@misc{Kleefeld, author = {Kleefeld, Andreas}, title = {The transmission problem for the Helmholtz equation in R³}, series = {Computational Methods in Applied Mathematics}, volume = {12}, journal = {Computational Methods in Applied Mathematics}, number = {3}, issn = {1609-4840}, pages = {330 -- 350}, abstract = {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.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {A modified boundary integral equation for solving the exterior Robin problem for the Helmholtz equation in three dimensions}, series = {Applied Mathematics and Computation}, volume = {2012}, journal = {Applied Mathematics and Computation}, number = {219}, issn = {0096-3003}, doi = {10.1016/j.amc.2012.08.055}, pages = {2114 -- 2123}, abstract = {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.}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distribution}, series = {European Journal of Applied Mathematics}, volume = {23}, journal = {European Journal of Applied Mathematics}, number = {5}, issn = {0956-7925}, doi = {10.1017/S0956792512000137}, pages = {593 -- 610}, abstract = {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.}, 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} } @incollection{KleefeldBurgeth, author = {Kleefeld, Andreas and Burgeth, Bernhard}, title = {Morphology for Color Images via Loewner Order for Matrix Fields}, series = {Mathematical Morphology and Its Applications to Signal and Image Processing}, booktitle = {Mathematical Morphology and Its Applications to Signal and Image Processing}, editor = {Hendriks, Cris L. Luengo and Borgefors, Gunilla and Strand, Robin}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-642-38293-2}, doi = {10.1007/978-3-642-38294-9_21}, pages = {243 -- 254}, abstract = {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.}, language = {en} } @misc{KleefeldLin, author = {Kleefeld, Andreas and Lin, Tzu-Chu}, title = {A Global Galerkin Method for Solving the Exterior Neumann Problem for the Helmholtz Equation Using Panich's Integral Equation Approach}, series = {SIAM Journal on Scientific Computing}, volume = {35}, journal = {SIAM Journal on Scientific Computing}, number = {3}, issn = {1064-8275}, doi = {10.1137/120873066}, pages = {A1709 -- A1735}, abstract = {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.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {A numerical method to compute interior transmission eigenvalues}, series = {Inverse Problems}, journal = {Inverse Problems}, number = {29}, issn = {0266-5611}, doi = {10.1088/0266-5611/29/10/104012}, pages = {104012}, abstract = {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.}, language = {en} } @misc{AnagnostōpulosCharalambopoulosKleefeld, author = {Anagnostōpulos, Kōnstantinos N. and Charalambopoulos, Antonios and Kleefeld, Andreas}, title = {The factorization method for the acoustic transmission problem}, series = {Inverse Problems}, volume = {29}, journal = {Inverse Problems}, number = {11}, issn = {0266-5611}, doi = {10.1088/0266-5611/29/11/115015}, pages = {115015}, abstract = {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.}, language = {en} } @inproceedings{KleefeldJuTozzaetal., author = {Kleefeld, Andreas and Ju, Yong Chul and Tozza, Silvia and Breuß, Michael and Bruhn, Andr{\´e}s}, title = {Generalised Perspective Shape from Shading with Oren-Nayar Reflectance}, series = {Proceedings 24th British Machine Vision Conference, BMVC 2013, Bristol, UK, September 2013}, booktitle = {Proceedings 24th British Machine Vision Conference, BMVC 2013, Bristol, UK, September 2013}, editor = {Burghardt, Tilo and Damen, Dima and Mayol-Cuevas, Walterio and Mirmehdi, Majid}, publisher = {BMVA Press}, doi = {10.5244/C.27.42}, pages = {42.1 -- 42.11}, abstract = {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.}, language = {en} } @incollection{KleefeldBurgeth, author = {Kleefeld, Andreas and Burgeth, Bernhard}, title = {Order based morphology for color images via matrix fields}, series = {Visualization and Processing of Tensor Fields and Higher Order Descriptors for Multi-Valued Data}, booktitle = {Visualization and Processing of Tensor Fields and Higher Order Descriptors for Multi-Valued Data}, editor = {Burgeth, Bernhard and Vilanova, Anna and Westin, Carl-Fredrik}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-642-54300-5}, doi = {10.1007/978-3-642-54301-2_4}, pages = {75 -- 95}, abstract = {Mathematical morphology is a successful branch of image processing with a history of more than four decades. Its fundamental operations are dilation and erosion, which are based on the notion of supremum and infimum with respect to an order. From dilation and erosion one can build readily other useful elementary morphological operators and filters, such as opening, closing, morphological top-hats, derivatives, and shock filters. Such operators are available for grey value images, and recently useful analogs of these processes for matrix-valued images have been introduced by taking advantage of the so-called Loewner order. There is a number of approaches to morphology for vector-valued images, that is, color images based on various orders, however, each with its merits and shortcomings. In this chapter we propose an approach to (elementary) morphology for color images that relies on the existing order based morphology for matrix fields of symmetric 2 × 2-matrices. An RGB-image is embedded into a field of those 2 × 2-matrices by exploiting the geometrical properties of the order cone associated with the Loewner order. To this end a modification of the HSL-color model and a relativistic addition of matrices is introduced. The experiments performed with various morphological elementary operators on synthetic and real images provide results promising enough to serve as a proof-of-concept.}, language = {en} } @misc{KleefeldBurgeth, author = {Kleefeld, Andreas and Burgeth, Bernhard}, title = {An approach to color-morphology based on Einstein addition and Loewner order}, series = {Pattern Recognition Letters}, volume = {47}, journal = {Pattern Recognition Letters}, issn = {0167-8655}, doi = {10.1016/j.patrec.2014.01.018}, pages = {29 -- 39}, abstract = {In this article, a new method to process color images via mathematical morphology is presented. Precisely, each pixel of an image contained in the rgb-space is converted into a 2 x 2-matrix representing a point in a color bi-cone. The supremum and infimum needed for dilation and erosion, respectively, are calculated with respect to the Loewner order. Since the result can be outside the bi-cone, a map is proposed to ensure the algebraic closure in the bi-cone. A new addition and subtraction based on Einstein's Relativity Theory is used to define morphological operation such as top-hats, gradients, and morphological Laplacian. Since the addition and subtraction is defined in the unit ball, a suitable mapping between those two spaces is constructed. A comparison with the component-wise approach and the full ordering using lexicographical cascades approach demonstrate the feasibility and capabilities of the proposed approach.}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {Authors 'Reply to' Letter to the Editor regarding folded models and the paper by Brazauskas and Kleefeld (2011)}, series = {Scandinavian Actuarial Journal}, journal = {Scandinavian Actuarial Journal}, number = {8}, issn = {0346-1238}, doi = {10.1080/03461238.2013.798070}, pages = {753 -- 757}, language = {en} } @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} }