Morphological Operations on Matrix-Valued Images

The output of modern imaging techniques such as diffusion tensor MRI or the physical measurement of anisotropic behaviour in materials such as the stress-tensor consists of tensor-valued data. Hence adequate image processing methods for shape analysis, skeletonisation, denoising and segmentation are in demand. The goal of this paper is to extend the morphological operations of dilation, erosion, opening and closing to the matrix-valued setting. We show that naive approaches such as componentwise application of scalar morphological operations are unsatisfactory, since they violate elementary requirements such as invariance under rotation. This lead us to study an analytic and a geometric alternative which are rotation invariant. Both methods introduce novel non-component-wise definitions of a supremum and an infimum of a finite set of matrices. The resulting morphological operations incorporate information from all matrix channels simultaneously and preserve positive definiteness of the matrix field. Their properties and their performance are illustrated by experiments on diffusion tensor MRI data.