Mathematical morphology for matrix fields induced by the Loewner ordering in higher dimensions

Positive semidefinite matrix fields are becoming increasingly important in digital imaging. One reason for this tendency is the introduction of diffusion tensor magnetic resonance imaging (DT-MRI). In order to perform shape analysis, enhancement or segmentation of such tensor/matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the matrix-valued setting. We start by presenting novel definitions for the supremum and infimum of a set of matrices since these notions lie at the heart of the morphological operations. In contrast to naive approaches like the component-wise maximum or minimum of the matrix channels, our approach is based on the Loewner ordering for symmetric matrices. The notions of supremum and infimum deduced from this partial ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. We introduce erosion, dilation, opening, closing, top hats, morphological derivatives, shock filters, and mid-range filters for positive semidefinite matrix-valued images. These morphological operations incorporate information simultaneously from all matrix channels rather than treating them independently. Experiments on DT-MRI images with ball- and rod-shaped structuring elements illustrate the properties and performance of our morphological operators for matrix-valued data.