Recent publications of the research group

	Geometry Processing - Discrete Geometry for Virtual Worlds (.pdf 0.8 MB) Konrad Polthier in: ICIAM - Zurich Intelligencer, Springer Verlag, 2007, pp. 42-43.

	Ulrich Bauer and Konrad Polthier: Parametric Reconstruction of Bent Tube Surfaces (pdf). in: CyberWorld 2007 Conference Proceedings, Workshop "New Advances in Shape Analysis and Geometric Modeling", IEEE 2007. We present a method for parametric reconstruction of a piecewise defined pipe surface, consisting of cylinder and torus segments, from an unorganized point set. Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G1 continuous circular arcs and line segments. Our algorithm accurately outputs the parametric data required for bending machines to create the reconstructed tube.
	QuadCover - Surface Parameterization using Branched Coverings (.pdf 1.9 MB) Felix Kälberer, Matthias Nieser and Konrad Polthier in: Computer Graphics Forum 26 (3), 2007, pp. 375-384. Presented at Eurographics 2007. We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2-manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2-manifold, and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.
	Constraint-based fairing of surface meshes (.pdf 6.9 MB) Klaus Hildebrandt and Konrad Polthier in: Symposium on Geometry Processing 2007, pp. 203-212. We propose a constraint-based method for the fairing of surface meshes. The main feature of our approach is that the resulting smoothed surface remains within a prescribed distance to the input mesh. For example, specifying the maximum distance in the order of the measuring precision of a laser scanner allows noise to be removed while preserving the accuracy of the scan. The approach is modeled as an optimization problem where a fairness measure is minimized subject to constraints that control the spatial deviation of the surface. The problem is efficiently solved by an active set Newton method.
	On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces (.pdf 0.4 MB) Klaus Hildebrandt, Konrad Polthier and Max Wardetzky Geometria Dedicata 123 (2006), pp. 89-112. We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in R3. 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 show 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.
	FreeLence - Coding with free valences (.pdf 5.6 MB) Felix Kälberer, Konrad Polthier, Ulrich Reitebuch and Max Wardetzky in: Computer Graphics Forum 24 (3), 2005, pp.469-478. Presented at Eurographics 2005. We introduce FreeLence, a novel and simple single-rate compression coder for triangle manifold meshes.  Our method uses free valences and exploits geometric information for connectivity encoding. Furthermore, we introduce a novel linear prediction scheme for geometry compression of 3D meshes. Together, these approaches yield a significant entropy reduction for mesh encoding with an average of 30% over leading single-rate region-growing coders, both for connectivity and geometry.
	Anisotropic Fairing of Point Sets (.pdf, 3.3 MB, videos of simulations) Carsten Lange and Konrad Polthier, ZIB-Preprint 05-16. in: Special Issue of CAGD 2005 (U. Reif Ed.) 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.
	Smooth Feature Lines on Surface Meshes (.pdf, 1.6 MB) Klaus Hildebrandt, Konrad Polthier and Max Wardetzky, in: Symposium on Geometry Processing 2005, M. Desbrun and H. Pottmann (Eds.). Presented at SGP 2005. Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two novel concepts for a more stable algorithm producing visually more pleasing feature lines: First, a new computation scheme based on discrete differential geometry is presented, avoiding costly computations of higher order approximating surfaces. Secondly, this scheme is augmented by a filtering method for higher order surface derivatives to improve both the stability of the extraction of feature lines and the smoothness of their appearance.
	Evolution of 3d Curves under Strict Spatial Constraints (pdf). Klaus Hildebrandt, Konrad Polthier and Eike Preuss Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005). 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.
	Anisotropic Filtering of Non-Linear Surface Features (.pdf 2.5 MB, videos of simulations) Klaus Hildebrandt and Konrad Polthier, ZIB Preprint, 04-25. in: Computer Graphics Forum, 23 (3), 2004. - Best Student Paper Award at Eurographics 2004 - 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 edges and surface regions. Our method uses a non-linear anisotropic geometric diffusion flow for polyhedral surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on polyhedral surfaces that is fully consistent with the known discrete mean curvature representation, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behavior along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow converging to surfaces with prescribed mean curvature which preserves non-linear features. Additionally our discrete flow is very well suited to prevent boundary shrinkage at constrained and free boundary segments.
	Computational Aspects of Discrete Minimal Surfaces (.pdf 1.4 MB) Konrad Polthier in: Global Theory of Minimal Surfaces, Proc. of the Clay Mathematics Institute Summer School, D. Hoffman (Ed.), CMI/AMS, (2005). In differential geometry the study of smooth submanifolds with distinguished curvature properties has a long history and belongs to the central themes of this field. Modern work on smooth submanifolds, and on surfaces in particular, relies heavily on geometric and analytic machinery which has evolved over hundreds of years. However, non-smooth surfaces are also natural mathematical objects, even though there is less machinery available for studying them. Consider, for example, the pioneering work on polyhedral surfaces by the Russian school around Alexandrov [Aleksandrov/Zalgaller67Intrinsic], or Gromov's approach of doing geometry using only a set with a measure and a measurable distance function [Gromov99Metric]. Also in other fields, for example in computer graphics and scientific computing, we nowadays encounter a strong need for a discrete differential geometry of arbitrary meshes. These tutorial notes introduce the theory and computation of discrete minimal surfaces which are characterized by variational properties, and are based on a part of the authors Habilitationsschrift [Polthier02Habilitationsschrift]. In Section we introduce simplicial surfaces and their function spaces. Laplace-Beltrami harmonic maps and the solution of the discrete Cauchy-Riemann equations are introduced on simplicial surfaces in Section . These maps are the basis for an iterative algorithm to compute discrete minimal and constant mean curvature surfaces which is discussed in Section . There we define the discrete mean curvature operator, derive the associate family of discrete minimal surfaces in terms of conforming and non-conforming triangles meshes, and present some recently discovered complete discrete surfaces, the family of discrete catenoids and helicoids.
:	Using MuPAD and JavaView to Visualize Mathematics on the Internet (.pdf 0.6 MB) Mirek Majewski and Konrad Polthier in: Proc. of the 9th Asian Technology Conference in Mathematics, (2004), pp. 465-474. Mathematics education strongly benefits from the interactivity and advanced features of the Internet. The presentation of mathematical concepts on the Internet may go far beyond what we could demonstrate in traditional mathematics textbooks. In this paper we demonstrate, in a number of examples, the additional insight into complex mathematical concepts that can be gained from 3D interactive visualization embedded into web pages. Mathematical visualization is improving our teaching environments and the communication between teachers and students. We combine two mathematical software systems—MuPAD as a development platform and JavaView for computation and online visualization of interactive mathematics experiments. We discuss the practical aspects of online publications and show some technical details about how to develop mathematical experiments on your own. The conference presentation will demonstrate MuPAD and JavaView components in live-action.
	Visualizing Maple Plots with JavaViewLib (.pdf 3.5 MB, .ps.zip 2.6 MB) Steven P. Dugaro and Konrad Polthier in: Algebra, Geometry, and Software Systems, M. Joswig, N. Takayama (Eds) Springer Verlag (2003), pp. 255-275. JavaViewLib is a new Maple package combined with the JavaView visualization toolkit that adds new interactivity to Maple plots in both web pages and worksheets. It provides a superior viewing environment to enhance plots in Maple by adding several features to plots' interactivity, such as mouse-controlled scaling, translation, rotation in 2d, 3d, and 4d, auto-view modes, animation, picking, material colors, texture and transparency. The arc-ball rotation makes geometry viewing smoother and less directionally constrained than in Maple. Furthermore, it offers geometric modeling features that allow plots to be manipulated and imported into a worksheet. Several commands are available to export Maple plots to interactive web pages while keeping interactivity. JavaViewLib is available as an official Maple Powertool.
	Identifying Vector Field Singularities using a Discrete Hodge Decomposition (.pdf 2.4 MB, .ps.zip 3.2 MB) Konrad Polthier and Eike Preuß in: Visualization and Mathematics III, Eds: H.C. Hege, K. Polthier, Springer Verlag (2003), pp. 113-134. We derive a Hodge decomposition of discrete vector fields on polyhedral surfaces, and apply it to the identification of vector field singularities. This novel approach allows us to easily detect and analyze singularities as critical points of corresponding potentials. Our method uses a global variational approach to independently compute two potentials whose gradient respectively co-gradient are rotation-free respectively divergence-free components of the vector field. The sinks and sources respectively vortices are then automatically identified as the critical points of the corresponding scalar-valued potentials. The global nature of the decomposition avoids the approximation problem of the Jacobian and higher order tensors used in local methods, while the two potentials plus a harmonic flow component are an exact decomposition of the vector field containing all information.
	Unstable Periodic Discrete Minimal Surfaces (.pdf 2.7 MB, .ps.zip 6.2 MB) Konrad Polthier in: Nonlinear Partial Differential Equations, S. Hildebrandt and H. Karcher (Eds.) Springer Verlag (2002), pp. 127-143. In this paper we define the new alignment energy for non-conforming triangle meshes, and describes its use to compute unstable conforming discrete minimal surfaces. Our algorithm makes use of the duality between conforming and non-conforming discrete minimal surfaces which was observed earlier. In first experiments the new algorithm allows us the computation of unstable periodic discrete minimal surfaces of high numerical precision. The extraordinary precision of the discrete mesh enables us to compute the index of several triply periodic minimal surfaces.

© 1996-2009 Last modified: 18.10.2009 --- Konrad Polthier --- Freie Universität Berlin, Germany