@misc{SridharBreussKahra, author = {Sridhar, Vivek and Breuß, Michael and Kahra, Marvin}, title = {Fast Approximation of Color Morphology}, series = {Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II}, journal = {Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II}, editor = {Bebis, George}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-90435-7}, doi = {10.1007/978-3-030-90436-4_39}, pages = {488 -- 499}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. The algorithmic complexity of fast algorithms that realise dilation and erosion for color images usually depends on size and shape of the structuring element. In this paper we propose and investigate an easy and fast way to make use of the fast Fourier transform for an approximate computation of dilation and erosion for color images. Similarly in construction as many other fast algorithms, the method extends a recent scheme proposed for single-channel filtering. It is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity only depends on the number of pixels in the filtered images. We analyse here some important aspects of the approximation, and we show experimentally that we obtain results of very reasonable quality while the method has very attractive computational properties.}, language = {en} } @misc{KahraSridharBreuss, author = {Kahra, Marvin and Sridhar, Vivek and Breuß, Michael}, title = {Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform}, series = {Scale Space and Variational Methods in Computer Vision}, journal = {Scale Space and Variational Methods in Computer Vision}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-75548-5}, issn = {978-3-030-75549-2}, doi = {10.1007/978-3-030-75549-2_6}, pages = {65 -- 77}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. Existing fast algorithms that realise dilation and erosion for grey value images are often limited with respect to size or shape of the structuring element. Usually their algorithmic complexity depends on these aspects. Many fast methods only address flat morphology. In this paper we propose a novel way to make use of the fast Fourier transform for the computation of dilation and erosion. Our method is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity does not depend on size or shape of the structuring element, but only on the number of pixels in the filtered images. We show experimentally that we obtain results of very reasonable quality with the proposed method.}, language = {en} } @misc{KahraBreuss, author = {Kahra, Marvin and Breuß, Michael}, title = {Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation}, series = {Journal of Mathematical Imaging and Vision}, volume = {65}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, issn = {0924-9907}, doi = {10.1007/s10851-022-01138-3}, pages = {577 -- 591}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a structuring element that is usually shifted pixel-wise over an image, together with a comparison process that takes place within the corresponding mask. This comparison is made in the grey value case by means of maximum or minimum formation. Hence, there is easy access to max-plus algebra and, by means of an algebra change, also to the theory of linear algebra. We show that an approximation of the maximum function forms a commutative semifield (with respect to multiplication) and corresponds to the maximum again in the limit case. In this way, we demonstrate a novel access to the logarithmic connection between the Fourier transform and the slope transformation. In addition, we prove that the dilation by means of a fast Fourier transform depends only on the size of the structuring element used. Moreover, we derive a bound above which the Fourier approximation yields results that are exact in terms of grey value quantisation.}, language = {en} } @misc{KahraBreussKleefeldetal., author = {Kahra, Marvin and Breuß, Michael and Kleefeld, Andreas and Welk, Martin}, title = {An Approach to Colour Morphological Supremum Formation using the LogSumExp Approximation}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2312.13792}, pages = {1 -- 12}, abstract = {Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric 2×2 matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.}, language = {en} } @misc{KoehlerKahraBreuss, author = {K{\"o}hler, Alexander and Kahra, Marvin and Breuß, Michael}, title = {A First Approach to Quantum Logical Shape Classification Framework}, series = {Mathematics}, volume = {12}, journal = {Mathematics}, number = {11}, publisher = {MDPI AG}, address = {Basel}, issn = {2227-7390}, doi = {10.3390/math12111646}, abstract = {Quantum logic is a well-structured theory, which has recently received some attention because of its fundamental relation to quantum computing. However, the complex foundation of quantum logic borrowing concepts from different branches of mathematics as well as its peculiar settings have made it a non-trivial task to devise suitable applications. This article aims to propose for the first time an approach using quantum logic in image processing for shape classification. We show how to make use of the principal component analysis to realize quantum logical propositions. In this way, we are able to assign a concrete meaning to the rather abstract quantum logical concepts, and we are able to compute a probability measure from the principal components. For shape classification, we consider encrypting given point clouds of different objects by making use of specific distance histograms. This enables us to initiate the principal component analysis. Through experiments, we explore the possibility of distinguishing between different geometrical objects and discuss the results in terms of quantum logical interpretation.}, language = {en} } @misc{KahraBreussKleefeldetal., author = {Kahra, Marvin and Breuß, Michael and Kleefeld, Andreas and Welk, Martin}, title = {An Approach to Colour Morphological Supremum Formation Using the LogSumExp Approximation}, series = {Lecture Notes in Computer Science}, volume = {14605}, journal = {Lecture Notes in Computer Science}, publisher = {Springer Nature Switzerland}, address = {Cham}, isbn = {9783031577925}, issn = {0302-9743}, doi = {10.1007/978-3-031-57793-2_25}, pages = {325 -- 337}, abstract = {Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.}, language = {en} } @misc{KoehlerKahraBreuss, author = {K{\"o}hler, Alexander and Kahra, Marvin and Breuß, Michael}, title = {A First Approach to Quantum Logical Shape Classification Framework}, series = {Preprints.org}, journal = {Preprints.org}, publisher = {MDPI AG}, doi = {10.20944/preprints202402.1042.v1}, abstract = {Quantum logic is a well-structured theory, which has recently received some attention because of its fundamental relation to quantum computing. However, the complex foundation of quantum logic borrowing concepts from different branches of mathematics as well as its peculiar settings have made it a non-trivial task to device suitable applications. This article aims to propose for the first time an approach to use quantum logic in image processing at hand of a process for shape classification. We show how to make use of the principal component analysis to realize quantum logical propositions. In this way we are able to assign a concrete meaning to the rather abstract quantum logical concepts, and we are able to compute a probability measure from the principal components. For shape classification we consider encrypting given point clouds of different objects by making use of specific distance histograms. This enables to initiate the principal component analysis. At hand of experiments, we explore the possibility to distinguish between different geometrical objects and discuss the results in terms of quantum logical interpretation.}, language = {en} } @misc{KahraBreuss, author = {Kahra, Marvin and Breuß, Michael}, title = {Colour morphological distance ordering based on the Log-Exp-Supremum}, series = {arXiv}, journal = {arXiv}, doi = {https://doi.org/10.48550/arXiv.2503.11329}, pages = {1 -- 13}, abstract = {Mathematical morphology, a field within image processing, includes various filters that either highlight, modify, or eliminate certain information in images based on an application's needs. Key operations in these filters are dilation and erosion, which determine the supremum or infimum for each pixel with respect to an order of the tonal values over a subset of the image surrounding the pixel. This subset is formed by a structuring element at the specified pixel, which weighs the tonal values. Unlike grey-scale morphology, where tonal order is clearly defined, colour morphology lacks a definitive total order. As no method fully meets all desired properties for colour, because of this difficulty, some limitations are always present. This paper shows how to combine the theory of the log-exp-supremum of colour matrices that employs the Loewner semi-order with a well-known colour distance approach in the form of a pre-ordering. The log-exp-supremum will therefore serve as the reference colour for determining the colour distance. To the resulting pre-ordering with respect to these distance values, we add a lexicographic cascade to ensure a total order and a unique result. The objective of this approach is to identify the original colour within the structuring element that most closely resembles a supremum, which fulfils a number of desired properties. Consequently, this approach avoids the false-colour problem. The behaviour of the introduced operators is illustrated by application examples of dilation and closing for synthetic and natural images.}, language = {en} } @incollection{KahraBreuss, author = {Kahra, Marvin and Breuß, Michael}, title = {Colour morphological distance ordering based on the log-exp-supremum}, series = {Scale space and variational methods in computer vision : 10th international conference, SSVM 2025, Dartington, UK, May 18-22, 2025, proceedings, part II}, booktitle = {Scale space and variational methods in computer vision : 10th international conference, SSVM 2025, Dartington, UK, May 18-22, 2025, proceedings, part II}, publisher = {Springer Nature Switzerland}, address = {Cham}, isbn = {9783031923685}, issn = {0302-9743}, doi = {https://doi.org/10.1007/978-3-031-92369-2_20}, pages = {258 -- 270}, abstract = {Mathematical morphology, a field within image processing, incorporates a variety of filters that either highlight, modify, or eliminate specific information within a mask that traverses the image. Key operations in these filters are dilation and erosion, which determine the supremum or infimum for each pixel with respect to an order of the tonal values over a subset of the image surrounding the pixel. This subset is formed by a structuring element at the specified pixel, which weighs the tonal values. Unlike grey-scale morphology, where tonal order is clearly defined, colour morphology lacks a definitive total order. As no method fully meets all desired properties for colour, because of this difficulty, some limitations are always present. This paper shows how to combine the theory of the log-exp-supremum of colour matrices that employs the Loewner semi-order with a well-known colour distance approach in the form of a pre-ordering. The log-exp-supremum will therefore serve as the reference colour for determining the colour distance. To the resulting pre-ordering with respect to these distance values, we add a lexicographic cascade to ensure a total order and a unique result. The objective of this approach is to identify the original colour within the structuring element that most closely resembles a supremum, which fulfils a number of desired properties. Consequently, this approach avoids the false-colour problem. The behaviour of the introduced operators is illustrated by application examples of dilation and closing for synthetic and natural images.}, language = {en} } @misc{ShabaniBreussKahraetal., author = {Shabani, Shima and Breuß, Michael and Kahra, Marvin and Teiser, Jens and V{\"o}lke, Gretha Swantje and Wenders, Nico}, title = {Morphological granulometric analysis of particle imagery from microgravity experiments}, series = {arXiv}, journal = {arXiv}, publisher = {Cornell University}, address = {Ithaca, NY}, doi = {10.48550/arXiv.2508.06593}, pages = {1 -- 12}, abstract = {The aim of our work is to analyze size distributions of particles and their agglomerates in imagery from astrophysical microgravity experiments. The data acquired in these experiments are given by sequences consisting of several hundred images. It is desirable to establish an automated routine that helps to assess size distributions of important image structures and their dynamics in a statistical way. The main technique we adopt to this end is the morphological granulometry. After preprocessing steps that facilitate granulometric analysis, we show how to extract useful information on size of particle agglomerates as well as underlying dynamics. At hand of the discussion of two different microgravity key experiments we demonstrate that the granulometric analysis enables to assess important experimental aspects. We conjecture that our developments are a useful basis for the quantitative assessment of microgravity particle experiments.}, language = {en} } @incollection{ShabaniBreussKahraetal., author = {Shabani, Shima and Breuß, Michael and Kahra, Marvin and Teiser, Jens and V{\"o}lke, Gretha Swantje and Wenders, Nico}, title = {Morphological granulometric analysis of particle imagery from microgravity experiments}, series = {Discrete geometry and mathematical morphology : 4th International Joint Conference, DGMM 2025, Groningen, The Netherlands, November 3-6, 2025, proceedings}, booktitle = {Discrete geometry and mathematical morphology : 4th International Joint Conference, DGMM 2025, Groningen, The Netherlands, November 3-6, 2025, proceedings}, publisher = {Springer Nature Switzerland}, address = {Cham}, isbn = {978-3-032-09544-2}, issn = {0302-9743}, doi = {https://doi.org/10.1007/978-3-032-09544-2_35}, pages = {487 -- 500}, abstract = {The aim of our work is to analyze size distributions of particles and their agglomerates in imagery from astrophysical microgravity experiments. The data acquired in these experiments are given by sequences consisting of several hundred images. It is desirable to establish an automated routine that helps to assess size distributions of important image structures and their dynamics in a statistical way. The main technique we adopt to this end is the morphological granulometry. After preprocessing steps that facilitate granulometric analysis, we show how to extract useful information on size of particle agglomerates as well as underlying dynamics. At hand of the discussion of two different microgravity key experiments we demonstrate that the granulometric analysis enables to assess important experimental aspects. We conjecture that our developments are a useful basis for the quantitative assessment of microgravity particle experiments.}, 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} }