@misc{KleefeldBurgeth, author = {Kleefeld, Andreas and Burgeth, Bernhard}, title = {An approach to color-morphology based on Einstein addition and Loewner order}, series = {Pattern Recognition Letters}, volume = {47}, journal = {Pattern Recognition Letters}, issn = {0167-8655}, doi = {10.1016/j.patrec.2014.01.018}, pages = {29 -- 39}, abstract = {In this article, a new method to process color images via mathematical morphology is presented. Precisely, each pixel of an image contained in the rgb-space is converted into a 2 x 2-matrix representing a point in a color bi-cone. The supremum and infimum needed for dilation and erosion, respectively, are calculated with respect to the Loewner order. Since the result can be outside the bi-cone, a map is proposed to ensure the algebraic closure in the bi-cone. A new addition and subtraction based on Einstein's Relativity Theory is used to define morphological operation such as top-hats, gradients, and morphological Laplacian. Since the addition and subtraction is defined in the unit ball, a suitable mapping between those two spaces is constructed. A comparison with the component-wise approach and the full ordering using lexicographical cascades approach demonstrate the feasibility and capabilities of the proposed approach.}, language = {en} } @misc{KahraBreussKleefeldetal., author = {Kahra, Marvin and Breuß, Michael and Kleefeld, Andreas and Welk, Martin}, title = {Matrix-valued LogSumExp approximation for colour morphology}, series = {Journal of mathematical imaging and vision}, volume = {67}, journal = {Journal of mathematical imaging and vision}, number = {5}, publisher = {Springer US}, address = {New York}, issn = {0924-9907}, doi = {10.1007/s10851-025-01267-5}, abstract = {Mathematical morphology is a part of image processing that employs a moving window to modify pixel values through the application of specific operations. The supremum and infimum are pivotal concepts, yet defining them in a general sense for high-dimensional data such as colour is a challenging endeavour. As a result, a number of different approaches have been taken to try to find a solution, with certain compromises being made along the way. In this paper, we present an analysis of a novel approach that replaces the supremum within a morphological operation with the LogExp approximation of the maximum for matrix-valued colours. This approach has the advantage of extending the associativity of dilation from the one-dimensional to the higher-dimensional case. Furthermore, the minimality property is investigated and a relaxation specified to ensure that the approach is continuously dependent on the input data.}, language = {en} }