Nonlinear regularization using constrained edges in image reconstruction

This paper deals with edge-preserving regularization for image reconstruction. We use a non-quadratic regularization term involving a φ-function applied on the intensity gradient modulus. During the process, small gradients are smoothed while high gradients are preserved. In order to take into account the noise more specifically, we propose to use the explicit version of the regularization term involving the edge variable. We add a nonlinear constraint on this edge variable, in order to remove the noise. It allows edge enhancement while smoothing the noise, even in the case where the edge and noise generate the same high gradient modulus. We have previously proposed a model composed of two coupled partial differential equations (PDE) on the image intensity and image edges. In this paper, we show that, for a particular regularization intensity function, the two coupled PDE can be slightly modified in order to correspond to the Euler equations associated with the minimization of a global criterion. This new criterion contains a nonlinear regularization term on both the intensity and the edges. We use convergence towards Mumford and Shah (1989) functional to improve our results