Distortion analysis on discrete Laplacian operators by introducing random images

The expected distortions of the discrete Laplacian operators by random images, whose operators play important roles in image processing and diffusion equations, are investigated on a hexagonal grid and a squared grid. By introducing the continuous two-dimensional random images, whose images are approximated by polynomials with finite degrees, those distortions are evaluated by taking the expectations for squared distortions. We can obtain the expected distortions with simple expressions, and the obtained results offer a guide for the evaluation of discrete Laplacian operators. Moreover, based on the results of discrete two-dimensional Laplacian operators, the distortion of a discrete three-dimensional Laplacian operator is investigated.