Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions

In this work we address the topic of image processing using an atomic function (AF) in a representation of quaternionic algebra. Our approach is based on the most important AF, the up(x)function. The main reason to use the atomic function up(x) is that this function can express analytically multiple operations commonly used in image processing such as low-pass filtering, derivatives, local phase, and multiscale and steering filters. Therefore, the modelling process in low level-processing becomes easy using this function. The quaternionic algebra can be used in image analysis because lines (even), edges (odd) and the symmetry of some geometric objects in R-2 are enhanced. The applications show an example of how up(x) can be applied in some basic operations in image processing and for quaternionic phase computation.