Publications
		Interactive spacetime control of deformable objects (Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier). To appear in ACM Transactions on Graphics 31(4) (SIGGRAPH 2012). (preprint) (supplementary video) Abstract: 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 framework for controlling the motion of deformable objects that offers interactive response times. 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 times through a new fast and robust numerical scheme for solving the one-dimensional problems that is based on a closed-form representation of the wiggly splines.
		Interactive Surface Modeling using Modal Analysis (Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier). ACM Transactions on Graphics, Volume 30, Issue 5, October 2011, pages 119:1-119:11. (supplementary video) Will be presented at SIGGRAPH 2012 DOI:10.1145/2019627.2019638 Abstract: 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].
		Modal Shape Analysis beyond Laplacian (Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier). Computer Aided Geometric Design, Volume 29, Issue 5, June 2012, Pages 204–218. DOI:10.1016/j.cagd.2012.01.001. (preprint) (supplementary video) Abstract: 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.
		Generalized Shape Operators on Polyhedral Surfaces (Klaus Hildebrandt and Konrad Polthier). Computer Aided Geometric Design, Volume 28, Issue 5, June 2011, pages 321-343, DOI:10.1016/j.cagd.2011.05.001. (preprint) Abstract: This work concerns the approximation of the shape operator of smooth surfaces in R3 from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions. The estimates are confirmed by numerical experiments.
		On approximation of the Laplace–Beltrami operator and the Willmore energy of surfaces (Klaus Hildebrandt and Konrad Polthier). Computer Graphics Forum, Volume 30, Issue 5, August 2011, pages 1513-1520. Proceedings of ACM Siggraph/Eurographics Symposium on Geometry Processing 2011. DOI: 10.1111/j.1467-8659.2011.02025.x 1st prize best paper award at SGP 2011 Abstract: Discrete Laplace–Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace–Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace–Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace–Beltrami operators.
		Koiter’s Thin Shells on Catmull–Clark Limit Surfaces (Anna Wawrzinek, Klaus Hildebrandt, and Konrad Polthier). Proceedings of the 16th International Workshop on Vision, Modeling, and Visualization 2011. DOI:10.2312/PE/VMV/VMV11/113-120 Abstract: We present a discretization of Koiter’s model of elastic thin shells based on a finite element that employs limit surfaces of Catmull–Clark’s subdivision scheme. The discretization can directly be applied to control grids of Catmull–Clark subdivision surfaces, and, therefore, integrates modeling of Catmull–Clark subdivision surfaces with analysis and optimization of elastic thin shells. To test the discretization, we apply it to standard examples for physical simulation of thin shells and compute free vibration modes of thin shells. Furthermore, we use the discrete shell model to set up a deformation-based modeling system for Catmull–Clark subdivision surfaces. This system integrates modeling of subdivision surfaces with deformation-based modeling and allows to switch back and forth between the two different approaches to modeling.
		Eigenmodes of surface energies for shape analysis (Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier), Advances in Geometric Modeling and Processing (Proceedings of Geometric Modeling and Processing 2010), Lecture Notes in Computer Science 6130, Springer, pages 296-314 Abstract: 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.
		Constraint-based fairing of surface meshes (Klaus Hildebrandt and Konrad Polthier), Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (2007). Abstract: 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 (Klaus Hildebrandt, Konrad Polthier, and Max Wardetzky), Geometriae Dedicata, 123, 89-112, 2006. Abstract: 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 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.
		Smooth Feature Lines on Surface Meshes (Klaus Hildebrandt, Konrad Polthier, and Max Wardetzky), Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (2005),  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.
		Anisotropic Filtering of Non-Linear Surface Features  (Klaus Hildebrandt and Konrad Polthier), Computer Graphics Forum, 23(3), 391-400, 2004. (view simulations) 1st prize best student paper award at Eurographics Abstract: A new method for noise removal of arbitrary surface 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.

Member of the DFG Research Center MATHEON "Mathematics for key technologies".

Address:
Klaus Hildebrandt
Freie Universität Berlin
Mathematical Geometry Processing
Arnimallee 6
D-14195 Berlin
Germany

Room: 111
email: klaus.hildebrandt "at" fu-berlin.de