Klaus Hildebrandt


Delft University of Technology 
Research Interests 
Geometric Modeling, Geometry Processing, Computer Graphics, Computer Animation,
Physical Simulation, Discrete Differential Geometry


Publications 


Fast Approximation of Laplace–Beltrami Eigenproblems
Ahmad Nasikun, Christopher Brandt, Klaus Hildebrandt
Computer Graphics Forum 37(5) (Symposium on Geometry Processing 2018)
Abstract: The spectrum and eigenfunctions of the LaplaceBeltrami operator are at the heart of effective schemes for a variety of problems in geometry processing. A burden attached to these spectral methods is that they need to numerically solve a largescale eigenvalue problem, which results in costly precomputation. In this paper, we address this problem by proposing a fast approximation algorithm for the lowest part of the spectrum of the LaplaceBeltrami operator. Our experiments indicate that the resulting spectra wellapproximate reference spectra, which are computed with stateoftheart eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.
[preprint][supp. document][video 1][video 2]



HyperReduced Projective Dynamics
Christopher Brandt, Elmar Eisemann, Klaus Hildebrandt
ACM Transactions on Graphics 37(4) Article 80 (SIGGRAPH 2018)
Abstract: We present a method for the realtime simulation of deformable objects that combines the robustness, generality, and high performance of Projective Dynamics with the efficiency and scalability offered by model reduction techniques. The method decouples the cost for time integration from the mesh resolution and can simulate large meshes in realtime. The proposed hyperreduction of Projective Dynamics combines a novel fast approximation method for constraint projections and a scalable construction of sparse subspace bases. The resulting system achieves realtime rates for large subspaces enabling rich dynamics and can resolve general user interactions, collision constraints, external forces and changes to the materials. The construction of the hyperreduced system does not require userinteraction and refrains from using training data or modal analysis, which results in a fast preprocessing stage.
[preprint][video][video (low res.)][supp. material][demo][code][doi]



Modeling nSymmetry Vector Fields using HigherOrder Energies
Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, Klaus Hildebrandt
ACM Transactions on Graphics 37(2) Article 18, 2018
To be presented at SIGGRAPH 2018
Abstract: We introduce a variational approach for modeling nsymmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while providing realtime responses. The approach is based on novel biharmonic and mharmonic energies for nfields on surface meshes and the integration of hard constraints to the resulting optimization problems. Realtime computation rates are achieved by a model reduction approach employing a Fourierlike nvector field decomposition, which associates frequencies and modes to nvector fields on surfaces. To demonstrate the benefits of the proposed nfield modeling approach, we use it for controlling stroke directions in lineart drawings of surfaces and for the modeling of anisotropic BRDFs which define the reflection behavior of surfaces.
[preprint][video][doi]



A Formalization of Relative Local Tempo Variations in Collections of Performances
Jeroen Peperkamp, Klaus Hildebrandt, Cynthia C. S. Liem
Proceedings of International Society for Music Information Retrieval Conference (ISMIR) 2017 Selected for oral presentation
Abstract: Multiple performances of the same piece share similarities, but also show relevant dissimilarities. With regard to the latter, analyzing and quantifying variations in collections of performances is useful to understand how a musical piece is typically performed, how naturally sounding new interpretations could be rendered, or what is peculiar about a particular performance. However, as there is no formal ground truth as to what these variations should look like, it is a challenge to provide and validate analysis methods for this. In this paper, we focus on relative local tempo variations in collections of performances. We propose a way to formally represent relative local tempo variations, as encoded in warping paths of aligned performances, in a vector space. This enables using statistics for analyzing tempo variations in collections of performances. We elaborate the computation and interpretation of the mean variation and the principal modes of variation. To validate our analysis method despite the absence of a ground truth, we present results on artificially generated data, representing several categories of local tempo variations. Finally, we show how our method can be used for analyzing to realworld data and discuss potential applications.
[paper]



Compressed Vibration Modes of Elastic Bodies
Christopher Brandt, Klaus Hildebrandt
Computer Aided Geometric Design 52–53 (Geometric Modeling and Processing 2017), pages 297–312
Abstract: The natural vibration modes of deformable objects are a fundamental physical phenomenon. In this paper, we introduce compressed vibration modes, which, in contrast to the natural vibration modes, are localized ("sparse") deformations. The localization is achieved by augmenting the objective which has the vibration modes as minima by a L^{1} term. As a result the compressed modes form a compromise between localization and optimal energy efficiency of the deformations. We introduce a scheme for computing bases of compressed modes by solving sequences of convex optimization problems. Our experiments demonstrate that the resulting bases are wellsuited for reducedorder shape deformation and for guiding the segmentation of objects into functional parts.
[preprint][video][doi]



Spectral Processing of Tangential Vector Fields
Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, Klaus Hildebrandt
Computer Graphics Forum 36(6), pages 338–353, 2017
Presented at Eurographics 2017
Abstract: We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fouriertype representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent cotan discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a splinetype editor for modeling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for realtime modeling of tangential vector fields.
[preprint][video][supp. material][doi]



Visualization and Extraction of Carvings for Heritage Conservation
Kai Lawonn, Erik Trostmann, Bernhard Preim, Klaus Hildebrandt
IEEE Transactions on Visualization and Computer Graphics 23(1), pages 801–810, 2017 (IEEE Visualization 2016)
Abstract: We present novel techniques for visualizing, illustrating, analyzing, and generating carvings in surfaces. In particular, we consider the carvings in the plaster of the cloister of the Magdeburg cathedral, which dates to the 13th century. Due to aging and weathering, the carvings have flattened. Historians and restorers are highly interested in using digitalization techniques to analyze carvings in historic artifacts and monuments and to get impressions and illustrations of their original shape and appearance. Moreover, museums and churches are interested in such illustrations for presenting them to visitors. The techniques that we propose allow for detecting, selecting, and visualizing carving structures. In addition, we introduce an examplebased method for generating carvings. The resulting tool, which integrates all techniques, was evaluated by three experienced restorers to assess the usefulness and applicability. Furthermore, we compared our approach with exaggerated shading and other stateoftheart methods.
[preprint][doi]



Optimized Subspaces for DeformationBased Modeling and Shape Interpolation
Philipp von Radziewsky, Elmar Eisemann, HansPeter Seidel, Klaus Hildebrandt
Computers & Graphics 58 (Shape Modeling International 2016), pages 128–138
Best Paper Award at Shape Modeling International 2016
Abstract: We propose a novel construction of subspaces for realtime deformationbased modeling and shape interpolation. Our goal is to construct the subspaces that bestapproximate the manifold of deformations relevant for a specific modeling or interpolation problem. The idea is to automatically sample the deformation manifold and construct the subspace that bestapproximates the snapshots. This is realized by writing the shape modeling and interpolation problems as parametrized optimization problems with few parameters. The snapshots are generated by sampling the parameter domain and computing the corresponding minimizers. Finally, the optimized subspaces are constructed using a massdependent principle component analysis. The optimality provided by this scheme contrasts it from alternative approaches, which aim at constructing spaces containing lowfrequency deformations. The benefit of this construction is that compared to alternative approaches a similar approximation quality is achieved with subspaces of significantly smaller dimension. This is crucial because the runtimes and memory requirements of the realtime shape modeling and interpolation schemes mainly depend on the dimensions of the subspaces.
[preprint][video][doi]



Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects
Christopher Brandt, Christoph von Tycowicz, Klaus Hildebrandt
Computer Graphics Forum 35(2) (Eurographics 2016), pages 295–305
Best Paper Honorable Mention Award at Eurographics 2016
Abstract: We introduce techniques for the processing of motion and animations of nonrigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in Rn to the processing of motion of nonrigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reducedorder method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct nonlinear Bézier curves by executing de Casteljau's algorithm in shape space.
[preprint][video][doi]



Directional Field Synthesis, Design, and Processing
Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, Mirela BenChen
Computer Graphics Forum 35(2) (STAR  Eurographics 2016), pages 545–572
Abstract: Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications incentivized the definition of many types of directional fields, from vector and tensor fields, over line and cross fields, to frame and vectorset fields.
[preprint][doi] 


Foldio: Digital Fabrication of Interactive and ShapeChanging Objects With Foldable Printed Electronics
Simon Olberding, Sergio Soto Ortega, Klaus Hildebrandt, Jürgen Steimle
ACM Symposium on User Interface Software and Technology (UIST) 2015
Best Paper Award at UIST 2015
Selected for the UIST reprise session at SIGGRAPH 2016
Abstract: Foldios are foldable interactive objects with embedded input sensing and output capabilities. Foldios combine the advantages of folding for thin, lightweight and shapechanging objects with the strengths of thinfilm printed electronics for embedded sensing and output. To enable designers and endusers to create highly custom interactive foldable objects, we contribute a new design and fabrication approach. It makes it possible to design the foldable object in a standard 3D environment and to easily add interactive highlevel controls, eliminating the need to manually design a fold pattern and lowlevel circuits for printed electronics. Second, we contribute a set of printable user interface controls for touch input and display output on folded objects. Moreover, we contribute controls for sensing and actuation of shapechangeable objects. We demonstrate the versatility of the approach with a variety of interactive objects that have been fabricated with this framework.
[preprint][video][doi] 


Animating Articulated Characters using Wiggly Splines
Christian Schulz, Christoph von Tycowicz, HansPeter Seidel, Klaus Hildebrandt
ACM SIGGRAPH/Eurographics Symposium on Computer Animation 2015, pages 101–109
Abstract: We propose a new framework for spacetime optimization that can generate artistic motion with a long planning horizon for complex virtual characters. The scheme can be used for generating general types of motion and neither requires motion capture data nor an initial motion that satisfies the constraints. Our modeling of the spacetime optimization combines linearized dynamics and a novel warping scheme for articulated characters. We show that the optimal motions can be described using a combination of vibration modes, wiggly splines, and our warping scheme. This enables us to restrict the optimization to lowdimensional spaces of explicitly parametrized motions. Thereby the computation of an optimal motion is reduced to a lowdimensional nonlinear least squares problem, which can be solved with standard solvers. We show examples of motions created by specifying only a few constraints for positions and velocities.
[preprint][video][doi] 


RealTime Nonlinear Shape Interpolation
Christoph von Tycowicz, Christian Schulz, HansPeter Seidel, Klaus Hildebrandt
ACM Transactions on Graphics 34(3) Article No. 34
Presented at SIGGRAPH 2015
Abstract: We introduce a scheme for realtime nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation problem, in particular, the fact that the set of all possible interpolated shapes is a lowdimensional object in a highdimensional shape space. The interpolated shapes are defined as the minimizers of a nonlinear objective functional on the shape space. Our approach is to construct a reduced optimization problem that approximates its unreduced counterpart and can be solved in milliseconds. To achieve this, we restrict the optimization to a lowdimensional subspace that is specifically designed for the shape interpolation problem. The construction of the subspace is based on two components: a formula for the calculation of derivatives of the interpolated shapes and a Krylovtype sequence that combines the derivatives and the Hessian of the objective functional. To make the computational cost for solving the reduced optimization problem independent of the resolution of the example shapes, we combine the dimensional reduction with schemes for the efficient approximation of the reduced nonlinear objective functional and its gradient. In our experiments, we obtain rates of 20100 interpolated shapes per second even for the largest examples which have 500k vertices per example shape. [preprint][video][doi]



Optimal Spline Approximation via l_{0}Minimization
Christopher Brandt, HansPeter Seidel, Klaus Hildebrandt
Computer Graphics Forum 34(2) (Eurographics 2015), pages 617–626.
Abstract: Splines are part of the standard toolbox for the approximation of functions and curves in R^{d}. Still, the problem of finding the spline that best approximates an input function or curve is illposed, since in general this yields a "spline" with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an l_{0}regularized quadratic problem. This gives us a notion of optimal approximating splines that depends on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including Bsplines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and spline conversion of motion capture data. [preprint][video][doi]



3D Model Retargeting Using Offset Statistics
Xiaokun Wu, Chuan Li, Michael Wand, Klaus Hildebrandt, Silke Jansen, HansPeter Seidel
Proceedings of the 2nd International Conference on 3D Vision 2014, IEEE, pages 353–360
Abstract: Texture synthesis is a versatile tool for creating and editing 2D images. However, applying it to 3D content creation is difficult due to the higher demand of model accuracy and the large search space that also contains many implausible shapes. Our paper explores offset statistics for 3D shape retargeting. We observe that the offset histograms between similar 3D features are sparse, in particular for manmade objects such as buildings and furniture. We employ sparse offset statistics to improve 3D shape retargeting (i.e. rescaling in different directions). We employ a graphcut texture synthesis method that iteratively stitches model fragments shifted by the detected sparse offsets. The offsets reveal important structural redundancy which leads to more plausible results and more efficient optimization. Our method is fully automatic, while intuitive user control can be incorporated for interactive modeling in realtime. We empirically evaluate the sparsity of offset statistics across a wide range of subjects, and show our statistics based retargeting significantly improves quality and efficiency over conventional MRF models.
[preprint][video][doi]



RealTime SymmetryPreserving Deformation
Xiaokun Wu, Michael Wand, Klaus Hildebrandt, Pushmeet Kohli, HansPeter Seidel
Computer Graphics Forum 33(7) (Pacific Graphics 2014), pages
229–238.
Abstract: In this paper, we address the problem of structureaware shape deformation: We specifically consider deformations that preserve symmetries of the shape being edited. While this is an elegant approach for obtaining plausible shape variations from minimal assumptions, a straightforward optimization is numerically expensive and poorly conditioned. Our paper introduces an explicit construction of bases of linear spaces of shape deformations that exactly preserve symmetries for any userdefined level of detail. This permits the construction of lowdimensional spaces of lowfrequency deformations that preserve the symmetries. We obtain substantial speedups over alternative approaches for symmetrypreserving shape editing due to (i) the subspace approach, which permits lowres editing, (ii) the removal of redundant, symmetric information, and (iii) the simplification of the numerical formulation due to hardcoded symmetry preservation. We demonstrate the utility in practice by applying our framework to symmetrypreserving corotated iterative Laplace surface editing of models with complex symmetry structure, including partial and nested symmetry.
[preprint][video][doi]



Animating Deformable Objects using Sparse Spacetime Constraints
Christian Schulz, Christoph von Tycowicz, HansPeter Seidel, Klaus Hildebrandt
ACM Transactions on Graphics 33(4) (SIGGRAPH 2014) Article No. 109
Abstract: We propose a scheme for animating deformable objects based on spacetime optimization. The main feature is that it robustly and quickly (within a few seconds) generates interesting motion from a sparse set of spacetime constraints. Providing only partial (as opposed to full) keyframes for positions and velocities is sufficient. The computed motion satisfies the constraints and the remaining degrees of freedom are determined by physical principles using elasticity and the spacetime constraints paradigm. Our modeling of the spacetime optimization problem combines dimensional reduction, modal coordinates, wiggly splines, and rotation strain warping. Controlling the warped motion requires the derivative of the warp map. We derive a representation of the derivative that can be efficiently and robustly evaluated. Our solver is based on a theorem that characterizes the solutions of the optimization problem and allows us to restrict the optimization to very lowdimensional search spaces. This treatment of the optimization problem avoids a time discretization and the resulting method can robustly deal with sparse input and wiggly motion.
[preprint][video][supplementary material][doi]



An Efficient Construction of Reduced Deformable Objects
Christoph von Tycowicz, Christian Schulz, HansPeter Seidel, Klaus Hildebrandt
ACM Transactions on Graphics 32(6) (SIGGRAPH Asia 2013) Article No. 213
Abstract: Many efficient computational methods for physical simulation are based on model reduction. We propose new model reduction techniques for the approximation of reduced forces and for the construction of reduced shape spaces of deformable objects that accelerate the construction of a reduced dynamical system, increase the accuracy of the approximation, and simplify the implementation of model reduction. Based on the techniques, we introduce schemes for realtime simulation of deformable objects and interactive deformationbased editing of triangle or tet meshes. We demonstrate the effectiveness of the new techniques in different experiments with elastic solids and shells and compare them to alternative approaches.
[preprint][video][doi]



Consistent discretizations of the
Laplace–Beltrami operator and the
Willmore energy of surfaces
Klaus Hildebrandt and Konrad
Polthier
Oberwolfach Reports, Workshop
1228: Discrete Differential Geometry,
2012.
Abstract: A fundamental aspect when
translating classical concepts from
smooth differential
geometry, such as differential operators
or geometric functionals, to
corresponding
discrete notions is consistency.
Here,
we are concerned with the construction
of consistent discrete counterparts to the
Laplace–Beltrami operator and the Willmore
energy on polyhedral surfaces.
[link] 


Interactive spacetime
control of deformable objects
Klaus Hildebrandt, Christian
Schulz, Christoph von Tycowicz, and
Konrad Polthier
ACM
Transactions on Graphics 31(4) (SIGGRAPH
2012) Article No. 71.
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
onedimensional 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 onedimensional
problems that is based on a closedform
representation of the wiggly splines.
[preprint][video][doi]



Interactive Surface Modeling using Modal Analysis
Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier
ACM Transactions on Graphics 30(5), pages
119:1–119:11, October 2011.
Presented at SIGGRAPH 2012
Abstract: We propose a framework for deformationbased surface modeling that is interactive, robust and intuitive to use.
The deformations are described by a nonlinear 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 quasiNewton 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 lowdimensional 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 quasiNewton 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].
[preprint][video][doi] 


Modal Shape Analysis beyond Laplacian
Klaus Hildebrandt, Christian
Schulz, Christoph von Tycowicz, and
Konrad Polthier
Computer Aided
Geometric Design 29(5), pages 204–218, June 2012.
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.
[preprint][video][doi] 


Generalized Shape Operators on Polyhedral Surfaces
Klaus
Hildebrandt and Konrad Polthier
Computer Aided Geometric Design
28(5), pages 321–343, June 2011.
Abstract: This work concerns the
approximation of the shape operator of
smooth surfaces in R^{3} from polyhedral surfaces. We introduce two generalized shape operators that
are vectorvalued 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.
[preprint][doi] 


On
approximation of the Laplace–Beltrami
operator and the Willmore energy of
surfaces
Klaus Hildebrandt and Konrad Polthier
Computer Graphics Forum 30(5), August 2011, pages
1513–1520.
Proceedings of ACM Siggraph/Eurographics
Symposium on Geometry Processing 2011.
1stPrize
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 wellstudied, 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.
[preprint][doi] 


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.
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 deformationbased
modeling system for Catmull–Clark
subdivision surfaces. This system
integrates modeling of subdivision
surfaces with deformationbased modeling
and allows to switch back and forth
between the two different approaches to
modeling.
[preprint][doi] 


Eigenmodes of surface energies for shape analysis
Klaus Hildebrandt, Christian Schulz, Christoph von
Tycowicz, and Konrad Polthier
Advances in Geometric Modeling and
Processing, Lecture Notes in Computer
Science 6130, Springer, pages 296–314.
Proceedings of Geometric Modeling and Processing 2010.
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.
[preprint][doi] 


Constraintbased fairing of surface
meshes
Klaus Hildebrandt and Konrad Polthier
Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing 2007,
203–212.
Abstract: We propose a constraintbased
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.
[preprint][doi] 


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 Euclidean
3space. 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 LaplaceBeltrami operators.
Additionally, we derive convergence of
minimizing geodesics,
mean curvature vectors, and solutions to
the Dirichlet problem.
[preprint][doi] 


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.
[preprint][doi] 


Anisotropic Filtering of NonLinear Surface
Features
Klaus Hildebrandt and Konrad Polthier
Computer Graphics Forum, 23(3), 391–400,
2004
1stPrize
Best Student Paper Award
at Eurographics
2004
Abstract: A new method
for noise removal of arbitrary surface
meshes is presented which focuses on the
preservation and sharpening of nonlinear
geometric features such as curved edges and
surface regions. Our method uses a
nonlinear 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 nonlinear features. Additionally
our discrete flow is very well suited to
prevent boundary shrinkage at constrained
and free boundary segments.
[preprint][videos][doi] 

