Search Results

Now showing 1 - 10 of 13
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    Young Researcher Award 2007
    (The Eurographics Association and Blackwell Publishing Ltd, 2007) Botsch, Mario
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    The Diamond Laplace for Polygonal and Polyhedral Meshes
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Bunge, Astrid; Botsch, Mario; Alexa, Marc; Digne, Julie and Crane, Keenan
    We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element - an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non-zero coefficients that depends on the degree of the mesh elements.
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    Constructing L∞ Voronoi Diagrams in 2D and 3D
    (The Eurographics Association and John Wiley & Sons Ltd., 2022) Bukenberger, Dennis R.; Buchin, Kevin; Botsch, Mario; Campen, Marcel; Spagnuolo, Michela
    Voronoi diagrams and their computation are well known in the Euclidean L2 space. They are easy to sample and render in generalized Lp spaces but nontrivial to construct geometrically. Especially the limit of this norm with p -> ∞ lends itself to many quad- and hex-meshing related applications as the level-set in this space is a hypercube. Many application scenarios circumvent the actual computation of L∞ diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis-aligned sites. Our novel algorithm allows for the construction of generalized L∞ Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level-set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non-cell geometry is filtered out based on the L∞ Voronoi criterion, leaving only the non-convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L∞ diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm.
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    Robust Articulated-ICP for Real-Time Hand Tracking
    (The Eurographics Association and John Wiley & Sons Ltd., 2015) Tagliasacchi, Andrea; Schröder, Matthias; Tkach, Anastasia; Bouaziz, Sofien; Botsch, Mario; Pauly, Mark; Mirela Ben-Chen and Ligang Liu
    We present a robust method for capturing articulated hand motions in realtime using a single depth camera. Our system is based on a realtime registration process that accurately reconstructs hand poses by fitting a 3D articulated hand model to depth images. We register the hand model using depth, silhouette, and temporal information. To effectively map low-quality depth maps to realistic hand poses, we regularize the registration with kinematic and temporal priors, as well as a data-driven prior built from a database of realistic hand poses. We present a principled way of integrating such priors into our registration optimization to enable robust tracking without severely restricting the freedom of motion. A core technical contribution is a new method for computing tracking correspondences that directly models occlusions typical of single-camera setups. To ensure reproducibility of our results and facilitate future research, we fully disclose the source code of our implementation.
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    Polyhedral Finite Elements Using Harmonic Basis Functions
    (The Eurographics Association and Blackwell Publishing Ltd, 2008) Martin, Sebastian; Kaufmann, Peter; Botsch, Mario; Wicke, Martin; Gross, Markus
    Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.
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    TailorMe: Self-Supervised Learning of an Anatomically Constrained Volumetric Human Shape Model
    (The Eurographics Association and John Wiley & Sons Ltd., 2024) Wenninger, Stephan; Kemper, Fabian; Schwanecke, Ulrich; Botsch, Mario; Bermano, Amit H.; Kalogerakis, Evangelos
    Human shape spaces have been extensively studied, as they are a core element of human shape and pose inference tasks. Classic methods for creating a human shape model register a surface template mesh to a database of 3D scans and use dimensionality reduction techniques, such as Principal Component Analysis, to learn a compact representation. While these shape models enable global shape modifications by correlating anthropometric measurements with the learned subspace, they only provide limited localized shape control. We instead register a volumetric anatomical template, consisting of skeleton bones and soft tissue, to the surface scans of the CAESAR database. We further enlarge our training data to the full Cartesian product of all skeletons and all soft tissues using physically plausible volumetric deformation transfer. This data is then used to learn an anatomically constrained volumetric human shape model in a self-supervised fashion. The resulting TAILORME model enables shape sampling, localized shape manipulation, and fast inference from given surface scans.
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    Polygon Laplacian Made Robust
    (The Eurographics Association and John Wiley & Sons Ltd., 2024) Bunge, Astrid; Bukenberger, Dennis R.; Wagner, Sven Dominik; Alexa, Marc; Botsch, Mario; Bermano, Amit H.; Kalogerakis, Evangelos
    Discrete Laplacians are the basis for various tasks in geometry processing. While the most desirable properties of the discretization invariably lead to the so-called cotangent Laplacian for triangle meshes, applying the same principles to polygon Laplacians leaves degrees of freedom in their construction. From linear finite elements it is well-known how the shape of triangles affects both the error and the operator's condition. We notice that shape quality can be encapsulated as the trace of the Laplacian and suggest that trace minimization is a helpful tool to improve numerical behavior. We apply this observation to the polygon Laplacian constructed from a virtual triangulation [BHKB20] to derive optimal parameters per polygon. Moreover, we devise a smoothing approach for the vertices of a polygon mesh to minimize the trace. We analyze the properties of the optimized discrete operators and show their superiority over generic parameter selection in theory and through various experiments.
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    Polygon Laplacian Made Simple
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Bunge, Astrid; Herholz, Philipp; Kazhdan, Misha; Botsch, Mario; Panozzo, Daniele and Assarsson, Ulf
    The discrete Laplace-Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de-facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non-convex, and even non-planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] - while being simpler to compute.
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    Example‐Driven Deformations Based on Discrete Shells
    (The Eurographics Association and Blackwell Publishing Ltd., 2011) Fröhlich, Stefan; Botsch, Mario; Eduard Groeller and Holly Rushmeier
    Despite the huge progress made in interactive physics‐based mesh deformation, manipulating a geometrically complex mesh or posing a detailed character is still a tedious and time‐consuming task. Example‐driven methods significantly simplify the modelling process by incorporating structural or anatomical knowledge learned from example poses. However, these approaches yield counter‐intuitive, non‐physical results as soon as the shape space spanned by the example poses is left. In this paper, we propose a modelling framework that is both example‐driven and physics‐based and thereby overcomes the limitations of both approaches. Based on an extension of the discrete shell energy we derive mesh deformation and mesh interpolation techniques that can be seamlessly combined into a simple and flexible mesh‐based inverse kinematics system.
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    Adaptive Space Deformations Based on Rigid Cells
    (The Eurographics Association and Blackwell Publishing Ltd, 2007) Botsch, Mario; Pauly, Mark; Wicke, Martin; Gross, Markus
    We propose a new adaptive space deformation method for interactive shape modeling. A novel energy formulation based on elastically coupled volumetric cells yields intuitive detail preservation even under large deformations. By enforcing rigidity of the cells, we obtain an extremely robust numerical solver for the resulting nonlinear optimization problem. Scalability is achieved using an adaptive spatial discretization that is decoupled from the resolution of the embedded object. Our approach is versatile and easy to implement, supports thin-shell and solid deformations of 2D and 3D objects, and is applicable to arbitrary sample-based representations, such as meshes, triangle soups, or point clouds.