Directional hypercomplex diffusion

Methods based on partial differential equations (PDE) become increasingly one of the methods of image processing. Recently a diffusion 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 nonlinear 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 interests 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.