A discrete approximation of Blake and Zisserman energy in image denoising with optimal choice of regularization parameters

Abstract

We introduce and study a multi‐scale approximation of a nonlinear elliptic functional introduced by Bake and Zisserman (BZ) for solving image restoration problems. The functional is a fourth‐order PDE variational model that aims to locate geometric features that are singularities of first kind and second kind in the image. The method consists in introducing a family of linear discrete energies approximating BZ functional in the sense of the Γ‐convergence. These functionals are built from an adaptive finite element method that reflects how the corresponding high‐order diffusion operators act on image restoration with preserving contours and corners. The initial nonlinear functional includes low‐dimensional measures (Hausdorff) of the singular sets, whereas its linear discrete counterpart keep a simple structure of the underlying PDEs system easy to solve. The resulting approach allows us to capture that singularities of the image and its gradient better than the second‐order methods. We perform the analysis of the proposed method in the framework of the Γ‐convergence, and we derive a new algorithm to solve the minimization of the BZ functional. We consider and implement the algorithm and a variant based on the alternating direction method of multipliers (ADMM) to enhance the convergence. We give some numerical results in agreement with the theoretical analysis to show the performances of the method.