Morphological operators for images valued on the sphere

The lack of a natural ordering on the sphere presents an inherent problem when defining morphological operators extended to unit sphere. We analyze here the notion of averaging over the unit sphere to obtain a local origin which can used to formulate ordering based operators. The notion of local supremum and infimum is introduced, which allows to define the dilation and erosion for images valued on the sphere. The algorithms are illustrated using polarimetric images.