Image Edge Respecting Denoising with Edge Denoising by a Designer Nonisotropic Structure Tensor Method

We consider image denoising as the problem of removing spurious oscillations due to noise while preserving edges in the images. We will suggest here how to directly make infinitesimal adjustment to standard variational methods of image denoising, to enhance desirable target assumption of the noiseless image. The standard regularization method is used to define a suitable energy functional to penalize the data fidelity and the smoothness of the solution. This energy functional is tailored so that the region with small gradient is isotropically smoothed whereas in a neighborhood of an edge presented by a large gradient smoothing is allowed only along the edge contour. The regularized solution that arises in this fashion is then the solution of a variational principle. To this end the associated Euler — Lagrange equation needs to be solved numerically and the half-quadratic minimization is generally used to linearize the equation and to derive an iterative scheme. We describe here a method to modify Euler — Largrange equation from commonly used energy functionals, in a way to enhance certain desirable preconceived assumptions of the image, such as edge preservation. From an algorithmic point of view, we may deem this algorithm as a smoothing by a local average with an adaptive gradient-based weight. However, this algorithm may result in noisy edges although the edge is preserved and noise is suppressed in the low-gradient regions of the image. The main focus here is to present an edge-preserving regularization in the aforementioned view point, and to provide an alternative and simple way to modify the existing algorithm to mitigate the phenomena of noisy edges without explicitly defining step where we specify an energy functional to be minimized.