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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - CHAP A1 - Kleefeld, Andreas A1 - Burgeth, Bernhard ED - Burgeth, Bernhard ED - Vilanova, Anna ED - Westin, Carl-Fredrik T1 - Order based morphology for color images via matrix fields T2 - Visualization and Processing of Tensor Fields and Higher Order Descriptors for Multi-Valued Data N2 - 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. KW - Mathematical Morphology KW - Color image KW - Matrix field Y1 - 2014 UR - http://link.springer.com/chapter/10.1007%2F978-3-642-54301-2_4 SN - 978-3-642-54300-5 SN - 978-3-642-54301-2 U6 - https://doi.org/10.1007/978-3-642-54301-2_4 SP - 75 EP - 95 PB - Springer CY - Berlin ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Burgeth, Bernhard ED - Hendriks, Cris L. Luengo ED - Borgefors, Gunilla ED - Strand, Robin T1 - Morphology for Color Images via Loewner Order for Matrix Fields T2 - Mathematical Morphology and Its Applications to Signal and Image Processing N2 - Mathematical morphology is a very 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 a maximum and a minimum with respect to an order. Many operators constructed from dilation and erosion 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 has been a number of approaches to morphology for vector-valued images, that is, colour images based on various orders, however, each with its merits and shortcomings. In this article we propose an approach to (elementary) morphology for colour 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-colour model and a relativistic addition of matrices is introduced. The experiments performed with various morphological elementary operators on synthetic and real images demonstrate the capabilities and restrictions of the novel approach. KW - matrix field KW - tensor field KW - symmetric matrix KW - colour images KW - dilation KW - erosion KW - colour space KW - Einstein addition Y1 - 2013 UR - http://link.springer.com/chapter/10.1007/978-3-642-38294-9_21 SN - 978-3-642-38293-2 SN - 978-3-642-38294-9 U6 - https://doi.org/10.1007/978-3-642-38294-9_21 SP - 243 EP - 254 PB - Springer CY - Berlin ER - TY - GEN A1 - Anagnostōpulos, Kōnstantinos N. A1 - Charalambopoulos, Antonios A1 - Kleefeld, Andreas T1 - The factorization method for the acoustic transmission problem T2 - Inverse Problems N2 - In this work, the shape reconstruction problem of acoustically penetrable bodies from the far-field data corresponding to time-harmonic plane wave incidence is investigated within the framework of the factorization method. Although the latter technique has received considerable attention in inverse scattering problems dealing with impenetrable scatterers and it has not been elaborated for inverse transmission problems with the only exception being a work by the first two authors and co-workers. We aim to bridge this gap in the field of acoustic scattering; the paper on one hand focuses on establishing rigorously the necessary theoretical framework for the application of the factorization method to the inverse acoustic transmission problem. The main outcome of the investigation undertaken is the derivation of an explicit formula for the scatterer's characteristic function, which depends solely on the far-field data feeding the inverse scattering scheme. Extended numerical examples in three dimensions are also presented, where a variety of different surfaces are successfully reconstructed by the factorization method, thus, complementing the method's validation from the computational point of view. KW - integral equations KW - Factorization method KW - scattering KW - inverse problem Y1 - 2013 UR - http://iopscience.iop.org/0266-5611/29/11/115015/ U6 - https://doi.org/10.1088/0266-5611/29/11/115015 SN - 0266-5611 SN - 1361-6420 VL - 29 IS - 11 SP - 115015 ER - TY - GEN A1 - Kleefeld, Andreas T1 - A numerical method to compute interior transmission eigenvalues T2 - Inverse Problems N2 - In this paper the numerical calculation of eigenvalues of the interior transmission problem arising in acoustic scattering for constant contrast in three dimensions is considered. From the computational point of view existing methods are very expensive, and are only able to show the existence of such transmission eigenvalues. Furthermore, they have trouble finding them if two or more eigenvalues are situated closely together. We present a new method based on complex-valued contour integrals and the boundary integral equation method which is able to calculate highly accurate transmission eigenvalues. So far, this is the first paper providing such accurate values for various surfaces different from a sphere in three dimensions. Additionally, the computational cost is even lower than those of existing methods. Furthermore, the algorithm is capable of finding complex-valued eigenvalues for which no numerical results have been reported yet. Until now, the proof of existence of such eigenvalues is still open. Finally, highly accurate eigenvalues of the interior Dirichlet problem are provided and might serve as test cases to check newly derived Faber–Krahn type inequalities for larger transmission eigenvalues that are not yet available. KW - interior transmission eigenvalues Y1 - 2013 UR - http://iopscience.iop.org/0266-5611/29/10/104012/article U6 - https://doi.org/10.1088/0266-5611/29/10/104012 SN - 0266-5611 SN - 1361-6420 IS - 29 SP - 104012 ER - TY - GEN A1 - Kleefeld, Andreas A1 - Lin, Tzu-Chu T1 - A Global Galerkin Method for Solving the Exterior Neumann Problem for the Helmholtz Equation Using Panich's Integral Equation Approach T2 - SIAM Journal on Scientific Computing N2 - We describe a boundary integral equation that solves the exterior Neumann problem for the Helmholtz equation in three dimensions for smooth obstacles that can be described globally in spherical coordinates. The unique solution is found by approximating the Fredholm integral equation of the second kind with the Galerkin method, where the basis functions are spherical harmonics. This leads to a fast method for small and medium wave numbers, since the system of equations is of size smaller than 64. To smooth the integrand a new map is introduced which significantly improves the accuracy. Numerical results for several smooth surfaces are presented. KW - Fredholm integral equation of the second kind KW - Helmholtz equation KW - exterior Neumann problem KW - Galerkin method Y1 - 2013 U6 - https://doi.org/10.1137/120873066 SN - 1064-8275 SN - 1095-7197 VL - 35 IS - 3 SP - A1709 EP - A1735 ER - TY - GEN A1 - Kleefeld, Andreas T1 - The transmission problem for the Helmholtz equation in R³ T2 - Computational Methods in Applied Mathematics N2 - The problem under consideration is the three-dimensional transmission problem for time harmonic acoustic waves for two homogeneous media. A simply-connected and bounded region with sufficiently smooth boundary is immersed in an infinite medium. Each medium is c haracterized by the space independent wave number κ and the density μ. The system of boundary integral equations is reviewed as well as an existence and uniqueness result. The system is approximated by the boundary element collocation method and consistency, stability, and convergence is proved. In addition, superconvergence is proved and numerical results illustrate the agreement with these theoretical results. No numerical results seem to be reported for this method yet. KW - Helmholtz's equation; integral equation; transmission problem; boundary element method; superconvergence Y1 - 2012 UR - http://www.degruyter.com/view/j/cmam.2012.12.issue-3/cmam-2012-0008/cmam-2012-0008.xml?format=INT SN - 1609-4840 VL - 12 IS - 3 SP - 330 EP - 350 ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Ju, Yong Chul A1 - Tozza, Silvia A1 - Breuß, Michael A1 - Bruhn, Andrés ED - Burghardt, Tilo ED - Damen, Dima ED - Mayol-Cuevas, Walterio ED - Mirmehdi, Majid T1 - Generalised Perspective Shape from Shading with Oren-Nayar Reflectance T2 - Proceedings 24th British Machine Vision Conference, BMVC 2013, Bristol, UK, September 2013 N2 - In spite of significant advances in Shape from Shading (SfS) over the last years, it is still a challenging task to design SfS approaches that are flexible enough to handle a wide range of input scenes. In this paper, we address this lack of flexibility by proposing a novel model that extends the range of possible applications. To this end, we consider the class of modern perspective SfS models formulated via partial differential equations (PDEs). By combining a recent spherical surface parametrisation with the advanced non-Lambertian Oren-Nayar reflectance model, we obtain a robust approach that allows to deal with an arbitrary position of the light source while being able to handle rough surfaces and thus more realistic objects at the same time. To our knowledge, the resulting model is currently the most advanced and most flexible approach in the literature on PDE-based perspective SfS. Apart from deriving our model, we also show how the corresponding set of sophisticated Hamilton-Jacobi equations can be efficiently solved by a specifically tailored fast marching scheme. Experiments with medical real-world data demonstrate that our model works in practice and that is offers the desired flexibility. Y1 - 2013 UR - http://www.bmva.org/bmvc/2013/Papers/paper0042/index.html U6 - https://doi.org/10.5244/C.27.42 SP - 42.1 EP - 42.11 PB - BMVA Press ER - TY - GEN A1 - Kleefeld, Andreas A1 - Brazauskas, Vytaras T1 - A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distribution T2 - European Journal of Applied Mathematics N2 - In this paper, we revisit the quantile mechanics approach, which was introduced by Steinbrecher and Shaw (Steinbrecher, G. & Shaw, W. T. (2008) Quantile mechanics. European. J. Appl. Math. 19, 87–112). Our objectives are (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student's t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving objective (i). To accomplish the second goal, we rely on the general large sample results for MTMs, which were established by Brazauskas et al. (Brazauskas, V., Jones, B. L. & Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. J. Stat. Plan. Inference 139, 2028–2043), and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method, which usually yields fully efficient but non-robust estimators, the MTM estimators are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student's t distributions in such outlier-prone areas as insurance and finance. KW - Point estimation (62F10); Asymptotic properties of estimators (62F12); Robustness and adaptive procedures (62F35); Method of trimmed moments (62F99); Computational problems in statistics (65C60) Y1 - 2012 UR - http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8678315 U6 - https://doi.org/10.1017/S0956792512000137 SN - 0956-7925 VL - 23 IS - 5 SP - 593 EP - 610 ER - TY - GEN A1 - Kleefeld, Andreas T1 - A modified boundary integral equation for solving the exterior Robin problem for the Helmholtz equation in three dimensions T2 - Applied Mathematics and Computation N2 - A boundary integral equation with modified fundamental solution to solve the exterior Robin problem for Helmholtz’s equation is considered. A specific choice of the coefficients of these added terms ensure the unique solvability and, in addition, they can be chosen to ensure the minimization of the least-squares difference of the modified and the exact Green’s function or the minimization of the condition number. Numerical results are reported showing the robustness and the superconvergence of the new method. KW - Boundary integral equation; Modified fundamental solution; Helmholtz equation; Exterior Robin problem; Superconvergence Y1 - 2012 UR - http://www.sciencedirect.com/science/article/pii/S009630031200848X# U6 - https://doi.org/10.1016/j.amc.2012.08.055 SN - 0096-3003 VL - 2012 IS - 219 SP - 2114 EP - 2123 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 -