Further, we show that for both energies the solution of the weak problem tends to the minimizer of the energy as $t \to \infty$. Finally, we find the rate of convergence of the weak solution of the regularized problem to that of the original one as $\epsilon \downarrow 0$.