TY - CHAP A1 - Kleefeld, Andreas A1 - Breuß, Michael A1 - Welk, Martin A1 - Burgeth, Bernhard ED - Trémeau, Alain ED - Schettini, Raimondo ED - Tominaga, Shoji T1 - Adaptive Filters for Color Images: Median Filtering and Its Extensions T2 - Computational Color Imaging N2 - In this paper we are concerned with robust structure-preserving denoising filters for color images. We build on a recently proposed transformation from the RGB color space to the space of symmetric 2×2 matrices that has already been used to transfer morphological dilation and erosion concepts from matrix-valued data to color images. We investigate the applicability of this framework to the construction of color-valued median filters. Additionally, we introduce spatial adaptivity into our approach by morphological amoebas that offer excellent capabilities for structure-preserving filtering. Furthermore, we define color-valued amoeba M-smoothers as a generalization of the median-based concepts. Our experiments confirm that all these methods work well with color images. They demonstrate the potential of our approach to define color processing tools based on matrix field techniques. KW - Matrix field KW - Color image KW - Median filter KW - M-smoother KW - Amoeba filter Y1 - 2015 UR - http://link.springer.com/chapter/10.1007/978-3-319-15979-9_15# SN - 978-3-319-15978-2 U6 - https://doi.org/10.1007/978-3-319-15979-9_15 SP - 149 EP - 158 PB - Springer CY - Berlin ER - TY - CHAP A1 - Sharifi Boroujerdi, Ali A1 - Breuß, Michael A1 - Burgeth, Bernhard A1 - Kleefeld, Andreas ED - Aujol, Jean-François ED - Nikolova, Mila ED - Papadakis, Nicolas T1 - PDE-Based Color Morphology Using Matrix Fields T2 - Scale Space and Variational Methods in Computer Vision, 5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31 - June 4, 2015, Proceedings N2 - In this work, we propose a novel way for performing operations of mathematical morphology on color images. To this end, we convert pixelwise the rgb-values into symmetric 2x2 matrices. The new color space can be interpreted geometrically as a biconal color space structure. Motivated by the formulation of the fundamental morphological operations dilation and erosion in terms of partial differential equations (PDEs), we show how to define finite difference schemes making use of the matrix field formulation. The computation of a pseudo supremum and a pseudo infimum of three color matrices is a crucial step for setting up advanced PDE-based methods. We show that this can be achieved for our goal by an algebraic technique. We investigate our approach by dedicated experiments and confirm useful properties of the new PDE-based color morphology operations. KW - PDE-based morphology KW - Matrix fields KW - Color morphology KW - Finite difference schemes KW - FCT scheme KW - Pseudo supremum KW - Pseudo infimum Y1 - 2015 SN - 978-3-319-18460-9 U6 - https://doi.org/10.1007/978-3-319-18461-6_37 SP - 461 EP - 473 PB - Springer CY - Berlin ER - TY - THES A1 - Kleefeld, Andreas T1 - Numerical methods for acoustic and electromagnetic scattering: Transmission boundary-value problems, interior transmission eigenvalues, and the factorization method Y1 - 2015 ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Veste, Maik T1 - Robuste und strukturerhaltende Bildentrauschung via Medianfilterung und Bildsegmentierung T2 - 20. und 21. Workshop Computer-Bildanalyse in der Landwirtschaft, 26. Mai 2014, Osnabrück N2 - Ein Medianfilter ist ein Werkzeug, das benutzt werden kann, um Bilder zu entrauschen. Allerdings verschwimmen vorhandene Strukturen, und eine gute Bildsegmentierung gestaltet sich schwierig. Daher wird hier ein Medianfilteralgorithmus vorgestellt, der in der Lage ist, Bilder zu entrauschen und gleichzeitig vorhandene Strukturen beizubehalten. Wir zeigen die Anwendbarkeit dieser Methode auf Bilder, die durch ein automatisches Minirhizotronsystem aufgenommen worden sind. Dieses dient zur zerstörungsfreien Observierung von Wurzelwachstum. Die verbesserten Bilder werden dann zur Bildsegmentierung benutzt, um automatisch die Wurzeln in den Bildern zu entdecken. KW - Medianfilter KW - Bildsegmentierung KW - automatisches Minirhizotronsystem Y1 - 2015 UR - http://www.atb-potsdam.de/?id=161 SN - 0947-7314 SP - 92 EP - 100 PB - Leibniz-Institut für Agrartechnik Potsdam-Bornim e.V. CY - Potsdam ER - TY - CHAP A1 - Welk, Martin A1 - Kleefeld, Andreas A1 - Breuß, Michael ED - Benediktsson, Jón Atli ED - Chanussot, Jocelyn ED - Najman, Laurent ED - Talbot, Hugues T1 - Non-adaptive and Amoeba Quantile Filters for Colour Images T2 - Mathematical Morphology and Its Applications to Signal and Image Processing N2 - Quantile filters, or rank-order filters, are local image filters which assign quantiles of intensities of the input image within neighbourhoods as output image values. Combining a multivariate quantile definition developed in matrix-valued morphology with a recently introduced mapping between the RGB colour space and the space of symmetric 2×2 matrices, we state a class of colour image quantile filters, along with a class of morphological gradient filters derived from these. Using amoeba structuring elements, we devise image-adaptive versions of both filter classes. Experiments demonstrate the favourable properties of the filters. KW - Quantile KW - Rank-order filter KW - Color image KW - Matrix field KW - Amoebas Y1 - 2015 SN - 978-3-319-18719-8 SN - 978-3-319-18720-4 U6 - https://doi.org/10.1007/978-3-319-18720-4_34 SP - 398 EP - 409 PB - Springer International Publishing CY - Berlin ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Meyer-Baese, Anke A1 - Burgeth, Bernhard ED - Benediktsson, Jón Atli ED - Chanussot, Jocelyn ED - Najman, Laurent ED - Talbot, Hugues T1 - Elementary Morphology for SO(2)- and SO(3)-Orientation Fields T2 - Mathematical Morphology and Its Applications to Signal and Image Processing N2 - In this article, techniques to process fields of special orthogonal matrices by means of mathematical morphology are proposed. Since the group structure of SO(2)- resp. SO(3)-fields is not suitable to establish useful notions of infimum and supremum, they are transformed into scalar resp. Sym(2)-fields utilizing a matrix-valued version of the Cayley transform. However, for symmetric 2×2 matrices, that is for Sym(2)-fields, elementary morphological operations are available, see [6] and [7]. Several examples and numerical results are presented to show the merits and the limitations of this novel approach. Additionally, the series of transformations utilized in the proposed methods open up possibilities to visualize SO(2)- and SO(3)-fields. KW - Orientation field KW - Tensor field KW - Loewner order KW - Lie group KW - Lie algebra KW - Cayley transform KW - Symmetric matrix KW - Erosion KW - Dilation KW - Morphological Laplacian KW - Beucher gradient Y1 - 2015 UR - http://link.springer.com/chapter/10.1007%2F978-3-319-18720-4_39 SN - 978-3-319-18719-8 SN - 978-3-319-18720-4 U6 - https://doi.org/10.1007/978-3-319-18720-4_39 SP - 458 EP - 469 PB - Springer International Publishing CY - Berlin ER - TY - CHAP A1 - Meyer-Baese, Anke A1 - Barbu, Adrian A1 - Lobbes, Marc A1 - Hoffmann, Sebastian A1 - Burgeth, Bernhard A1 - Kleefeld, Andreas A1 - Meyer-Baese, Uwe ED - Szu, Harold H. ED - Dai, Liyi ED - Zheng, Yufeng T1 - Computer-aided diagnosis of breast MRI with high accuracy optical flow estimation T2 - Independent Component Analyses, Compressive Sampling, Large Data Analyses (LDA), Neural Networks, Biosystems, and Nanoengineering XIII N2 - Non-mass enhancing lesions represent a challenge for the radiological reading. They are not well-defined in both morphology (geometric shape) and kinetics (temporal enhancement) and pose a problem to lesion detection and classification. To enhance the discriminative properties of an automated radiological workflow, the correct preprocessing steps need to be taken. In an usual computer-aided diagnosis (CAD) system, motion compensation plays an important role. To this end, we employ a new high accuracy optical flow based motion compensation algorithm with robustification variants. An automated computer-aided diagnosis system evaluates the atypical behavior of these lesions, and additionally considers the impact of non-rigid motion compensation on a correct diagnosis. Y1 - 2015 U6 - https://doi.org/10.1117/12.2181815 PB - SPIE CY - Bellingham, Wash. ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Burgeth, Bernhard ED - Hotz, Ingrid ED - Schultz, Thomas T1 - Processing Multispectral Images via Mathematical Morphology T2 - Visualization and Processing of Higher Order Descriptors for Multi-Valued Data Y1 - 2015 UR - http://www.springer.com/us/book/9783319150895 SN - 978-3-319-15089-5 SN - 978-3-319-15090-1 U6 - https://doi.org/10.1007/978-3-319-15090-1 SP - 129 EP - 148 PB - Springer International Publishing CY - Berlin ER - TY - GEN A1 - Welk, Martin A1 - Kleefeld, Andreas A1 - Breuß, Michael T1 - Quantile Filtering of Colour Images via Symmetric Matrices T2 - Mathematical Morphology - Theory and Applications N2 - Quantile filters, or rank-order filters, are local image filters which assign quantiles of intensities of the input image within neighbourhoods as output image values. Combining a multivariate quantile definition developed in matrix-valued morphology with a recently introduced mapping between the RGB colour space and the space of symmetric 2 × 2 matrices, we state a class of colour image quantile filters, along with a class of morphological gradient filters derived from these.We consider variants of these filters based on three matrix norms – the nuclear, Frobenius, and spectral norm – and study their differences. We investigate the properties of the quantile and gradient filters and their links to dilation and erosion operators. Using amoeba structuring elements,we devise image-adaptive versions of our quantile and gradient filters. Experiments are presented to demonstrate the favourable properties of the filters, and compare them to existing approaches in colour morphology. KW - quantile KW - rank-order filter KW - colour image KW - matrix field KW - amoebas Y1 - 2016 U6 - https://doi.org/10.1515/mathm-2016-0008 SN - 2353-3390 VL - 1 IS - 1 SP - 136 EP - 174 ER - TY - CHAP A1 - Hoeltgen, Laurent A1 - Harris, I. A1 - Breuß, Michael A1 - Kleefeld, Andreas T1 - Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian T2 - Scale Space and Variational Methods in Computer Vision, 6th International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2017, Kolding, Denmark, June 2017), proceedings N2 - Partial differential equations are well suited for dealing with image reconstruction tasks such as inpainting. One of the most successful mathematical frameworks for image reconstruction relies on variations of the Laplace equation with different boundary conditions. In this work we analyse these formulations and discuss the existence and uniqueness of solutions of corresponding boundary value problems, as well as their regularity from an analytic point of view. Our work not only sheds light on useful aspects of the well posedness of several standard problem formulations in image reconstruction but also aggregates them in a common framework. In addition, the performed analysis guides us to specify two new formulations of the classic image reconstruction problem that may give rise to new developments in image reconstruction. KW - Partial differential equations Laplace equation Mixed boundary conditions Image reconstruction Image inpainting Y1 - 2017 SN - 978-3-319-58770-7 U6 - https://doi.org/10.1007/978-3-319-58771-4_6 SP - 66 EP - 79 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - Breuß, Michael A1 - Hoeltgen, Laurent A1 - Kleefeld, Andreas T1 - Matrix-Valued Levelings for Colour Images T2 - Mathematical morphology and its applications to signal and image processing, 13th international symposium, ISMM 2017, Fontainebleau, France, May 15-17, 2017, proceedings N2 - Morphological levelings represent a useful tool for the decomposition of an image into cartoon and texture components. Moreover, they can be used to construct a morphological scale space. However, the classic construction of levelings is limited to the use of grey scale images, since an ordering of pixel values is required. In this paper we propose an extension of morphological levelings to colour images. To this end, we consider the formulation of colour images as matrix fields and explore techniques based on the Loewner order for formulating morphological levelings in this setting. Using the matrix-valued colours we study realisations of levelings relying on both the completely discrete construction and the formulation using a partial differential equation. Experimental results confirm the potential of our matrix-based approaches for analysing texture in colour images and for extending the range of applications of levelings in a convenient way to colour image processing. KW - Mathematical morphology Dilation Erosion Loewner order Leveling Y1 - 2017 SN - 978-3-319-57239-0 U6 - https://doi.org/10.1007/978-3-319-57240-6_24 SP - 296 EP - 308 PB - Springer International Publishing CY - Cham ER - TY - GEN A1 - Hoeltgen, Laurent A1 - Kleefeld, Andreas A1 - Harries, Isaac A1 - Breuß, Michael T1 - Theoretical foundation of the weighted laplace inpainting problem T2 - Journal of Applications of Mathematics N2 - Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup. KW - image inpainting KW - image reconstruction KW - Laplace equation KW - Laplace interpolation KW - mixed boundary condition KW - partial differential equation KW - weighted Sobolev space Y1 - 2019 U6 - https://doi.org/10.21136/AM.2019.0206-18 SN - 1572-9109 SN - 0862-7940 VL - 64 IS - 3 SP - 281 EP - 300 ER - TY - GEN A1 - Breuß, Michael A1 - Kleefeld, Andreas T1 - Implicit Monotone Difference Methods for Scalar Conservation Laws with Source Terms T2 - Acta Mathematica Vietnamica N2 - In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests. KW - Conservation laws KW - Finite difference methods KW - Implicit methods KW - Monotone methods KW - Source term KW - Entropy solution Y1 - 2020 U6 - https://doi.org/10.1007/s40306-019-00354-1 SN - 2315-4144 SN - 0251-4184 VL - 45 IS - 3 SP - 709 EP - 738 ER - TY - GEN A1 - Kahra, Marvin A1 - Breuß, Michael A1 - Kleefeld, Andreas A1 - Welk, Martin T1 - An Approach to Colour Morphological Supremum Formation using the LogSumExp Approximation T2 - arXiv N2 - 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. Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2312.13792 SP - 1 EP - 12 ER - TY - GEN A1 - Kahra, Marvin A1 - Breuß, Michael A1 - Kleefeld, Andreas A1 - Welk, Martin T1 - An Approach to Colour Morphological Supremum Formation Using the LogSumExp Approximation T2 - Lecture Notes in Computer Science N2 - 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. KW - mathematical morphology KW - colour image KW - matrix-valued image KW - symmetric matrix KW - transitivity Y1 - 2024 SN - 9783031577925 U6 - https://doi.org/10.1007/978-3-031-57793-2_25 SN - 0302-9743 VL - 14605 SP - 325 EP - 337 PB - Springer Nature Switzerland CY - Cham ER - TY - GEN A1 - Kleefeld, Andreas A1 - Gypser, Stella A1 - Herppich, Werner B. A1 - Bader, Georg A1 - Veste, Maik T1 - Identification of spatial pattern of photosynthesis hotspots in moss- and lichen-dominated biological soil crusts by combining chlorophyll fluorescence imaging and multispectral BNDVI images T2 - Pedobiologia N2 - Although biological soil crusts can be found in open landscapes worldwide, their species composition depends on soil properties such as texture and pH, on microclimate, and their respective developmental stage. In addition, local variations in water holding capacity and/or chemical properties of soils influence the formation of spatial patterns and different types of biocrusts on the landscape level. For the evaluation of biocrusts functions and their impact on soil carbon pools, the analysis of the interrelationship between photosynthetic activity and the variations of spatial distribution pattern and types of biocrust is indispensable. For this purpose, an image processing approach was applied that combined chlorophyll fluorescence analyses and multispectral BNDVI to comprehensively characterize the spatial patterns of photosynthetic hotspots in biological soil crusts. For image analysis, five biological soil crust samples with different ratios of substrate, mosses and lichens were collected on an inland dune system in Lieberose, dominated by the moss Polytrichum piliferum, and the lichens Cladonia fimbriata and C. coccifera. RGB-images of the biocrusts were taken with a standard consumer camera Nikon 5200, BNDVI images with a modified Canon S110 NIR camera and chlorophyll fluorescence images with a modular open FluorCAM FC 800-O/1010, respectively. BNDVI and Fv/Fm were nearly in the same range for all biocrust samples related to the total surface area. Although mosses showed higher BNDVI than lichens within the separate biocrust samples. F0 and Fm increased with species coverage and with advancing biocrust development. Overlapping of BNDVI with F0 and Fm images showed that not all crustal organisms contribute to BNDVI and chlorophyll fluorescence. The overlapping areas of BNDVI and F0 ranged between 13% and 29%, that of BNDVI and Fm between 17% and 47%. Matching of RGB, BNDVI and CFI allows visualizing spatial pattern with high or low photosynthesis in biocrusts. Y1 - 2018 U6 - https://doi.org/10.1016/j.pedobi.2018.04.001 SN - 1873-1511 VL - 68 SP - 1 EP - 11 ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Veste, Maik T1 - Robuste and strukturerhaltende Bildentrauschung via Medianfilterung und Bildsegmentierung T2 - 20. und 21. Workshop Computer-Bildanalyse in der Landwirtschaft, 3. Workshop Unbemannte Autonom Fliegende Systeme (UAS) in der Landwirtschaft, 26. Mai 2014, Osnabrück 07. Mai 2015, Braunschweig Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:186-opus4-75951 SP - 92 EP - 100 PB - ATB, Leibniz-Institut für Agrartechnik CY - Potsdam-Bornim ER - TY - GEN A1 - Kahra, Marvin A1 - Breuß, Michael A1 - Kleefeld, Andreas A1 - Welk, Martin T1 - Matrix-valued LogSumExp approximation for colour morphology T2 - Journal of mathematical imaging and vision N2 - 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. KW - Mathematical morphology KW - Colour image KW - Matrix-valued image KW - Positive definite matrix KW - Symmetric matrix KW - Supremum Y1 - 2025 U6 - https://doi.org/10.1007/s10851-025-01267-5 SN - 0924-9907 SN - 1573-7683 VL - 67 IS - 5 PB - Springer US CY - New York ER -