Morphology for Higher-Dimensional Tensor Data Via Loewner Ordering

Bernhard Burgeth, Nils Papenberg, Andrés Bruhn, Martin Welk, Christian Feddern, Joachim Weickert

Abstract: The operators of greyscale morphology rely on the notions of maximum and minimum which regrettably are not directly available for tensor-valued data since the straightforward component-wise approach fails. This paper aims at the extension of the maximum and minimum operations to the tensor-valued setting by employing the Loewner ordering for symmetric matrices. This prepares the ground for matrix-valued analogs of the basic morphological operations. The novel definitions of maximal/minimal matrices are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DT-MRI data. Furthermore, they depend continuously on the input data which makes them viable for the design of morphological derivatives such as the Beucher gradient or a morphological Laplacian. Experiments on DT-MRI images illustrate the properties and performance of our morphological operators.