Complex Di usion Processes for Image Filtering

A framework that naturally uni es smoothing and enhancement processes is presented. We generalize the linear and nonlinear scale spaces in the complex domain, by combining the di usion equation with the simpli ed schrodinger equation. A fundamental solution for the linear case is developed. Preliminary analysis of the complex di usion shows that the generalized di usion has properties of both forward and inverse di usion. 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 di usion coeÆcient approaches the real axis. Based on this observation, a nonlinear complex process for ramp preserving denoising is developed.