TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Wardetzky, Max T1 - On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces N2 - We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidian $3$-space. The notion of totally normal convergence is shown to be equivalent to the convergence of either one of the following: surface area, intrinsic metric, and Laplace-Beltrami operators. We further s how that totally normal convergence implies convergence results for shortest geodesics, mean curvature, and solutions to the Dirichlet problem. This work provides the justification for a discrete theory of differential geometric operators defined on polyhedral surfaces based on a variational formulation. T3 - ZIB-Report - 05-24 KW - Discrete Geometry KW - Differential Geometry KW - Numerical Analysis Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8587 ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Preuß, Eike T1 - Evolution of 3d Curves under Strict Spatial Constraints N2 - We present a new algorithm for fairing of space curves with respect spatial constraints based on a vector valued curvature function. Smoothing with the vector valued curvature function is superior to standard Frenet techniques since the individual scalar components can be modeled similar to curvature-based curve smoothing techniques in 2d. This paper describes a curve smoothing flow that satisfies strict spatial constraints and allows simultaneous control of both curvature functions. T3 - ZIB-Report - 05-33 KW - curve fairing KW - curve smoothing KW - curvature smoothing flow Y1 - 2005 N1 - kein Volltext, Artikel ist in Verlagspublikation erschienen, der Preprint wurde offensichtlich vor längerer Zeit zurückgezogen (keine urn). Kossick 20. 11. 13 ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Polthier, Konrad T1 - Anisotropic filtering of non-linear surface features JF - Computer Graphics Forum Y1 - 2004 VL - 23(3) SP - 391 EP - 400 CY - Grenoble, France ER - TY - GEN A1 - Brandt, Christopher A1 - Tycowicz, Christoph von A1 - Hildebrandt, Klaus T1 - Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects N2 - 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 R^n 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. T3 - ZIB-Report - 16-29 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59504 SN - 1438-0064 ER - TY - JOUR A1 - Brandt, Christopher A1 - Tycowicz, Christoph von A1 - Hildebrandt, Klaus T1 - Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects JF - Computer Graphics Forum N2 - 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. Y1 - 2016 U6 - https://doi.org/10.1111/cgf.12832 VL - 35 IS - 2 ER - TY - JOUR A1 - Tycowicz, Christoph von A1 - Schulz, Christian A1 - Seidel, Hans-Peter A1 - Hildebrandt, Klaus T1 - Real-time Nonlinear Shape Interpolation JF - ACM Transactions on Graphics N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1145/2729972 VL - 34 IS - 3 SP - 34:1 EP - 34:10 ER - TY - CHAP A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Seidel, Hans-Peter A1 - Hildebrandt, Klaus T1 - Animating articulated characters using wiggly splines T2 - ACM SIGGRAPH / Eurographics Symposium on Computer Animation N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1145/2786784.2786799 SP - 101 EP - 109 ER - TY - CHAP A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Wardetzky, Max T1 - Smooth feature lines on surface meshes T2 - Third Eurographics Symposium on Geometry Processing Y1 - 2005 SP - 85 EP - 90 ER - TY - CHAP A1 - Hildebrandt, Klaus A1 - Polthier, Konrad A1 - Preuß, Eike T1 - Evolution of 3D curves under strict spatial constraints T2 - IEEE Proceedings on Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005) Y1 - 2005 U6 - https://doi.org/10.1109/CAD-CG.2005.38 SP - 40 EP - 45 ER - TY - CHAP A1 - Polthier, Konrad A1 - Bobenko, Alexander A1 - Hildebrandt, Klaus A1 - Kornhuber, Ralf A1 - Tycowicz, Christoph von A1 - Yserentant, Harry A1 - Ziegler, Günter M. T1 - Geometry processing T2 - MATHEON - Mathematics for Key Technologies Y1 - 2014 SN - 978-3-03719-137-8 U6 - https://doi.org/10.4171/137 VL - 1 SP - 341 EP - 355 PB - EMS Publishing House ER - TY - JOUR A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Seidel, Hans-Peter A1 - Hildebrandt, Klaus T1 - Animating Deformable Objects Using Sparse Spacetime Constraints JF - ACM Transactions on Graphics N2 - 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. Y1 - 2014 U6 - https://doi.org/10.1145/2601097.2601156 VL - 33 IS - 4 SP - 109:1 EP - 109:10 ER - TY - JOUR A1 - Tycowicz, Christoph von A1 - Schulz, Christian A1 - Seidel, Hans-Peter A1 - Hildebrandt, Klaus T1 - An Efficient Construction of Reduced Deformable Objects JF - ACM Transactions on Graphics N2 - 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. Y1 - 2013 U6 - https://doi.org/10.1145/2508363.2508392 VL - 32 IS - 6 SP - 213:1 EP - 213:10 PB - ACM ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Interactive spacetime control of deformable objects JF - ACM Transactions on Graphics N2 - 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 technique that controls the motion of deformable objects and offers an interactive response. 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 using a new fast and robust numerical scheme for solving a set of one-dimensional problems based on an explicit representation of the wiggly splines. Y1 - 2012 U6 - https://doi.org/10.1145/2185520.2185567 VL - 31 IS - 4 SP - 71:1 EP - 71:8 PB - ACM ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Interactive surface modeling using modal analysis JF - ACM Transactions on Graphics N2 - 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]. Y1 - 2011 U6 - https://doi.org/10.1145/2019627.2019638 VL - 30 IS - 5 SP - 119:1 EP - 119:11 ER - TY - JOUR A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad T1 - Modal Shape Analysis beyond Laplacian JF - Computer Aided Geometric Design N2 - 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. Y1 - 2012 U6 - https://doi.org/10.1016/j.cagd.2012.01.001 VL - 29 IS - 5 SP - 204 EP - 2018 ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad ED - Mourrain, Bernard ED - Schaefer, Scott ED - Xu, Guoliang T1 - Eigenmodes of Surface Energies for Shape Analysis T2 - Advances in Geometric Modeling and Processing (Proceedings of Geometric Modeling and Processing 2010) N2 - 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. Y1 - 2010 U6 - https://doi.org/10.1007/978-3-642-13411-1_20 VL - 6130 SP - 296 EP - 314 PB - Springer Berlin / Heidelberg ER - TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad T1 - Anisotropic Filtering of Non-Linear Surface Features N2 - A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well-known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non-linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments. T3 - ZIB-Report - 04-25 KW - mesh smoothing KW - principal curvature KW - feature lines KW - anisotropic smoothing KW - prescribed mean curvature Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8003 ER -