Image enhancement using fourth order partial differential equations

Abstract

A class of fourth order partial differential equations (PDE) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a functional which is an increasing function of the absolute value of the Laplacian of the image intensity function, hence it generates a family of images of increasing degree of smoothness. Since the Laplacian of a plane image is zero, the stationary points of this functional or this class of PDEs are images whose intensity functions are a union of plane images of various boundaries. This kind of images look more natural than step images which are the stationary points of anisotropic diffusion (second order PDEs), so the proposed PDEs are able to achieve comparable degree of noise removal while avoiding the blocky effects widely seen in images processed by anisotropic diffusion. However, the proposed fourth order PDEs tend to develop speckle artifacts which may be characterized as isolated white and/or black dots, but they can be easily alleviated by simple despeckle algorithms such as the one shown in this paper.