TY - GEN A1 - Hege, Hans-Christian A1 - Polthier, Konrad T1 - Visualization and Mathematics '97 N2 - This report contains paper abstracts of the workshop "Visualization and Mathematics" held in Berlin-Dahlem in September 1997. The meeting serves as a forum for an international community of researchers and practitioners on the application of visualization techniques in mathematics and on mathematical concepts in visualization. It is the second symposium in a series of workshops bringing together mathematicians and experts from scientific visualization. The themes of the workshop include: \begin{itemize} \item - applications in differential geometry and partial differential equations \item - algorithmic aspects of adaptive and hierarchical techniques in space and time \item - time control of animated objects and corresponding algorithms \item - algorithmic representation of objects for display, storage and exchange \item - new visualization techniques for mathematical structures \item - integration of visualization with symbolic and numerical computation. \end{itemize} T3 - ZIB-Report - TR-97-06 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5480 ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Wardetzky, Max T1 - On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces N2 - We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidian $3$-space. The notion of totally normal convergence is shown to be equivalent to the convergence of either one of the following: surface area, intrinsic metric, and Laplace-Beltrami operators. We further s how that totally normal convergence implies convergence results for shortest geodesics, mean curvature, and solutions to the Dirichlet problem. This work provides the justification for a discrete theory of differential geometric operators defined on polyhedral surfaces based on a variational formulation. T3 - ZIB-Report - 05-24 KW - Discrete Geometry KW - Differential Geometry KW - Numerical Analysis Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8587 ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Preuß, Eike T1 - Evolution of 3d Curves under Strict Spatial Constraints N2 - We present a new algorithm for fairing of space curves with respect spatial constraints based on a vector valued curvature function. Smoothing with the vector valued curvature function is superior to standard Frenet techniques since the individual scalar components can be modeled similar to curvature-based curve smoothing techniques in 2d. This paper describes a curve smoothing flow that satisfies strict spatial constraints and allows simultaneous control of both curvature functions. T3 - ZIB-Report - 05-33 KW - curve fairing KW - curve smoothing KW - curvature smoothing flow Y1 - 2005 N1 - kein Volltext, Artikel ist in Verlagspublikation erschienen, der Preprint wurde offensichtlich vor längerer Zeit zurückgezogen (keine urn). Kossick 20. 11. 13 ER - TY - GEN A1 - Hege, Hans-Christian A1 - Polthier, Konrad T1 - Visualization and Mathematics. International Workshop N2 - International Workshop with support of the Deutsche Forschungsgemeinschaft (DFG) and Max-Planck-Gesellschaft (MPG) Berlin (Dahlem), Germany, May 30 - June 2, 1995 T3 - ZIB-Report - TR-95-02 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5179 ER - TY - GEN A1 - Götschel, Sebastian A1 - von Tycowicz, Christoph A1 - Polthier, Konrad A1 - Weiser, Martin T1 - Reducing Memory Requirements in Scientific Computing and Optimal Control N2 - In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data. T3 - ZIB-Report - 13-64 KW - optimal control KW - trajectory storage KW - mesh compression KW - compression Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42695 SN - 1438-0064 ER - TY - CHAP A1 - Majewski, Mirek A1 - Polthier, Konrad T1 - Using MuPAD and JavaView to visualize mathematics on the Internet T2 - Proc. of the 9th Asian Technology Conference in Mathematics Y1 - 2004 SP - 465 EP - 474 ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Polthier, Konrad T1 - Anisotropic filtering of non-linear surface features JF - Computer Graphics Forum Y1 - 2004 VL - 23(3) SP - 391 EP - 400 CY - Grenoble, France ER - TY - GEN A1 - Hege, Hans-Christian A1 - Polthier, Konrad A1 - Rumpf, Martin T1 - MathInsight 2002 Y1 - 2001 ER - TY - BOOK A1 - Hege, Hans-Christian A1 - Polthier, Konrad T1 - VideoMath - Festival at ICM 98 T3 - Springer VideoMath Y1 - 1998 SN - 978-3-540-21385-7 ER - TY - GEN A1 - Polthier, Konrad A1 - Sullivan, John A1 - Ziegler, Günter M. A1 - Hege, Hans-Christian ED - Deuflhard, Peter ED - et al., T1 - Visualization T2 - MATHEON - Mathematics for Key Technologies Y1 - 2014 SN - 978-3-03719-137-8 U6 - https://doi.org/10.4171/137 SP - 335 EP - 339 PB - European Mathematical Society ER - TY - JOUR A1 - Kälberer, Felix A1 - Polthier, Konrad A1 - Reitebuch, Ulrich A1 - Wardetzky, Max T1 - Free Lence - Coding with free valences JF - Computer Graphics Forum Y1 - 2005 U6 - https://doi.org/10.1111/j.1467-8659.2005.00872.x VL - 24 IS - Issue 3 SP - 469 EP - 478 ER - TY - CHAP A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Wardetzky, Max T1 - Smooth feature lines on surface meshes T2 - Third Eurographics Symposium on Geometry Processing Y1 - 2005 SP - 85 EP - 90 ER - TY - CHAP A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Preuß, Eike T1 - Evolution of 3D curves under strict spatial constraints T2 - IEEE Proceedings on Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005) Y1 - 2005 U6 - https://doi.org/10.1109/CAD-CG.2005.38 SP - 40 EP - 45 ER - TY - CHAP A1 - Polthier, Konrad ED - Hass, J. ED - Hoffman, D. ED - Jaffe, A. ED - Rosenberg, H. ED - Schoen, R. ED - Wol, M. T1 - Computational aspects of discrete minimal surfaces T2 - Global Theory of Minimal Surfaces, Proc. of the Clay Mathematics Institute Summer School, CMI/AMS Y1 - 2005 ER - TY - JOUR A1 - Lange, Carsten A1 - Polthier, Konrad T1 - Anisotropic fairing of point sets JF - Special Issue of CAGD 2005 Y1 - 2005 U6 - https://doi.org/10.1016/j.cagd.2005.06.010 ER - TY - CHAP A1 - Götschel, Sebastian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad A1 - Weiser, Martin ED - Carraro, T. ED - Geiger, M. ED - Koerkel, S. ED - Rannacher, R. T1 - Reducing Memory Requirements in Scientific Computing and Optimal Control T2 - Multiple Shooting and Time Domain Decomposition Methods Y1 - 2015 SP - 263 EP - 287 PB - Springer ER - TY - CHAP A1 - Polthier, Konrad A1 - Bobenko, Alexander A1 - Hildebrandt, Klaus A1 - Kornhuber, Ralf A1 - Tycowicz, Christoph von A1 - Yserentant, Harry A1 - Ziegler, Günter M. T1 - Geometry processing T2 - MATHEON - Mathematics for Key Technologies Y1 - 2014 SN - 978-3-03719-137-8 U6 - https://doi.org/10.4171/137 VL - 1 SP - 341 EP - 355 PB - EMS Publishing House ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Interactive spacetime control of deformable objects JF - ACM Transactions on Graphics N2 - Creating motions of objects or characters that are physically plausible and follow an animator’s intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we propose a technique that controls the motion of deformable objects and offers an interactive response. This is achieved by a model reduction of the underlying variational problem, which combines dimension reduction, multipoint linearization, and decoupling of ODEs. After a preprocess, the cost for creating or editing a motion is reduced to solving a number of one-dimensional spacetime problems, whose solutions are the wiggly splines introduced by Kass and Anderson [2008]. We achieve interactive response using a new fast and robust numerical scheme for solving a set of one-dimensional problems based on an explicit representation of the wiggly splines. Y1 - 2012 U6 - https://doi.org/10.1145/2185520.2185567 VL - 31 IS - 4 SP - 71:1 EP - 71:8 PB - ACM ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Interactive surface modeling using modal analysis JF - ACM Transactions on Graphics N2 - We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008]. Y1 - 2011 U6 - https://doi.org/10.1145/2019627.2019638 VL - 30 IS - 5 SP - 119:1 EP - 119:11 ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Modal Shape Analysis beyond Laplacian JF - Computer Aided Geometric Design N2 - In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to the discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies, which operate on a function space on the surface, or of deformation energies, which operate on a shape space. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces. Y1 - 2012 U6 - https://doi.org/10.1016/j.cagd.2012.01.001 VL - 29 IS - 5 SP - 204 EP - 2018 ER - TY - JOUR A1 - Tycowicz, Christoph von A1 - Kälberer, Felix A1 - Polthier, Konrad T1 - Context-based Coding of Adaptive Multiresolution Meshes JF - Computer Graphics Forum N2 - Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression scheme for these adaptive structures that exploits the parent-child relationships inherent to the mesh hierarchy. We use the rules that correspond to the adaptive refinement scheme and store bits only where some freedom of choice is left, leading to compact codes that are free of redundancy. Moreover, we extend the coder to sequences of meshes with varying refinement. The connectivity compression ratio of our method exceeds that of state-of-the-art coders by a factor of 2 to 7. For efficient compression of vertex positions we adapt popular wavelet-based coding schemes to the adaptive triangular and quadrangular cases to demonstrate the compatibility with our method. Akin to state-of-the-art coders, we use a zerotree to encode the resulting coefficients. Using improved context modeling we enhanced the zerotree compression, cutting the overall geometry data rate by 7% below those of the successful Progressive Geometry Compression. More importantly, by exploiting the existing refinement structure we achieve compression factors that are 4 times greater than those of coders which can handle irregular meshes. Y1 - 2011 U6 - https://doi.org/10.1111/j.1467-8659.2011.01972.x VL - 30 IS - 8 SP - 2231 EP - 2245 ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad ED - Mourrain, Bernard ED - Schaefer, Scott ED - Xu, Guoliang T1 - Eigenmodes of Surface Energies for Shape Analysis T2 - Advances in Geometric Modeling and Processing (Proceedings of Geometric Modeling and Processing 2010) N2 - In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Furthermore, we design a quadratic energy, such that the eigenmodes of the Hessian of this energy are sensitive to the extrinsic curvature of the surface. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of the surface. In addition, we discuss a spectral quadrangulation scheme for surfaces. Y1 - 2010 U6 - https://doi.org/10.1007/978-3-642-13411-1_20 VL - 6130 SP - 296 EP - 314 PB - Springer Berlin / Heidelberg ER - TY - CHAP A1 - Kälberer, Felix A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Lossless Compression of Adaptive Multiresolution Meshes T2 - XXII Brazilian Symposium on Computer Graphics and Image Processing N2 - We present a novel coder for lossless compression of adaptive multiresolution meshes that exploits their special hierarchical structure. The heart of our method is a new progressive connectivity coder that can be combined with leading geometry encoding techniques. The compressor uses the parent/child relationships inherent to the hierarchical mesh. We use the rules that accord to the refinement scheme and store bits only where it leaves freedom of choice, leading to compact codes that are free of redundancy. To illustrate our scheme we chose the widespread red-green refinement, but the underlying concepts can be directly transferred to other adaptive refinement schemes as well. The compression ratio of our method exceeds that of state-of-the-art coders by a factor of 2 to 3 on most of our benchmark models. Y1 - 2009 U6 - https://doi.org/10.1109/SIBGRAPI.2009.53 SP - 80 EP - 87 ER - TY - JOUR A1 - Nava-Yazdani, Esfandiar A1 - Polthier, Konrad T1 - De Casteljau's Algotithm on Manifolds JF - Computer Aided Geometric Design N2 - This paper proposes a generalization of the ordinary de Casteljau algorithm to manifold-valued data including an important special case which uses the exponential map of a symmetric space or Riemannian manifold. We investigate some basic properties of the corresponding Bézier curves and present applications to curve design on polyhedra and implicit surfaces as well as motion of rigid body and positive definite matrices. Moreover, we apply our approach to construct canal and developable surfaces. Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69096 VL - 30 IS - 7 SP - 722 EP - 732 PB - CAGD ER - TY - GEN A1 - Kälberer, Felix A1 - Polthier, Konrad A1 - Reitebuch, Ulrich A1 - Wardetzky, Max T1 - Compressing Triangle Meshes using Geometric Infomation N2 - We introduce FreeLence, a lossless single-rate connectivity compression algorithm for triangle surface meshes. Based upon a geometry-driven traversal scheme we present two novel and simple concepts: free-valence connectivity encoding and entropy coding based on geometric context. Together these techniques yield significantly smaller rates for connectivity compression than current state of the art approaches - valence-based algorithms and Angle- Analyzer, with an average of $36\%$ improvement over the former and an average of $18\%$ over the latter on benchmark 3D models, combined with the ability to well adapt to the regularity of meshes. We also prove that our algorithm exhibits a smaller worst case entropy for a class of "'well-behaved"' triangle meshes than valence-driven connectivity encoding approaches. T3 - ZIB-Report - 04-26 KW - Mesh Compression KW - Connectivity Encoding KW - Polygonal Meshes KW - Context-based Entropy Coding Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8016 ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad T1 - Anisotropic Filtering of Non-Linear Surface Features N2 - A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well-known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non-linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments. T3 - ZIB-Report - 04-25 KW - mesh smoothing KW - principal curvature KW - feature lines KW - anisotropic smoothing KW - prescribed mean curvature Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8003 ER - TY - GEN A1 - Lange, Carsten A1 - Polthier, Konrad T1 - Anisotropic Smoothing of Point Sets N2 - The use of point sets instead of meshes became more popular during the last years. We present a new method for anisotropic fairing of a point sampled surface using an anisotropic geometric mean curvature flow. The main advantage of our approach is that the evolution removes noise from a point set while it detects and enhances geometric features of the surface such as edges and corners. We derive a shape operator, principal curvature properties of a point set, and an anisotropic Laplacian of the surface. This anisotropic Laplacian reflects curvature properties which can be understood as the point set analogue of Taubin's curvature-tensor for polyhedral surfaces. We combine these discrete tools with techniques from geometric diffusion and image processing. Several applications demonstrate the efficiency and accuracy of our method. T3 - ZIB-Report - 05-16 KW - point sets KW - anisotropic mean curvature KW - anisotropic sampling KW - shape operator KW - principal curvatures KW - anisotropic Laplace KW - mean curvature flow KW - f Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8508 ER -