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Item 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, KeenanWe 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.Item A Compact Patch-Based Representation for Technical Mesh Models(The Eurographics Association, 2020) Kammann, Lars; Menzel, Stefan; Botsch, Mario; Krüger, Jens and Niessner, Matthias and Stückler, JörgWe present a compact and intuitive geometry representation for technical models initially given as triangle meshes. For CADlike models the defining features often coincide with the intersection between smooth surface patches. Our algorithm therefore first segments the input model into patches of constant curvature. The intersections between these patches are encoded through Bézier curves of adaptive degree, the patches enclosed by them are encoded by their (constant) mean and Gaussian curvatures. This sparse geometry representation enables intuitive understanding and editing by manipulating either the patches' curvature values and/or the feature curves. During decoding/reconstruction we exploit remeshing and hence are independent of the underlying triangulation, such that besides the feature curve topology no additional connectivity information has to be stored. We also enforce discrete developability for patches with vanishing Gaussian curvature in order to obtain straight ruling lines.Item 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, MichelaVoronoi 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.Item VMV 2022: Frontmatter(The Eurographics Association, 2022) Bender, Jan; Botsch, Mario; Keim, Daniel A.; Bender, Jan; Botsch, Mario; Keim, Daniel A.Item 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, EvangelosHuman 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.Item 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, EvangelosDiscrete 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.Item 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, UlfThe 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.Item NePHIM: A Neural Physics-Based Head-Hand Interaction Model(The Eurographics Association and John Wiley & Sons Ltd., 2025) Wagner, Nicolas; Schwanecke, Ulrich; Botsch, Mario; Bousseau, Adrien; Day, AngelaDue to the increasing use of virtual avatars, the animation of head-hand interactions has recently gained attention. To this end, we present a novel volumetric and physics-based interaction simulation. In contrast to previous work, our simulation incorporates temporal effects such as collision paths, respects anatomical constraints, and can detect and simulate skin pulling. As a result, we can achieve more natural-looking interaction animations and take a step towards greater realism. However, like most complex and computationally expensive simulations, ours is not real-time capable even on high-end machines. Therefore, we train small and efficient neural networks as accurate approximations that achieve about 200 FPS on consumer GPUs, about 50 FPS on CPUs, and are learned in less than four hours for one person. In general, our focus is not to generalize the approximation networks to low-resolution head models but to adapt them to more detailed personalized avatars. Nevertheless, we show that these networks can learn to approximate our head-hand interaction model for multiple identities while maintaining computational efficiency. Since the quality of the simulations can only be judged subjectively, we conducted a comprehensive user study which confirms the improved realism of our approach. In addition, we provide extensive visual results and inspect the neural approximations quantitatively. All data used in this work has been recorded with a multi-view camera rig. Code and data are available at https://gitlab.cs.hs-rm.de/cvmr_releases/HeadHand.Item A Survey on Discrete Laplacians for General Polygonal Meshes(The Eurographics Association and John Wiley & Sons Ltd., 2023) Bunge, Astrid; Botsch, Mario; Bousseau, Adrien; Theobalt, ChristianThe Laplace Beltrami operator is one of the essential tools in geometric processing. It allows us to solve numerous partial differential equations on discrete surface meshes, which is a fundamental building block in many computer graphics applications. Discrete Laplacians are typically limited to standard elements like triangles or quadrilaterals, which severely constrains the tessellation of the mesh. But in recent years, several approaches were able to generalize the Laplace Beltrami and its closely related gradient and divergence operators to more general meshes. This allows artists and engineers to work with a wider range of elements which are sometimes required and beneficial in their field. This paper discusses the different constructions of these three ubiquitous differential operators on arbitrary polygons and analyzes their individual advantages and properties in common computer graphics applications.