TY - GEN A1 - Kleefeld, Andreas A1 - Kirsch, Andreas T1 - The factorization method for a conductive boundary condition T2 - Journal of Integral Equations and Applications N2 - 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. Y1 - 2012 UR - http://projecteuclid.org/euclid.jiea/1362669522 U6 - https://doi.org/10.1216/JIE-2012-24-4-575 SN - 0897-3962 VL - 24 IS - 4 SP - 575 EP - 601 ER - TY - GEN A1 - Kleefeld, Andreas A1 - Reißel, Martin T1 - The Levenberg-Marquardt method applied to a parameter estimation problem arising from electrical resistivity tomography T2 - Journal of Applied Mathematics and Computation N2 - 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. KW - Levenberg-Marquardt method KW - Fréchett derivative KW - nonlinear and ill-posed problem KW - estimation problem KW - parameter estimation problem Y1 - 2011 SN - 0096-3003 VL - 217 IS - 9 SP - 4490 EP - 4501 ER - TY - GEN A1 - Kleefeld, Andreas A1 - Brazauskas, Vytaras T1 - Folded and log-folded-t distributions as models for insurance loss data N2 - 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. Y1 - 2011 U6 - https://doi.org/http://www.tandfonline.com/doi/abs/10.1080/03461230903424199 ER - TY - GEN A1 - Kleefeld, Andreas T1 - Linear Fredholm integral equations of the first kind over surfaces in 3D N2 - 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. Y1 - 2011 UR - http://www.tandfonline.com/doi/abs/10.1080/00207160.2011.554543 ER - TY - GEN A1 - Kleefeld, Andreas A1 - Lin, Tzu-Chu T1 - The nonlinear Landweber method applied to an inverse scattering problem for sound-soft obstacles in 3D N2 - 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. Y1 - 2011 UR - http://www.sciencedirect.com/science/article/pii/S0010465511002724 ER - TY - GEN A1 - Kleefeld, Andreas A1 - Lin, Tzu-Chu T1 - Boundary element collocation method for solving the exterior Neumann problem for Helmholtz's equation in three dimensions T2 - Electronic transactions on numerical analysis N2 - 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. KW - Fredholm integral equation of the second kind, Helmholtz's equation, exterior Neumann problem, boundary element collocation method, superconvergence Y1 - 2012 UR - http://etna.mcs.kent.edu/vol.39.2012/pp113-143.dir/pp113-143.pdf SN - 1068-9613 VL - 39 SP - 113 EP - 143 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 - 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 - 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 - The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions T2 - International Journal of Computer Mathematics N2 - 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. KW - Helmholtz equation, integral equation, mixed boundary conditions, boundary element collocation method, superconvergence Y1 - 2012 UR - http://www.tandfonline.com/doi/full/10.1080/00207160.2012.709932 U6 - https://doi.org/10.1080/00207160.2012.709932 SN - 0020-7160 SN - 1029-0265 VL - 89 IS - 17 SP - 2392 EP - 2409 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 - 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 - 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 - 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 - 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 - 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 - GEN A1 - Kleefeld, Andreas A1 - Burgeth, Bernhard T1 - An approach to color-morphology based on Einstein addition and Loewner order T2 - Pattern Recognition Letters N2 - 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. KW - Color space KW - Loewner order KW - Einstein addition KW - Mathematical morphology Y1 - 2014 UR - http://www.sciencedirect.com/science/journal/01678655/47 U6 - https://doi.org/10.1016/j.patrec.2014.01.018 SN - 0167-8655 VL - 47 SP - 29 EP - 39 ER - TY - GEN A1 - Kleefeld, Andreas A1 - Brazauskas, Vytaras T1 - Authors 'Reply to' Letter to the Editor regarding folded models and the paper by Brazauskas and Kleefeld (2011) T2 - Scandinavian Actuarial Journal Y1 - 2014 UR - http://www.tandfonline.com/doi/abs/10.1080/03461238.2013.798070#.U9YDOrExFIl U6 - https://doi.org/10.1080/03461238.2013.798070 SN - 0346-1238 IS - 8 SP - 753 EP - 757 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 - 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 -