Complex Diffusion Processes for Image Filtering

A framework that naturally unifies smoothing and enhancement processes is presented. We generalize the linear and nonlinear scale spaces in the complex domain, by combining the diffusion equation with the simplified Schrödinger equation. A fundamental solution for the linear case is developed. Preliminary analysis of the complex diffusion shows that the generalized diffusion has properties of both forward and inverse diffusion. An important observation, supported theoretically and numerically, is that the imaginary part can be regarded as an edge detector (smoothed second derivative), after rescaling by time, when the complex diffusion coefficient approaches the real axis. Based on this observation, a nonlinear complex process for ramp preserving denoising is developed.