Mathematical morphology on directional data

We define morphological operators and filters for directional images whose pixel values are vectors on the unit sphere. This requires an ordering relation for unit vectors which is obtained by using depth functions. They provide a centre-outward ordering with respect to a specified centre vector. We apply our operators on synthetic directional images and compare them with classical morphological operators for grey-scale images. As application example, we enhance the fault region in a compressed glass foam.