Split Bregman iteration and infinity Laplacian for image decomposition

In this paper, we address the issue of decomposing a given real-textured image into a cartoon/geometric part and an oscillatory/texture part. The cartoon component is modeled by a function of bounded variation, while, motivated by the works of Meyer [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22 of University Lecture Series, AMS, 2001], we propose to model the oscillating component vv by a function of the space GG of oscillating functions, which is, in some sense, the dual space of BV(Ω)BV(Ω). To overcome the issue related to the definition of the GG-norm, we introduce auxiliary variables that naturally emerge from the Helmholtz–Hodge decomposition for smooth fields, which yields to the minimization of the L∞L∞-norm of the gradients of the new unknowns. This constrained minimization problem is transformed into a series of unconstrained problems by means of Bregman Iteration. We prove the existence of minimizers for the involved subproblems. Then a gradient descent method is selected to solve each subproblem, becoming related, in the case of the auxiliary functions, to the infinity Laplacian. Existence/Uniqueness as well as regularity results of the viscosity solutions of the PDE introduced are proved.