Max Wardetzky

I have the pleasure of heading the Discrete Differential Geometry Lab at the University of Goettingen, with its wonderful people.

Research interests

Discrete Differential Geometry, Geometry Processing, Computational Topology, Physical Simulation, Computer Graphics.

Contact

Institute of Num. and Appl. Math University of Göttingen Lotzestr. 16-18 37083 Göttingen, Germany

Vox: +49 (0)551 39-22235 Fax: +49 (0)551 39-33944 Net: lastname (at) math.uni-goettingen.de

Publications: Refereed Articles

	Time-Discrete Geodesics in the Space of Shells Behrend Heeren, Martin Rumpf, Max Wardetzky and Benedikt Wirth. Comput. Graph. Forum 31(5):1755–1764, 2012. Abstract: Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so-called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path. [pdf]
	Flexible Developable Surfaces Justin Solomon, Etienne Vouga, Max Wardetzky, Eitan Grinspun. Comput. Graph. Forum 31(5):1567–1576, 2012. Abstract: We introduce a discrete paradigm for developable surface modeling. Unlike previous attempts at interactive developable surface modeling, our system is able to enforce exact developability at every step, ensuring that users do not inadvertently suggest configurations that leave the manifold of admissible folds of a flat two-dimensional sheet. With methods for navigation of this highly nonlinear constraint space in place, we show how to formulate a discrete mean curvature bending energy measuring how far a given discrete developable surface is from being flat. This energy enables relaxation of user-generated configurations and suggests a straightforward subdivision scheme that produces admissible smoothed versions of bent regions of our discrete developable surfaces. [pdf]
	Optimal topological simplification of discrete functions on surfaces Ulrich Bauer, Carsten Lange, and Max Wardetzky. Discrete and Computational Geometry 47:2 (2012), 347–377. Abstract: Given a function f on a surface and a tolerance δ > 0, we construct a function fδ subject to ‖fδ - f‖∞ ≤ δ such that fδ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤ 2δ from the input function f. The number of critical points of the resulting simplified function fδ achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed. [pdf] [doi]
	Discrete Laplacians on General Polygonal Meshes Marc Alexa and Max Wardetzky, ACM Transaction on Graphics 30:4 (SIGGRAPH), pp. 102:1--102:10, 2011. Abstract: While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace-Beltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes. [pdf] [code]
	Total Variation Meets Topological Persistence: A First Encounter Ulrich Bauer, Carola-Bibiane Schönlieb, and Max Wardetzky. Proceedings of ICNAAM 2010, pp. 1022-1026. Abstract: We present first insights into the relation between two popular yet apparently dissimilar approaches to denoising of one dimensional signals, based on (i) total variation (TV) minimization and (ii) ideas from topological persistence. While a close relation between (i) and (ii) might phenomenologically not be unexpected, our work appears to be the first to make this connection precise for one dimensional signals. We provide a link between (i) and (ii) that builds on the equivalence between TV-L2 regularization and taut strings and leads to a novel and efficient denoising algorithm that is contrast preserving and operates in O(nlogn) time, where n is the size of the input. [pdf]
	Discrete Viscous Threads Miklos Bergou, Basile Audoly, Etienne Vouga, Max Wardetzky, Eitan Grinspun, ACM Transaction on Graphics 29:4 (SIGGRAPH), pp. 116:1-116:10, 2010. Abstract: We present a continuum-based discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time symmetry, which sheds light on the role of time-parallel transport in eliminating - without approximation - all but an O(n) band of entries of the physical system's energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluidmechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics. [pdf] [Video]
	Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines Ulrich Bauer, Konrad Polthier, Max Wardetzky, Discrete and Computational Geometry 43:4, 798–823, 2010. Abstract: We study “Steiner-type” discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform pointwise convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on the limit surface and the shape regularity of the discrete net. [pdf] [doi]
	Discrete Elastic Rods Miklos Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, Eitan Grinspun, ACM Transaction on Graphics 27:3 (SIGGRAPH), pp. 63:1-63:12, 2008. Abstract: We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description - we represent the material frame by its angular deviation from the natural Bishop frame - as well as in the dynamical treatment - we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons. [pdf] [Video]
	Convergence of the Cotangent Formula: An Overview Max Wardetzky, in "Discrete Differential Geometry" (A. I. Bobenko, John M. Sullivan, Peter Schröder, Günter Ziegler, eds.), Birkhäuser Basel, 2008. Abstract: The cotangent formula constitutes an intrinsic discretization of the Laplace- Beltrami operator on polyhedral surfaces in a finite element sense. This note gives an overview of approximation and convergence properties of discrete Laplacians and mean curvature vectors for polyhedral surfaces located in the vicinity of a smooth surface in Euclidean 3-space. In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L^2. [pdf]
	TRACKS: Toward Directable Thin Shells Miklos Bergou, Saurabh Mathur, Max Wardetzky, Eitan Grinspun, ACM Transaction on Graphics 26:3 (SIGGRAPH), pp. 50:1-50:10, 2007. Abstract: We combine the often opposing forces of artistic freedom and mathematical determinism to enrich a given animation or simulation of a surface with physically based detail. We present a process called tracking, which takes as input a rough animation or simulation and enhances it with physically simulated detail. Building on the foundation of constrained Lagrangian mechanics, we propose weak-form constraints for tracking the input motion. This method allows the artist to choose where to add details such as characteristic wrinkles and folds of various thin shell materials and dynamical effects of physical forces. We demonstrate multiple applications ranging from enhancing an artist's animated character to guiding a simulated inanimate object. [pdf] [Video]
	Discrete Quadratic Curvature Energies Max Wardetzky, Miklos Bergou, David Harmon, Denis Zorin, Eitan Grinspun, Computer Aided Geometric Design (CAGD) 24, 2007, pp. 499-518. Abstract: We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics with a two- to three-fold net speedup over existing nonlinear methods, and near-interactive rates for Willmore smoothing of large meshes. [pdf] [Video]
	Discrete Laplace operators: No free lunch Max Wardetzky, Saurabh Mathur, Felix Kälberer, Eitan Grinspun, Symposium on Geometry Processing, 2007, pp. 33-37. Abstract: Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators. [pdf]
	Cubic Shells Akah Garg, Eitan Grinspun, Max Wardetzky, Denis Zorin, Symposium on Computer Animation, 2007, pp. 91-98. Abstract: Hinge-based bending models are widely used in the physically-based animation of cloth, thin plates and shells. We propose a hinge-based model that is simpler to implement, more efficient to compute, and offers a greater number of effective material parameters than existing models. Our formulation builds on two mathematical observations: (a) the bending energy of curved flexible surfaces can be expressed as a cubic polynomial if the surface does not stretch; (b) a general class of anisotropic materials - those that are orthotropic - is captured by appropriate choice of a single stiffness per hinge. Our contribution impacts a general range of surface animation applications, from isotropic cloth and thin plates to orthotropic fracturing thin shells. [pdf] [Video]
	On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces Klaus Hildebrandt, Konrad Polthier, Max Wardetzky, in Geometriae Dedicata 123, 2006, pp. 89-112. Abstract: We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3-space. Under the assumption of convergence of surfaces in Hausdorff distance, we show that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace-Beltrami operators. Additionally, we derive convergence of minimizing geodesics, mean curvature vectors, and solutions to the Dirichlet problem. [pdf]
	A Quadratic Bending Model for Inextensible Surfaces Miklos Bergou, Max Wardetzky, David Harmon, Denis Zorin, Eitan Grinspun, Symposium on Geometry Processing, 2006, pp. 227-230. Abstract: Efficient computation of curvature-based energies is important for practical implementations of geometric modeling and physical simulation applications. Building on a simple geometric observation, we provide a version of a curvature-based energy expressed in terms of the Laplace operator acting on the embedding of the surface. The corresponding energy, being quadratic in positions, gives rise to a constant Hessian in the context of isometric deformations. The resulting isometric bending model is shown to significantly speed up common cloth solvers, and when applied to geometric modeling situations built on Willmore flow to provide runtimes which are close to interactive rates. [pdf] [Video]
	FreeLence - Coding with free Valences Felix Kälberer, Konrad Polthier, Ulrich Reitebuch, Max Wardetzky, Eurographics (Computer Graphics Forum), 2005, pp. 469-478. Abstract: 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 20-30% over leading single-rate region-growing coders, both for connectivity and geometry. [pdf]
	Smooth Feature Lines on Surface Meshes Klaus Hildebrandt, Konrad Polthier, Max Wardetzky, Symposium on Geometry Processing, 2005, pp. 85-90. Abstract: 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. [pdf]

Research Reports and Course Notes

	Convergence of Discrete Elastica Henrik Schumacher, Sebastian Scholtes, Max Wardetzky, Oberwolfach Reports No. 34/2012. Abstract: Using techniques related to the notions of epigraph distance and Attouch-Wets-convergence, we show that under appropriate boundary conditions discrete elastica (i.e., polygonal curves of some fixed length that minimize a certain discrete bending energy) converge to smooth elastica (i.e., smooth curves of some given length that minimize smooth bending energy). [pdf]
	Persistence Simplifiation of Discrete Morse Functions on Surfaces Ulrich Bauer, Carsten Lange, Max Wardetzky, Oberwolfach Reports, Volume 6, Issue 1, 2009. Abstract: We combine the concept of persistent homology with Forman's discrete Morse theory on regular 2-manifold CW complexes to solve the problem of minimizing the number of critical points among all functions within a prescribed distance from a given input function. We give a constructive proof of the tightness of the lower bound on the number of critical points provided by the Stability Theorem of persistent homology. [pdf]
	Geometric Aspects of Discrete Elastic Rods Max Wardetzky, Miklos Bergou, Stephen Robinson, Basile Audoly, Eitan Grinspun, Oberwolfach Reports, Volume 6, Issue 1, 2009. Abstract: Elastic rods are curve-like elastic bodies that have one dimension (length) much larger than the others (cross-section). Their elastic energy breaks down into three contributions: stretching, bending, and twisting. Stretching and bending are captured by the deformation of a space curve called the centerline, while twisting is captured by the rotation of a material frame associated to each point on the centerline. Building on the notions of framed curves, parallel transport, and holonomy, we present a smooth and a corresponding discrete theory that establishes an efficient model for simulating thin flexible rods with arbitrary cross section and undeformed configuration. [pdf]
	Algebraic Topology on Polyhedral Surfaces from Finite Elements Max Wardetzky, Klaus Hildebrandt, Konrad Polthier, Oberwolfach Reports 12/2006. Abstract: We report on a development using piecewise constant vector fields (or one-forms) on compact polyhedral surfaces. The function spaces corresponding to a discrete Hodge decomposition then turn out to be a mixture of conforming and nonconforming linear finite elements. For sequences of polyhedral surfaces whose positions and normals converge to the positions and normals of an embedded compact smooth surface, we report on a convergence result for the corresponding discrete Hodge decompositions and Hodge star operators. [pdf]
	SIGGRAPH ASIA 2008 COURSE NOTES on Discrete Differential Geometry Course Organizors: Eitan Grinspun and Max Wardetzky Abstract: This volume documents the full day course Discrete Differential Geometry: An Applied Introduction at SIGGRAPH Asia 2008 in Singapore on 12 December 2008. These notes supplement the lectures given by Mathieu Desbrun, Peter Schröder, and Max Wardetzky. These notes include contributions by Miklos Bergou, Mathieu Desbrun, Sharif Elcott, Akash Garg, Eitan Grinspun, David Harmon, Eva Kanso, Felix Kälberer, Saurabh Mathur, Ulrich Pinkall, Peter Schröder, Adrian Secord, Boris Springborn, Ari Stern, John M. Sullivan, Yiying Tong, Max Wardetzky, and Denis Zorin, and build on the ideas of many others. [pdf]

Thesis

Discrete Differential Operators on Polyhedral Surfaces - Convergence and Approximation, Max Wardetzky, Dissertation, Freie Universität Berlin, 2006. [dissertation online]

Miscellaneous

Kepler, oranges and shadows from the fourth dimension. ... a site for mathematical entertainment about densest ball packings, disk packings, and their connection to shadows from the 4th dimension. Prepared for the "Beliner Tag der Mathematik 2005" - a day for mathematically interested high-school students.

Blind Dates für die Wissenschaft Brigitte Lutz-Westphal und Max Wardetzky, Mitteilungen der DMV, 2008. Abstract: Wissenschaftlicher Austausch ist eine wichtige Basis für erfolgreiche und innovative Forschung. Das klassische Format von Tagungen und Workshops im Wissenschaftsbetrieb stößt hier immer wieder an seine Grenzen. Insbesondere ist es für den wissenschaftlichen Nachwuchs häufig schwierig, von Vorträgen aus anderen Fachgebieten zu profitieren und so in einen sinnvollen Austausch mit Kolleginnen und Kollegen treten zu können. Für das Berliner DFG-Forschungszentrum MATHEON entwickelten wir ein neues Konzept, um einen vertieften wissenschaftlichen Austausch besser zu fördern. [pdf]