Browsing by Author "Botsch, Mario"
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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 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 Multilinear Model for Bidirectional Craniofacial Reconstruction(The Eurographics Association, 2018) Achenbach, Jascha; Brylka, Robert; Gietzen, Thomas; Hebel, Katja zum; Schömer, Elmar; Schulze, Ralf; Botsch, Mario; Schwanecke, Ulrich; Puig Puig, Anna and Schultz, Thomas and Vilanova, Anna and Hotz, Ingrid and Kozlikova, Barbora and Vázquez, Pere-PauWe present a bidirectional facial reconstruction method for estimating the skull given a scan of the skin surface and vice versa estimating the skin surface given the skull. Our approach is based on a multilinear model that describes the correlation between the skull and the facial soft tissue thickness (FSTT) on the one hand and the head/face surface geometry on the other hand. Training this model requires to densely sample the Cartesian product space of skull shape times FSTT variation, which cannot be obtained by measurements alone. We generate this data by enriching measured data-volumetric computed tomography scans and 3D surface scans of the head-by simulating statistically plausible FSTT variations. We demonstrate the versatility of our novel multilinear model by estimating faces from given skulls as well as skulls from given faces within just a couple of seconds. To foster further research in this direction, we will make our multilinear model publicly available.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 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.Item VMV 2022: Frontmatter(The Eurographics Association, 2022) Bender, Jan; Botsch, Mario; Keim, Daniel A.; Bender, Jan; Botsch, Mario; Keim, Daniel A.