Directional Hypercomplex Diffusion

Overview & Synopsis

Methods based on partial differential equations (PDE) become in- creasingly one of the methods of image processing. Recently a dif- fusion method is appeared, it allows to generalize the diffusion to the complex domain by the injection of a complex number in the heat equation. For small phase angles, the linear process generates the Gaussian and Laplacian pyramids (scale-spaces) simultaneously, depicted in the real and imaginary parts, respectively. The imaginary value serves as a robust edge-detector with increasing confidence in time, thus handles noise well and may serve as a controller for non- linear processes. In this article we propose to extend this concept by introducing a notion of directionality in such a way as each equation of the system will correspond to a specific direction. It is in our in- terests to use higher order algebra to adapt the process to the four discrete directions. Then we will focus on the imaginary parts for developing a nonlinear scheme.