Klaus Hildebrandt 

Delft University of Technology


Klaus Hildebrandt
Assistant Professor
Computer Graphics and Visualization Group

Delft University of Technology
EEMCS - Dept. Intelligent Systems
Mekelweg 4, 2628 CD Delft
Room 11.270

Phone: +31 (0)15 27 81445
Email: k.a.hildebrandt -at- tudelft -dot- nl


Research Interests

Geometric Modeling, Geometry Processing, Computer Graphics, Computer Animation,
Physical Simulation, Discrete Differential Geometry


  Visualization and Extraction of Carvings for Heritage Conservation
Kai Lawonn, Erik Trostmann, Bernhard Preim, Klaus Hildebrandt
IEEE Transactions on Visualization and Computer Graphics (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 example-based 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 state-of-the-art methods.
  Spectral Processing of Tangential Vector Fields
Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, Klaus Hildebrandt
Computer Graphics Forum 35, 2016
Abstract: We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier-type 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 spline-type 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 real-time modeling of tangential vector fields.
[preprint][video][supp. material][doi]
  Optimized Subspaces for Deformation-Based Modeling and Shape Interpolation
Philipp von Radziewsky, Elmar Eisemann, Hans-Peter 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 real-time deformation-based modeling and shape interpolation. Our goal is to construct the subspaces that best-approximate 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 best-approximates 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 mass-dependent principle component analysis. The optimality provided by this scheme contrasts it from alternative approaches, which aim at constructing spaces containing low-frequency 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 run-times and memory requirements of the real-time shape modeling and interpolation schemes mainly depend on the dimensions of the subspaces.
  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 non-rigid 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 non-rigid 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 reduced-order 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 non-linear Bézier curves by executing de Casteljau's algorithm in shape space.
  Directional Field Synthesis, Design, and Processing
Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, Mirela Ben-Chen
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 vector-set fields.
  Foldio: Digital Fabrication of Interactive and Shape-Changing 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 shape-changing objects with the strengths of thin-film printed electronics for embedded sensing and output. To enable designers and end-users 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 high-level controls, eliminating the need to manually design a fold pattern and low-level 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 shape-changeable objects. We demonstrate the versatility of the approach with a variety of interactive objects that have been fabricated with this framework.
  Animating Articulated Characters using Wiggly Splines
Christian Schulz, Christoph von Tycowicz, Hans-Peter 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 low-dimensional spaces of explicitly parametrized motions. Thereby the computation of an optimal motion is reduced to a low-dimensional non-linear 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.
  Real-Time Nonlinear Shape Interpolation
Christoph von Tycowicz, Christian Schulz, Hans-Peter Seidel, Klaus Hildebrandt
ACM Transactions on Graphics 34(3) Article No. 34
Presented at SIGGRAPH 2015
Abstract: We introduce a scheme for real-time 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 low-dimensional object in a high-dimensional 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 low-dimensional 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 Krylov-type 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 20-100 interpolated shapes per second even for the largest examples which have 500k vertices per example shape.
  Optimal Spline Approximation via l0-Minimization
Christopher Brandt, Hans-Peter 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 Rd. Still, the problem of finding the spline that best approximates an input function or curve is ill-posed, 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 l0-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 B-splines 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.
  3D Model Retargeting Using Offset Statistics
Xiaokun Wu, Chuan Li, Michael Wand, Klaus Hildebrandt, Silke Jansen, Hans-Peter 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 man-made 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 graph-cut 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 real-time. 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.
  Real-Time Symmetry-Preserving Deformation
Xiaokun Wu, Michael Wand, Klaus Hildebrandt, Pushmeet Kohli, Hans-Peter Seidel
Computer Graphics Forum 33(7) (Pacific Graphics 2014), pages 229-238.
Abstract: In this paper, we address the problem of structure-aware 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 user-defined level of detail. This permits the construction of low-dimensional spaces of low-frequency deformations that preserve the symmetries. We obtain substantial speed-ups over alternative approaches for symmetry-preserving shape editing due to (i) the sub-space approach, which permits low-res editing, (ii) the removal of redundant, symmetric information, and (iii) the simplification of the numerical formulation due to hard-coded symmetry preservation. We demonstrate the utility in practice by applying our framework to symmetry-preserving co-rotated iterative Laplace surface editing of models with complex symmetry structure, including partial and nested symmetry.
  Animating Deformable Objects using Sparse Spacetime Constraints
Christian Schulz, Christoph von Tycowicz, Hans-Peter 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 low-dimensional 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, Hans-Peter 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 real-time simulation of deformable objects and interactive deformation-based 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.
Oberwolfach Reports
  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.
  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 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 30(5), pages 119:1-119:11, October 2011.
Presented at SIGGRAPH 2012
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 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.
  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 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 30(5), August 2011, pages 1513-1520.
Proceedings of ACM Siggraph/Eurographics Symposium on Geometry Processing 2011.
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.
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, 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.
  Constraint-based 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 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 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.
  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
1st-Prize 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 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.