Abstract

Edge Preservation in Nonlinear Diffusion Filtering

Mohammad Reza Hajiaboli, Ph.D.
Concordia University, 2012


Image denoising techniques, in which the process of noise diffusion is modeled as a nonlinear partial differential equation, are well known for providing low-complexity solutions to the denoising problem with a minimal amount of image artifacts. In discrete settings of these nonlinear models, the objective of providing a good noise removal while preserving the image edges is heavily dependent on the kernels, diffusion functions and the associated contrast parameters employed by these nonlinear diffusion techniques. This thesis makes an in-depth study of the roles of the kernels and contrast parameters with a view to providing an effective solution to the problem of denoising of the images contaminated with stationary and signal-dependent noise.

Within the above unified theme, this thesis has two major parts. In the first part of this study, the impact of anisotropic behavior of the Laplacian operator on the capabilities of nonlinear diffusion filters in preserving the image edges in different orientations is investigated. Based on this study, an analytical scheme is devised to obtain a spatially-varying kernel that adapt itself to the diffusivity function. The proposed edge-adaptive Laplacian kernel is then incorporated into various nonlinear diffusion filters for denoising of images contaminated by additive white Gaussian noise.

The performance optimality of the the existing nonlinear diffusion techniques is generally based on the assumption that the noise and signal are uncorrelated. However, in many applications, such as in medical imaging systems and in remote sensing where the images are degraded by Poisson noise, this assumption is not valid. As such, in the second part of the thesis, a study is undertaken for denoising of images contaminated by Poisson noise within the framework of the Perona-Malik nonlinear diffusion filter. Specifically, starting from a Skellam distribution model of the gradient of the Poisson-noise corrupted images and following the diffusion mechanism of the nonlinear filter, a spatially and temporally varying contrast parameter is designed. It is shown that the nonlinear diffusion filters employing the new Laplacian kernel supports the extremum principle and that the proposed contrast parameter satisfies the sufficient conditions for observance of the scale-space properties.

Extensive experiments are performed throughout the thesis to demonstrate the effectiveness and validity of the various schemes and techniques developed in this investigation. The simulation results of applying the new Laplacian kernel to a number of nonlinear diffusion filters show its distinctive advantages over the conventional Rosenfeld and Kak kernel, in terms of the filters' noise reduction and edge preservation capabilities for images corrupted by additive white Gaussian noise. The simulation results of incorporating the proposed spatially- and temporally-varying contrast parameter into the Perona-Malik nonlinear diffusion filter demonstrate a performance much superior to that provided by some of the other state-of-the-art techniques in denoising images corrupted by Poisson noise.