Short Resume


I am a tenured Assistant Professor at the Department of Intelligent Systems at Delft University of Technology. Previous to that I was a Senior Researcher at the Max Planck Institute for Informatics in Saarbrücken, where I headed the Applied Geometry group. I received my PhD from the Free University Berlin.
My research interests include Visual Computing, Geometric Data Processing, Physical Simulation, and Computational and Discrete Differential Geometry. My work received several best paper awards at international conferences including Eurographics, ACM UIST, Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, and Solid Modeling International. I cochaired the Eurographics Symposium on Geometry Processing in 2017 and the Eurographics StateoftheArt Reports program in 2018. I am associate editor of Computer Graphics Forum, Graphical Models, and Computer Animation and Virtual Worlds and Junior Fellow of the European Association of Computer Graphics (Eurographics).




Teaching


Courses I teach regularly at TU Delft:
 Geometric Data Processing
 Seminar Computer Graphics
 Research Methodology for Data Science
 Visual Data Processing
I serve as master coordinator for the Data Science and Technology track of the master's program Computer Science at TU Delft.




Publications



A Fast Geometric Multigrid Method for Curved Surfaces
Ruben Wiersma*, Ahmad Nasikun*, Elmar Eisemann, Klaus Hildebrandt
* both authors contributed equally
ACM SIGGRAPH 2023
Abstract: We introduce a geometric multigrid method for solving linear systems arising from variational problems on surfaces in geometry processing, Gravo MG. Our scheme uses point clouds as a reduced representation of the levels of the multigrid hierarchy to achieve a fast hierarchy construction and to extend the applicability of the method from triangle meshes to other surface representations like point clouds, nonmanifold meshes, and polygonal meshes. To build the prolongation operators, we associate each point of the hierarchy to a triangle constructed from points in the next coarser level. We obtain wellshaped candidate triangles by computing graph Voronoi diagrams centered around the coarse points and determining neighboring Voronoi cells. Our selection of triangles ensures that the connections of each point to points at adjacent coarser and finer levels are balanced in the tangential directions. As a result, we obtain sparse prolongation matrices with three entries per row and fast convergence of the solver.
[preprint][supplementary material][code]



Parametrizing Product Shape Manifolds by Composite Networks
Josua Sassen, Klaus Hildebrandt, Martin Rumpf, Benedikt Wirth
ICLR 2023 spotlight paper (notable top 25%)
Abstract: Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural network approximation. We show that this is indeed possible for shape spaces with a special product structure, namely those smoothly approximable by a direct sum of lowdimensional manifolds. Our proposed architecture leverages this structure by separately learning approximations for the lowdimensional factors and a subsequent combination. After developing the approach as a general framework, we apply it to a shape space of triangular surfaces. Here, typical examples of data manifolds are given through datasets of articulated models and can be factorized, for example, by a Sparse Principal Geodesic Analysis (SPGA). We demonstrate the effectiveness of our proposed approach with experiments on synthetic data as well as manifolds extracted from data via SPGA.
[preprint]



Improving Error Detection in Deep Learning Based Radiotherapy Autocontouring Using Bayesian Uncertainty
Prerak Mody, Nicolas ChavesdePlaza, Klaus Hildebrandt, Marius Staring
Uncertainty for Safe Utilization of Machine Learning in Medical Imaging, MICCAI 2022 workshop, Lecture Notes in Computer Science (volume 13563)
Abstract: Bayesian Neural Nets (BNN) are increasingly used for robust organ autocontouring. Uncertainty heatmaps extracted from BNNs have been shown to correspond to inaccurate regions. To help speed up the mandatory quality assessment (QA) of contours in radiotherapy, these heatmaps could be used as stimuli to direct visual attention of clinicians to potential inaccuracies. In practice, this is nontrivial to achieve since many accurate regions also exhibit uncertainty. To influence the output uncertainty of a BNN, we propose a modified accuracyversusuncertainty (AvU) metric as an additional objective during model training that penalizes both accurate regions exhibiting uncertainty as well as inaccurate regions exhibiting certainty. For evaluation, we use an uncertaintyROC curve that can help differentiate between Bayesian models by comparing the probability of uncertainty in inaccurate versus accurate regions. We train and evaluate a FlipOut BNN model on the MICCAI2015 Head and Neck Segmentation challenge dataset and on the DeepMindTCIA dataset, and observed an increase in the AUC of uncertaintyROC curves by 5.6% and 5.9%, respectively, when using the AvU objective. The AvU objective primarily reduced false positives regions (uncertain and accurate), drawing less visual attention to these regions, thereby potentially improving the speed of error detection.
[preprint][doi]



DeltaConv: Anisotropic Operators for Geometric Deep Learning on Point Clouds
Ruben Wiersma, Ahmad Nasikun, Elmar Eisemann, Klaus Hildebrandt
ACM Transactions on Graphics 41(4) (ACM SIGGRAPH 2022)
Abstract: Learning from 3D pointcloud data has rapidly gained momentum, motivated by the success of deep learning on images and the increased availability of 3D data. In this paper, we aim to construct anisotropic convolution layers that work directly on the surface derived from a point cloud. This is challenging because of the lack of a global coordinate system for tangential directions on surfaces. We introduce DeltaConv, a convolution layer that combines geometric operators from vector calculus to enable the construction of anisotropic filters on point clouds. Because these operators are defined on scalar and vectorfields, we separate the network into a scalar and a vectorstream, which are connected by the operators. The vector stream enables the network to explicitly represent, evaluate, and process directional information. Our convolutions are robust and simple to implement and match or improve on stateoftheart approaches on several benchmarks, while also speeding up training and inference.
[preprint][code][project page][doi]



The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems
Ahmad Nasikun, Klaus Hildebrandt
ACM Transactions on Graphics 41(2) Article 17, 2022
Presented at ACM SIGGRAPH 2022
Abstract: Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the LaplaceBeltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this paper, we introduce the Hierarchical Subspace Iteration Method (HSIM), a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the nonhierarchical Subspace Iteration Method. Our experiments show that HSIM can solve LaplaceBeltrami eigenproblems on meshes faster than stateoftheart methods based on Lanczos iterations, preconditioned conjugate gradients and subspace iterations.
[preprint][supplementary material][doi]



Towards fast humancentered contouring workflows for adaptive external beam radiotherapy
Nicolas F. ChavesdePlaza, Prerak Mody, Klaus Hildebrandt, Marius Staring, Eleftheria Astreinidou, Mischa de Ridder, Huib de Ridder, René van Egmond
Human Factors and Ergonomics Society Europe Chapter Annual Conference 2022
Abstract: Delineation of tumors and organsatrisk permits detecting and correcting changes in the patients' anatomy throughout the treatment, making it a core step of adaptive external beam radiotherapy. Although autocontouring technologies have sped up this process, the time needed to perform the quality assessment of the generated contours remains a bottleneck, taking clinicians between several minutes and an hour to complete. The authors of this article conducted several interviews and an observational study at two treatment centers in the Netherlands to identify challenges and opportunities for speeding up the delineation process in adaptive therapies. The study revealed three contextual variables that influence contouring performance: usable additional information, applicable domainspecific knowledge, and available editing capabilities in contouring software. In practice, clinicians leverage these variables to accelerate contouring in two ways. First, they use domainspecific knowledge and relevant clinical features such as the proximity of the organsatrisk to the tumor to enable targeted inspection of the delineation. Second, clinicians modulate editing precision depending on the effect they anticipate the edit will have on the patient outcome. By implementing these acceleration strategies in guidelines and contouring tools, developers and workflow builders could increase contouring efficiency and consistency without affecting the patient outcome.
[proceedings]



DCGrid: An Adaptive Grid Structure for MemoryConstrained Fluid Simulation on the GPU
Wouter Raateland, Torsten Hädrich, Jorge Alejandro Amador Herrera, Daniel Banuti, Wojciech Palubicki, Sören Pirk, Klaus Hildebrandt, Dominik Michels
Proc. ACM Comput. Graph. Interact. Tech. (I3D 2022)
Abstract: We introduce Dynamic Constrained Grid (DCGrid), a hierarchical and adaptive grid structure for fluid simulation combined with a scheme for effectively managing the grid adaptations. DCGrid is designed to be implemented on the GPU and used in highperformance simulations. Specifically, it allows us to efficiently vary and adjust the grid resolution across the spatial domain and to rapidly evaluate local stencils and individual cells in a GPU implementation. A special feature of DCGrid is that the control of the grid adaption is modeled as an optimization under a constraint on the maximum available memory, which addresses the memory limitations in GPUbased simulation. To further advance the use of DCGrid in highperformance simulations, we complement DCGrid with an efficient scheme for approximating collisions between fluids and static solids on cells with different resolutions. We demonstrate the effectiveness of DCGrid for smoke flows and complex cloud simulations in which terrainatmosphere interaction requires working with cells of varying resolution and rapidly changing conditions. Finally, we compare the performance of DCGrid to that of alternative adaptive grid structures.
[preprint][supplementary material][video][code][doi]



Deep vanishing point detection: Geometric priors make dataset variations vanish
Yancong Lin, Ruben Wiersma, Silvia Pintea, Klaus Hildebrandt, Elmar Eisemann, Jan van Gemert
IEEE CVPR 2022
Abstract: Deep learning has greatly improved vanishing point detection in images. Yet, deep networks require expensive annotated datasets trained on costly hardware and do not generalize to even slightly different domains and minor problem variants. Here, we address these issues by injecting deep vanishing point detection networks with prior knowledge. This prior knowledge no longer needs to be learned from data, saving valuable annotation efforts and compute, unlocking realistic fewsample scenarios, and reducing the impact of domain changes. Moreover, because priors are interpretable, it is easier to adapt deep networks to minor problem variations such as switching between Manhattan and nonManhattan worlds. We incorporate two endtoend trainable geometric priors: (i) Hough Transform  mapping image pixels to straight lines, and (ii) Gaussian sphere  mapping lines to great circles whose intersections denote vanishing points. Experimentally, we ablate our choices and show comparable accuracy as existing models in the largedata setting. We then validate that our model improves data efficiency, is robust to domain changes, and can easily be adapted to a nonManhattan setting.
[preprint][doi]



Comparing Bayesian Models for Organ Contouring in Head and Neck Radiotherapy
Prerak Mody, Nicolas ChavesdePlaza, Klaus Hildebrandt, Rene van Egmond, Huib de Ridder, Marius Staring
SPIE Medical Imaging 2022: Image Processing
Abstract: Deep learning models for organ contouring in radiotherapy are poised for clinical usage, but currently, there exist few tools for automated quality assessment (QA) of the predicted contours. Using Bayesian models and their associated uncertainty, one can potentially automate the process of detecting inaccurate predictions. We investigate two Bayesian models for autocontouring, DropOut and FlipOut, using a quantitative measure  expected calibration error (ECE) and a qualitative measure  regionbased accuracyvsuncertainty (RAvU) graphs. It is well understood that a model should have low ECE to be considered trustworthy. However, in a QA context, a model should also have high uncertainty in inaccurate regions and low uncertainty in accurate regions. Such behaviour could direct visual attention of expert users to potentially inaccurate regions, leading to a speed up in the QA process. Using RAvU graphs, we qualitatively compare the behaviour of different models in accurate and inaccurate regions. Experiments are conducted on the MICCAI2015 Head and Neck Segmentation Challenge and on the DeepMindTCIA CT dataset using three models: DropOutDICE, DropoutCE (Cross Entropy) and FlipOutCE. Quantitative results show that DropOutDICE has the highest ECE, while DropoutCE and FlipOutCE have the lowest ECE. To better understand the difference between DropOutCE and FlipOutCE, we use the RAvU graph which shows that FlipOutCE has better uncertainty coverage in inaccurate regions than DropOutCE. Such a combination of quantitative and qualitative metrics explores a new approach that helps to select which model can be deployed as a QA tool in clinical settings.
[preprint][doi]



Nonlinear Deformation Synthesis via Sparse Principal Geodesic Analysis
Josua Sassen, Klaus Hildebrandt, Martin Rumpf
Computer Graphics Forum 39(5) (Symposium on Geometry Processing 2020)
Abstract: This paper introduces the construction of a lowdimensional nonlinear space capturing the variability of a nonrigid shape from a data set of example poses. The core of the approach is a Sparse Principal Geodesic Analysis (SPGA) on the Riemannian manifold of discrete shells, in which a pose of a nonrigid shape is a point. The SPGA is invariant to rigid body motions of the poses and supports large deformation. Since the Riemannian metric measures the membrane and bending distortions of the shells, the sparsity term forces the modes to describe largely decoupled and localized deformations. This property facilitates the analysis of articulated shapes. The modes often represent characteristic articulations of the shape and usually come with a decomposing of the spanned subspace into lowdimensional widely decoupled subspaces. For example, for human models, one expects distinct, localized modes for the bending of elbow or knee whereas some more modes are required to represent shoulder articulation. The decoupling property can be used to construct useful starting points for the computation of the nonlinear deformations via a superposition of shape submanifolds resulting from the decoupling. In a preprocessing stage, samples of the individual subspaces are computed, and, in an online phase, these are interpolated multilinearly. This accelerates the construction of nonlinear deformations and makes the method applicable for interactive applications. The method is compared to alternative approaches and the benefits are demonstrated on different kinds of input data.
[preprint][video][presentation][doi]



CNNs on Surfaces using RotationEquivariant Features
Ruben Wiersma, Elmar Eisemann, Klaus Hildebrandt
ACM Transactions on Graphics 39(4) (SIGGRAPH 2020)
Abstract: This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vectorvalued, rotationequivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution layer. The resulting network is agnostic to the choices of coordinate systems for the tangent spaces on the surface. We implement our approach for triangle meshes. Based on circular harmonic functions, we introduce convolution filters for meshes that are rotationequivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.
[preprint][code][project page][doi]



Locally Supported Tangential Vector, nVector, and Tensor Fields
Ahmad Nasikun, Christopher Brandt, Klaus Hildebrandt
Computer Graphics Forum 39(2) (Eurographics 2020)
Abstract: We introduce a construction of subspaces of the spaces of tangential vector, nvector, and tensor fields on surfaces. The resulting subspaces can be used as the basis of fast approximation algorithms for design and processing problems that involve tangential fields. Important features of our construction are that it is based on a general principle, from which constructions for different types of tangential fields can be derived, and that it is scalable, making it possible to efficiently compute and store large subspace bases for large meshes. Moreover, the construction is adaptive, which allows for controlling the distribution of the degrees of freedom of the subspaces over the surface. We evaluate our construction in several experiments addressing approximation quality, scalability, adaptivity, computation times and memory requirements. Our design choices are justified by comparing our construction to possible alternatives. Finally, we discuss examples of how subspace methods can be used to build interactive tools for tangential field design and processing tasks.
[preprint][doi]



Geometric Optimization using Nonlinear RotationInvariant Coordinates
Josua Sassen, Behrend Heeren, Klaus Hildebrandt, Martin Rumpf
Computer Aided Geometric Design 77, 2020
Abstract: Geometric optimization problems are at the core of many applications in geometry processing. The choice of a representation fitting an optimization problem can considerably simplify solving the problem. We consider the Nonlinear RotationInvariant Coordinates (NRIC) that represent the nodal positions of a discrete triangular surface with fixed combinatorics as a vector that stacks all edge lengths and dihedral angles of the mesh. It is known that this representation associates a unique vector to an equivalence class of nodal positions that differ by a rigid body motion. Moreover, integrability conditions that ensure the existence of nodal positions that match a given vector of edge lengths and dihedral angles have been established. The goal of this paper is to develop the machinery needed to use the NRIC for solving geometric optimization problems. First, we use the integrability conditions to derive an implicit description of the space of discrete surfaces as a submanifold of an Euclidean space and a corresponding description of its tangent spaces. Secondly, we reformulate the integrability conditions using quaternions and provide explicit formulas for their first and second derivatives facilitating the use of Hessians in NRICbased optimization problems. Lastly, we introduce a fast and robust algorithm that reconstructs nodal positions from almost integrable NRIC. We demonstrate the benefits of this approach on a collection of geometric optimization problems. Comparisons to alternative approaches indicate that NRICbased optimization is particularly effective for problems involving nearisometric deformations.
[preprint][supp. document][video][doi]



The Reduced Immersed Method for RealTime FluidElastic Solid Interaction and Contact Simulation
Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, Klaus Hildebrandt
ACM Transactions on Graphics 38(6) (SIGGRAPH Asia 2019)
Abstract: We introduce the Reduced Immersed Method (RIM) for the realtime simulation of twoway coupled incompressible fluids and elastic solids and the interaction of multiple deformables with (self)collisions. Our framework is based on a novel discretization of the immersed boundary equations of motion, which model fluid and deformables as a single incompressible medium and their interaction as a unified system on a fixed domain combining Eulerian and Lagrangian terms. An advantage for realtime simulations resulting from this modeling is that twoway coupling phenomena can be faithfully simulated while avoiding costly calculations such as tracking the deforming fluidsolid interfaces and the associated fluid boundary conditions. Our discretization enables the combination of a PIC/FLIP fluid solver with a reducedorder Lagrangian elasticity solver. Crucial for the performance of RIM is the efficient transfer of information between the elasticity and the fluid solver and the synchronization of the Lagrangian and Eulerian settings. We introduce the concept of twin subspaces that enables an efficient reducedorder modeling of the transfer. Our experiments demonstrate that RIM handles complex meshes and highly resolved fluids for large time steps at high framerates on offtheshelf hardware, even in the presence of high velocities and rapid user interaction. Furthermore, it extends reducedorder elasticity solvers such as HyperReduced Projective Dynamics with natural collision handling.
[preprint][video][video (low res.)][code][doi]



A Geometric Optimization Approach for the Detection and Segmentation of Multiple Aneurysms
Kai Lawonn, Monique Meuschke, Ralph Wickenhoefer, Bernhard Preim, Klaus Hildebrandt
Computer Graphics Forum 38(3) (EuroVis 2019), pages 413–425
Abstract: We present a method for detecting and segmenting aneurysms in blood vessels that facilitates the assessment of risks associated with the aneurysms. The detection and analysis of aneurysms is important for medical diagnosis as aneurysms bear the risk of rupture with fatal consequences for the patient. For risk assessment and treatment planning, morphological descriptors, such as the height and width of the aneurysm, are used. Our system enables the fast detection, segmentation and analysis of single and multiple aneurysms. The method proceeds in two stages plus an optional third stage in which the user interacts with the system. First, a set of aneurysm candidate regions is created by segmenting regions of the vessels. Second, the aneurysms are detected by a classification of the candidates. The third stage allows users to adjust and correct the result of the previous stages using a brushing interface. When the segmentation of the aneurysm is complete, the corresponding ostium curves and morphological descriptors are computed and a report including the results of the analysis and renderings of the aneurysms is generated. The novelty of our approach lies in combining an analytic characterization of aneurysms and vessels to generate a list of candidate regions with a classifier trained on data to identify the aneurysms in the candidate list. The candidate generation is modeled as a global combinatorial optimization problem that is based on a local geometric characterization of aneurysms and vessels and can be efficiently solved using a graph cut algorithm. For the aneurysm classification scheme, we identified four suitable features and modeled appropriate training data. An important aspect of our approach is that the resulting system is fast enough to allow for user interaction with the global optimization by specifying additional constraints via a brushing interface.
[preprint][video][supp. document][doi]



Fast Approximation of Laplace–Beltrami Eigenproblems
Ahmad Nasikun, Christopher Brandt, Klaus Hildebrandt
Computer Graphics Forum 37(5) (Symposium on Geometry Processing 2018), pages 121–134
Abstract: The spectrum and eigenfunctions of the Laplace–Beltrami 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 Laplace–Beltrami operator. Our experiments indicate that the resulting spectra wellapproximate reference spectra, which are computed with stateoftheart eigensolvers. Moreover, we demonstrate that for different applications 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][code][doi]



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
Presented at ACM 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 R^{n} 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 ACM 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 ACM 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] 

