@incollection{KleefeldBurgeth, author = {Kleefeld, Andreas and Burgeth, Bernhard}, title = {Order based morphology for color images via matrix fields}, series = {Visualization and Processing of Tensor Fields and Higher Order Descriptors for Multi-Valued Data}, booktitle = {Visualization and Processing of Tensor Fields and Higher Order Descriptors for Multi-Valued Data}, editor = {Burgeth, Bernhard and Vilanova, Anna and Westin, Carl-Fredrik}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-642-54300-5}, doi = {10.1007/978-3-642-54301-2_4}, pages = {75 -- 95}, abstract = {Mathematical morphology is a successful branch of image processing with a history of more than four decades. Its fundamental operations are dilation and erosion, which are based on the notion of supremum and infimum with respect to an order. From dilation and erosion one can build readily other useful elementary morphological operators and filters, such as opening, closing, morphological top-hats, derivatives, and shock filters. Such operators are available for grey value images, and recently useful analogs of these processes for matrix-valued images have been introduced by taking advantage of the so-called Loewner order. There is a number of approaches to morphology for vector-valued images, that is, color images based on various orders, however, each with its merits and shortcomings. In this chapter we propose an approach to (elementary) morphology for color images that relies on the existing order based morphology for matrix fields of symmetric 2 × 2-matrices. An RGB-image is embedded into a field of those 2 × 2-matrices by exploiting the geometrical properties of the order cone associated with the Loewner order. To this end a modification of the HSL-color model and a relativistic addition of matrices is introduced. The experiments performed with various morphological elementary operators on synthetic and real images provide results promising enough to serve as a proof-of-concept.}, language = {en} } @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{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {Authors 'Reply to' Letter to the Editor regarding folded models and the paper by Brazauskas and Kleefeld (2011)}, series = {Scandinavian Actuarial Journal}, journal = {Scandinavian Actuarial Journal}, number = {8}, issn = {0346-1238}, doi = {10.1080/03461238.2013.798070}, pages = {753 -- 757}, language = {en} }