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.
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Burgeth, B., Papenberg, N., Bruhn, A., Welk, M., Feddern, C., Weickert, J. (2005). Morphology for Higher-Dimensional Tensor Data Via Loewner Ordering. In: Ronse, C., Najman, L., Decencière, E. (eds) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol 30. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3443-1_37
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DOI: https://doi.org/10.1007/1-4020-3443-1_37
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3442-8
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