@misc{KleefeldKirsch, author = {Kleefeld, Andreas and Kirsch, Andreas}, title = {The factorization method for a conductive boundary condition}, series = {Journal of Integral Equations and Applications}, volume = {24}, journal = {Journal of Integral Equations and Applications}, number = {4}, issn = {0897-3962}, doi = {10.1216/JIE-2012-24-4-575}, pages = {575 -- 601}, abstract = {In this article, the inverse scattering problem for obstacles with conductive boundary conditions is considered. The problem at hand is solved with the Factorization Method; that is, the operator and its factorization is derived. To obtain generic far field data for several obstacles the boundary element method is employed to solve the integral equation numerically. The far field data are highly accurate due to superconvergence. Finally, we show that we are able to reconstruct several obstacles and thus confirm the validity of the new operator factorization.}, language = {en} } @misc{KleefeldReissel, author = {Kleefeld, Andreas and Reißel, Martin}, title = {The Levenberg-Marquardt method applied to a parameter estimation problem arising from electrical resistivity tomography}, series = {Journal of Applied Mathematics and Computation}, volume = {217}, journal = {Journal of Applied Mathematics and Computation}, number = {9}, issn = {0096-3003}, pages = {4490 -- 4501}, abstract = {An efficient and robust electrical resistivity tomographic inversion algorithm based on the Levenberg-Marquardt method is considered to obtain quantities like grain size, source discrimination and particle size distribution. The corresponding model in two-dimensions is based on the Maxwell equations and leads to a partial differential equation with mixed Dirichlet-Neumann boundary conditions. The forward problem is solved numerically with the finite-difference method. However, the inverse problem at hand is a classic nonlinear and ill-posed parameter estimation problem. Linearizing it and applying the Tikhonov regularization method yields an iterative scheme, the Levenberg-Marquardt method. Several systems of equations of large size have to be solved efficiently in each iteration step which is accomplished by the conjugate gradient method without setting up the corresponding matrix. Instead fast matrix-vector multiplications are performed directly. Therefore, the derivative and its adjoint for the parameter-to-solution map are needed. Numerical results demonstrate the performance of our method as well as the possibility to reconstruct some of the desired parameters.}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {Folded and log-folded-t distributions as models for insurance loss data}, doi = {http://www.tandfonline.com/doi/abs/10.1080/03461230903424199}, abstract = {A rich variety of probability distributions has been proposed in the actuarial literature for fitting of insurance loss data. Examples include: lognormal, log-t, various versions of Pareto, loglogistic, Weibull, gamma and its variants, and generalized beta of the second kind distributions, among others. In this paper, we supplement the literature by adding the log-folded-normal and log-folded-t families. Shapes of the density function and key distributional properties of the ‘folded’ distributions are presented along with three methods for the estimation of parameters: method of maximum likelihood; method of moments; and method of trimmed moments. Further, large and small-sample properties of these estimators are studied in detail. Finally, we fit the newly proposed distributions to data which represent the total damage done by 827 fires in Norway for the year 1988. The fitted models are then employed in a few quantitative risk management examples, where point and interval estimates for several value-at-risk measures are calculated.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {Linear Fredholm integral equations of the first kind over surfaces in 3D}, abstract = {Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method - the Tikhonov method - to deal with this ill-conditioning and to obtain accurate numerical results.}, language = {en} } @misc{KleefeldLin, author = {Kleefeld, Andreas and Lin, Tzu-Chu}, title = {The nonlinear Landweber method applied to an inverse scattering problem for sound-soft obstacles in 3D}, abstract = {The inverse scattering problem for sound-soft obstacles is considered for both smooth and piecewise smooth surfaces in 3D. The nonlinear and ill-posed integral equation of the first kind is solved by the nonlinear Landweber method. It is an iterative regularization scheme to obtain approximations for the unknown boundary of the obstacle. It is stable with respect to noise and essentially no extra work is required to incorporate several incident waves. So far, it has only been applied to the two dimensional case. Two different integral equations are presented to obtain far-field data. Furthermore, the domain derivative and its adjoint are characterized. The integral equations of the second kind are approximated by a boundary element collocation method. The two-grid method is used to solve the large and dense linear systems. Numerical examples are illustrated to show that both smooth and piecewise smooth obstacles can be reconstructed with this method, where the latter case has not yet been reported.}, language = {en} } @misc{KleefeldLin, author = {Kleefeld, Andreas and Lin, Tzu-Chu}, title = {Boundary element collocation method for solving the exterior Neumann problem for Helmholtz's equation in three dimensions}, series = {Electronic transactions on numerical analysis}, volume = {39}, journal = {Electronic transactions on numerical analysis}, issn = {1068-9613}, pages = {113 -- 143}, abstract = {We describe a boundary integral equation that solves the exterior Neumann problem for the Helmholtz equation in three dimensions. The unique solution is found by approximating a Fredholm integral equation of the second kind with the boundary element collocation method. We prove uperconvergence at the collocation points, distinguishing the cases of even and odd interpolation. Numerical examples demonstrate the performance of the method solving the integral equation and confirm the superconvergence.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions}, series = {International Journal of Computer Mathematics}, volume = {89}, journal = {International Journal of Computer Mathematics}, number = {17}, issn = {0020-7160}, doi = {10.1080/00207160.2012.709932}, pages = {2392 -- 2409}, abstract = {In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.}, 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} } @phdthesis{Kleefeld, author = {Kleefeld, Andreas}, title = {Numerical methods for acoustic and electromagnetic scattering: Transmission boundary-value problems, interior transmission eigenvalues, and the factorization method}, pages = {165}, language = {en} } @inproceedings{KleefeldVeste, author = {Kleefeld, Andreas and Veste, Maik}, title = {Robuste und strukturerhaltende Bildentrauschung via Medianfilterung und Bildsegmentierung}, series = {20. und 21. Workshop Computer-Bildanalyse in der Landwirtschaft, 26. Mai 2014, Osnabr{\"u}ck}, booktitle = {20. und 21. Workshop Computer-Bildanalyse in der Landwirtschaft, 26. Mai 2014, Osnabr{\"u}ck}, publisher = {Leibniz-Institut f{\"u}r Agrartechnik Potsdam-Bornim e.V.}, address = {Potsdam}, issn = {0947-7314}, pages = {92 -- 100}, abstract = {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 zerst{\"o}rungsfreien Observierung von Wurzelwachstum. Die verbesserten Bilder werden dann zur Bildsegmentierung benutzt, um automatisch die Wurzeln in den Bildern zu entdecken.}, language = {de} } @incollection{WelkKleefeldBreuss, author = {Welk, Martin and Kleefeld, Andreas and Breuß, Michael}, title = {Non-adaptive and Amoeba Quantile Filters for Colour Images}, series = {Mathematical Morphology and Its Applications to Signal and Image Processing}, booktitle = {Mathematical Morphology and Its Applications to Signal and Image Processing}, editor = {Benediktsson, J{\´o}n Atli and Chanussot, Jocelyn and Najman, Laurent and Talbot, Hugues}, publisher = {Springer International Publishing}, address = {Berlin}, isbn = {978-3-319-18719-8}, doi = {10.1007/978-3-319-18720-4_34}, pages = {398 -- 409}, abstract = {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.}, 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} } @inproceedings{BreussHoeltgenKleefeld, author = {Breuß, Michael and Hoeltgen, Laurent and Kleefeld, Andreas}, title = {Matrix-Valued Levelings for Colour Images}, series = {Mathematical morphology and its applications to signal and image processing, 13th international symposium, ISMM 2017, Fontainebleau, France, May 15-17, 2017, proceedings}, booktitle = {Mathematical morphology and its applications to signal and image processing, 13th international symposium, ISMM 2017, Fontainebleau, France, May 15-17, 2017, proceedings}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-57239-0}, doi = {10.1007/978-3-319-57240-6_24}, pages = {296 -- 308}, abstract = {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.}, language = {en} } @inproceedings{HoeltgenHarrisBreussetal., author = {Hoeltgen, Laurent and Harris, I. and Breuß, Michael and Kleefeld, Andreas}, title = {Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian}, series = {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}, booktitle = {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}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-58770-7}, doi = {10.1007/978-3-319-58771-4_6}, pages = {66 -- 79}, abstract = {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.}, language = {en} } @inproceedings{SharifiBoroujerdiBreussBurgethetal., author = {Sharifi Boroujerdi, Ali and Breuß, Michael and Burgeth, Bernhard and Kleefeld, Andreas}, title = {PDE-Based Color Morphology Using Matrix Fields}, series = {Scale Space and Variational Methods in Computer Vision, 5th International Conference, SSVM 2015, L{\`e}ge-Cap Ferret, France, May 31 - June 4, 2015, Proceedings}, booktitle = {Scale Space and Variational Methods in Computer Vision, 5th International Conference, SSVM 2015, L{\`e}ge-Cap Ferret, France, May 31 - June 4, 2015, Proceedings}, editor = {Aujol, Jean-Fran{\c{c}}ois and Nikolova, Mila and Papadakis, Nicolas}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-319-18460-9}, doi = {10.1007/978-3-319-18461-6_37}, pages = {461 -- 473}, abstract = {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.}, language = {en} } @incollection{KleefeldBreussWelketal., author = {Kleefeld, Andreas and Breuß, Michael and Welk, Martin and Burgeth, Bernhard}, title = {Adaptive Filters for Color Images: Median Filtering and Its Extensions}, series = {Computational Color Imaging}, booktitle = {Computational Color Imaging}, editor = {Tr{\´e}meau, Alain and Schettini, Raimondo and Tominaga, Shoji}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-319-15978-2}, doi = {10.1007/978-3-319-15979-9_15}, pages = {149 -- 158}, abstract = {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.}, language = {en} } @inproceedings{MeyerBaeseBarbuLobbesetal., author = {Meyer-Baese, Anke and Barbu, Adrian and Lobbes, Marc and Hoffmann, Sebastian and Burgeth, Bernhard and Kleefeld, Andreas and Meyer-Baese, Uwe}, title = {Computer-aided diagnosis of breast MRI with high accuracy optical flow estimation}, series = {Independent Component Analyses, Compressive Sampling, Large Data Analyses (LDA), Neural Networks, Biosystems, and Nanoengineering XIII}, booktitle = {Independent Component Analyses, Compressive Sampling, Large Data Analyses (LDA), Neural Networks, Biosystems, and Nanoengineering XIII}, editor = {Szu, Harold H. and Dai, Liyi and Zheng, Yufeng}, publisher = {SPIE}, address = {Bellingham, Wash.}, doi = {10.1117/12.2181815}, abstract = {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.}, language = {en} } @incollection{KleefeldMeyerBaeseBurgeth, author = {Kleefeld, Andreas and Meyer-Baese, Anke and Burgeth, Bernhard}, title = {Elementary Morphology for SO(2)- and SO(3)-Orientation Fields}, series = {Mathematical Morphology and Its Applications to Signal and Image Processing}, booktitle = {Mathematical Morphology and Its Applications to Signal and Image Processing}, editor = {Benediktsson, J{\´o}n Atli and Chanussot, Jocelyn and Najman, Laurent and Talbot, Hugues}, publisher = {Springer International Publishing}, address = {Berlin}, isbn = {978-3-319-18719-8}, doi = {10.1007/978-3-319-18720-4_39}, pages = {458 -- 469}, abstract = {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.}, language = {en} } @incollection{KleefeldBurgeth, author = {Kleefeld, Andreas and Burgeth, Bernhard}, title = {Processing Multispectral Images via Mathematical Morphology}, series = {Visualization and Processing of Higher Order Descriptors for Multi-Valued Data}, booktitle = {Visualization and Processing of Higher Order Descriptors for Multi-Valued Data}, editor = {Hotz, Ingrid and Schultz, Thomas}, publisher = {Springer International Publishing}, address = {Berlin}, isbn = {978-3-319-15089-5}, doi = {10.1007/978-3-319-15090-1}, pages = {129 -- 148}, language = {en} }