Summary: 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 Y. Meyer [Oscillating patterns in image processing and nonlinear evolution equations. The fifteenth Dean Jacqueline B. Lewis memorial lectures. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0987.35003)], we propose to model the oscillating component v by a function of the space G of oscillating functions, which is, in some sense, the dual space of \(BV(\Omega)\). To overcome the issue related to the definition of the G-norm, we introduce auxiliary variables that naturally emerge from the Helmholtz-Hodge decomposition for smooth fields, which yields to the minimization of the \(L^\infty\)-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.