We introduce Geometric Multigrid Neural Networks (GMNN), a novel network structure for geometric deep learning on point clouds and surfaces. Convolutional neural networks face a common challenge: how can relevant features be communicated over longer distances? Our architecture facilitates long-distance communication with Geometric Multigrid Convolution (GMC) blocks, which apply convolutions in parallel to features defined on each level of a multigrid representation of the surface, and enable communication all the way up and down the hierarchy. We observe two major structural advantages of such a network: First, because each GMC operates on all levels of the multigrid hierarchy, even early stages can make use of coarse-scale information and receptive field grows rapidly with depth. Second, networks built with this backbone have the freedom to route information between different scales, including in ways not possible for other architectures. Because of these advantages, we find that a GMNN can combine the fast convergence of a shallow network with the greater expressiveness of a deeper, larger network. We build a GMNN from the components of a state-of-the-art U-Net, and find that on real tasks it can match or exceed the accuracy of the base network while using fewer epochs and roughly half the parameter count.

Geometric multigrid (GMG) methods are a fundamental tool for efficiently solving large sparse linear systems. A requirement for GMG is a hierarchy of grids; however, many practical volumetric domains are available only as single, irregular tetrahedral meshes, making the construction of a multigrid hierarchy necessary. Existing approaches often trade off speed against hierarchy quality: remeshing- or coarsening-based methods can be expensive to construct, whereas graph-based techniques are fast but often yield weaker multigrid performance. We introduce GravoTet, which bridges this gap by combining geometric structure with graph-based efficiency to construct fast and effective multigrid hierarchies. GravoTet builds a vertex hierarchy and then generates graph-Voronoi diagrams whose dual cells define coarse tetrahedra, enabling rapid construction of multigrid levels. Boundary elements are explicitly prioritized during both sampling and tet generation to preserve boundary. In our evaluation, we solve Poisson and biharmonic problems on irregular tetrahedral meshes and compare GravoTet against state-of-the-art geometric multigrid, algebraic multigrid and direct solvers, demonstrating superior performance, particularly on large meshes.

Correlation clustering (CC) offers an effective approach to analyze scalar field ensembles by detecting correlated regions and consistent structures, enabling the extraction of meaningful patterns. However, existing CC methods are computationally expensive, making them impractical for both interactive analysis and large-scale scalar fields. We introduce the Local-to-Global Correlation Clustering (LoGCC) framework, which accelerates pivot-based CC by leveraging the spatial structure of scalar fields and the weak transitivity of correlation. LoGCC operates in two stages: a local step that uses the neighborhood graph of the scalar field's spatial domain to build highly correlated local clusters, and a global step that merges them into global clusters. We implement the LoGCC framework for two well-known pivot-based CC methods, Pivot and CN-Pivot, demonstrating its generality. Our evaluation using synthetic and real-world meteorological and medical image segmentation datasets shows that LoGCC achieves speedups-up to 15 × for Pivot and 200 × for CN-Pivot-and improved scalability to larger scalar fields, while maintaining cluster quality. These contributions broaden the applicability of correlation clustering in large-scale and interactive analysis settings.

Abstract The contour depth methodology enables non‐parametric summarization of contour ensembles by extracting their representatives, confidence bands, and outliers for visualization (via contour boxplots) and robust downstream procedures. We address two shortcomings of these methods. Firstly, we significantly expedite the computation and recomputation of Inclusion Depth (ID), introducing a linear‐time algorithm for epsilon ID, a variant used for handling ensembles with contours with multiple intersections. We also present the inclusion matrix, which contains the pairwise inclusion relationships between contours, and leverage it to accelerate the recomputation of ID. Secondly, extending beyond the single distribution assumption, we present the Relative Depth (ReD), a generalization of contour depth for ensembles with multiple modes. Building upon the linear‐time eID, we introduce CDclust, a clustering algorithm that untangles ensemble modes of variation by optimizing ReD. Synthetic and real datasets from medical image segmentation and meteorological forecasting showcase the speed advantages, illustrate the use case of progressive depth computation and enable non‐parametric multimodal analysis. To promote research and adoption, we offer the contour‐depth Python package.

Ensembles of contours arise in various applications like simulation, computer-aided design, and semantic segmentation. Uncovering ensemble patterns and analyzing individual members is a challenging task that suffers from clutter. Ensemble statistical summarization can alleviate this issue by permitting analyzing ensembles' distributional components like the mean and median, confidence intervals, and outliers. Contour boxplots, powered by Contour Band Depth (CBD), are a popular non-parametric ensemble summarization method that benefits from CBD's generality, robustness, and theoretical properties. In this work, we introduce Inclusion Depth (ID), a new notion of contour depth with three defining characteristics. First, ID is a generalization of functional Half-Region Depth, which offers several theoretical guarantees. Second, ID relies on a simple principle: the inside/outside relationships between contours. This facilitates implementing ID and understanding its results. Third, the computational complexity of ID scales quadratically in the number of members of the ensemble, improving CBD's cubic complexity. This also in practice speeds up the computation enabling the use of ID for exploring large contour ensembles or in contexts requiring multiple depth evaluations like clustering. In a series of experiments on synthetic data and case studies with meteorological and segmentation data, we evaluate ID's performance and demonstrate its capabilities for the visual analysis of contour ensembles.

The need to understand the structure of hierarchical or high-dimensional data is present in a variety of fields. Hyperbolic spaces have proven to be an important tool for embedding computations and analysis tasks as their non-linear nature lends itself well to tree or graph data. Subsequently, they have also been used in the visualization of high-dimensional data, where they exhibit increased embedding performance. However, none of the existing dimensionality reduction methods for embedding into hyperbolic spaces scale well with the size of the input data. That is because the embeddings are computed via iterative optimization schemes and the computation cost of every iteration is quadratic in the size of the input. Furthermore, due to the non-linear nature of hyperbolic spaces, euclidean acceleration structures cannot directly be translated to the hyperbolic setting. This article introduces the first acceleration structure for hyperbolic embeddings, building upon a polar quadtree. We compare our approach with existing methods and demonstrate that it computes embeddings of similar quality in significantly less time. Implementation and scripts for the experiments can be found at https://graphics.tudelft.nl/accelerating-hyperbolic-tsne.

Deep learning has 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 few-sample scenarios, and reducing the impact of domain changes. Moreover, the interpretability of the priors allows to adapt deep networks to minor problem variations such as switching between Manhattan and non-Manhattan worlds. We seamlessly incorporate two 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 to existing models in the large-data setting. We validate our model's improved data efficiency, robustness to domain changes, adaptability to non-Manhattan settings.

Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami 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 article, 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 non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.

Learning from 3D point-cloud 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 vector-fields, we separate the network into a scalar- and a vector-stream, 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 state-of-the-art approaches on several benchmarks, while also speeding up training and inference.

Abstract We introduce a construction of subspaces of the spaces of tangential vector, n‐vector, 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.

Abstract This paper introduces the construction of a low‐dimensional nonlinear space capturing the variability of a non‐rigid 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 non‐rigid 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 low‐dimensional 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.

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 vector-valued, rotation-equivariant 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 rotation-equivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.

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.

We introduce the Reduced Immersed Method ( RIM ) for the real-time simulation of two-way 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. One advantage for real-time simulations resulting from this modeling is that two-way coupling phenomena can be faithfully simulated while avoiding costly calculations such as tracking the deforming fluid-solid interfaces and the associated fluid boundary conditions. Our discretization enables the combination of a PIC/FLIP fluid solver with a reduced-order 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 reduced-order modeling of the transfer. Our experiments demonstrate that RIM handles complex meshes and highly resolved fluids for large time steps at high framerates on off-the-shelf hardware, even in the presence of high velocities and rapid user interaction. Furthermore, it extends reduced-order elasticity solvers such as Hyper-Reduced Projective Dynamics with natural collision handling.

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 large‐scale 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 well‐approximate reference spectra, which are computed with state‐of‐the‐art eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.

We present a method for the real-time 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 real-time. The proposed hyper-reduction of Projective Dynamics combines a novel fast approximation method for constraint projections and a scalable construction of sparse subspace bases. The resulting system achieves real-time rates for large sub-spaces enabling rich dynamics and can resolve general user interactions, collision constraints, external forces and changes to the materials. The construction of the hyper-reduced system does not require user-interaction and refrains from using training data or modal analysis, which results in a fast preprocessing stage.

We introduce a variational approach for modeling n -symmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while providing real-time responses. The approach is based on novel biharmonic and m -harmonic energies for n -fields on surface meshes and the integration of hard constraints to the resulting optimization problems. Real-time computation rates are achieved by a model reduction approach employing a Fourier-like n -vector field decomposition, which associates frequencies and modes to n -vector fields on surfaces. To demonstrate the benefits of the proposed n -field modeling approach, we use it for controlling stroke directions in line-art drawings of surfaces and for the modeling of anisotropic BRDFs, which define the reflection behavior of surfaces.

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 prominentcotandiscretization 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 modelling 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 modelling of tangential vector fields.

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.

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 ℝnto 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.

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 resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector‐set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This report provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges.

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.

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 depend 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 to spline conversion of motion capture data.

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.

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.

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.

We propose a scheme for animating deformable objects based on spacetime optimization. The main feature is that it robustly and 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. Our solver is based on a theorem that characterizes the solutions of the optimization problem and allows us to restrict the optimization to 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.

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.

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.

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.

We propose a framework for deformation-based 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 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 Sorkine [2008].

In the past the computer architetures and technologies have only been optimized for speed - never for security. The aspects of IT-security were always disregarded. [1] shows that it is possible to write malware that can not be detected or removed from an infected system by software. Stealth-features are possible by special properties of the x86 architecture. This work describes a general hardware architecture, that will improve the security.

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.

Abstract 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.