Edge preserving image denoising with a closed form solution

This paper addresses the problem of image denoising which is still a valid challenge at the crossing of functional analysis and statistics. We herein propose a novel pixel-based algorithm, which formulates the image denoising problem as the maximum a posterior (MAP) estimation problem using Markov random fields (MRFs). Such an MAP estimation problem is equivalent to a maximum likelihood (ML) estimation constrained on spatial homogeneity and is NP-hard in discrete domain. To make it tractable, we convert it to a continuous label assignment problem based on a Gaussian MRF model and then obtain a closed form globally optimal solution. Since the Gaussian MRFs tend to over-smooth images and blur edges, our algorithm incorporates the pre-estimated image edge information into the energy function construction and therefore better preserves the image structures. In the algorithm, patch similarity based pairwise interaction is also involved to better preserve image details and make the algorithm more robust to noise. Based on the theoretical analysis on the deviation caused by the discretization from obtained continuous global optimum to discrete output, we demonstrate the guaranteed optimal property of our algorithm. Both quantitative and qualitative comparative experimental results are given to demonstrate the better performance of our algorithm over several existing state-of-the-art related algorithms.