|Ph.D Thesis||Department of Electrical Engineering|
|Supervisors:||Dr. Sochen Nir|
|Prof. Emeritus Zeevi Yehoshua|
Sharpening and denoising are contradictory requirements in image enhancement. It is shown how they can be reconciled by local adaptive mechanisms. For this purpose, two new nonlinear diffusion-type partial differential equations are proposed. The first process is based on a nonlinear diffusion coefficient that is locally adjusted according to image features such as edges, textures and moments. This adaptive diffusion flow enhances features while locally denoising smoother segments of the signal or image. The process is robust and insensitive to noise, and fits within a very general image degradation model. The method is further generalized for color processing via the Beltrami flow, by adaptively modifying the structure tensor that controls the non-linear diffusion process.
The second process is a generalization to the complex domain of the linear and nonlinear scale spaces, generated by the inherently real-valued diffusion equation. This is accomplished by incorporating the free Schrodinger equation. A fundamental solution for the linear case of the complex diffusion equation is developed for the first time. We prove that the imaginary part is a smoothed second derivative, scaled by time, when the complex diffusion coefficient approaches the real axis. Based on this observation, we develop two examples of nonlinear complex processes, useful in image processing: a regularized shock filter for image enhancement and a ramp preserving denoising process. Qualitative and quantitative experiments indicate that the new complex-valued filters outperform top state-of-the-art real-valued ones.
Finally, we suggest an improvement to current PDE-based denoising processes, and show how textures can be preserved by a relatively simple modification in which spatially varying power constraints are imposed instead of global ones.