An Adaptive Finite Difference Method for Total Variation Minimization

In this paper, we propose an adaptive finite difference scheme in order to numerically solve total variation type problems for image processing tasks. The automatic generation of the grid relies on indicators derived from a local estimation of the primal-dual gap error. This process leads in general to a non-uniform grid for which we introduce an adjusted finite difference method. Further we quantify the impact of the grid refinement on the respective discrete total variation. In particular, it turns out that a finer discretization may lead to a higher value of the discrete total variation for a given function. To compute a numerical solution on non-uniform grids we derive a semi-smooth Newton algorithm in 2D for scalar and vector-valued total variation minimization. We present numerical experiments for image denoising and the estimation of motion in image sequences to demonstrate the applicability of our adaptive scheme.