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Item Geometric Modeling Based on Triangle Meshes(The Eurographics Association, 2006) Botsch, Mario; Pauly, Mark; Rössl, Christian; Bischoff, Stephan; Kobbelt, Leif; Nadia Magnenat-Thalmann and Katja BühlerIn the last years triangle meshes have become increasingly popular and are nowadays intensively used in many different areas of computer graphics and geometry processing. In classical CAGD irregular triangle meshes developed into a valuable alternative to traditional spline surfaces, since their conceptual simplicity allows for more flexible and highly efficient processing. Moreover, the consequent use of triangle meshes as surface representation avoids error-prone conversions, e.g., from CAD surfaces to meshbased input data of numerical simulations. Besides classical geometric modeling, other major areas frequently employing triangle meshes are computer games and movie production. In this context geometric models are often acquired by 3D scanning techniques and have to undergo postprocessing and shape optimization techniques before being actually used in production.This course discusses the whole geometry processing pipeline based on triangle meshes. We will first introduce general concepts of surface representations and point out the advantageous properties of triangle meshes in Section 2, and present efficient data structures for their implementation in Section 3. The different sources of input data and types of geometric and topological degeneracies and inconsistencies are described in Section 4, as well as techniques for their removal, resulting in clean two-manifold meshes suitable for further processing. Mesh quality criteria measuring geometric smoothness and element shape together with the corresponding analysis techniques are presented in Section 6. Mesh smoothing reduces noise in scanned surfaces by generalizing signal processing techniques to irregular triangle meshes (Section 7). Similarly, the underlying concepts from differential geometry are useful for surface parametrization as well (Section 8). Due to the enormous complexity of meshes acquired by 3D scanning, mesh decimation techniques are required for error-controlled simplification (Section 9). The shape of triangles, which is important for the robustness of numerical simulations, can be optimized by general remeshing methods (Section 10). After optimizing meshes with respect to the different quality criteria, we finally present techniques for intuitive and interactive shape deformation (Section 11). Since solving linear systems is a commonly required component for many of the presented mesh processing algorithms, we will discuss their efficient solution and compare several existing libraries in Section 12.Item High-Resolution Volumetric Computation of Offset Surfaces with Feature Preservation(The Eurographics Association and Blackwell Publishing Ltd, 2008) Pavic, Darko; Kobbelt, LeifWe present a new algorithm for the efficient and reliable generation of offset surfaces for polygonal meshes. The algorithm is robust with respect to degenerate configurations and computes (self-)intersection free offsets that do not miss small and thin components. The results are correct within a prescribed ?-tolerance. This is achieved by using a volumetric approach where the offset surface is defined as the union of a set of spheres, cylinders, and prisms instead of surface-based approaches that generally construct an offset surface by shifting the input mesh in normal direction. Since we are using the unsigned distance field, we can handle any type of topological inconsistencies including non-manifold configurations and degenerate triangles. A simple but effective mesh operation allows us to detect and include sharp features (shocks) into the output mesh and to preserve them during post-processing (decimation and smoothing). We discretize the distance function by an efficient multi-level scheme on an adaptive octree data structure. The problem of limited voxel resolutions inherent to every volumetric approach is avoided by breaking the bounding volume into smaller tiles and processing them independently. This allows for almost arbitrarily high voxel resolutions on a commodity PC while keeping the output mesh complexity low. The quality and performance of our algorithm is demonstrated for a number of challenging examples.Item Freeform Shape Representations for Efficient Geometry Processing(Eurographics Association, 2003) Kobbelt, LeifThe most important concepts for the handling and storage of freeform shapes in geometry processing applications are parametric representations and volumetric representations. Both have their specific advantages and drawbacks. While the algebraic complexity of volumetric representations is independent from the shape complexity, the domain of a parametric representation usually has to have the same structure as the surface itself (which sometimes makes it necessary to update the domain when the surface is modified). On the other hand, the topology of a parametrically defined surface can be controlled explicitly while in a volumetric representation, the surface topology can change accidentally during deformation. A volumetric representation reduces distance queries or inside/outside tests to mere function evaluations but the geodesic neighborhood relation between surface points is difficult to resolve. As a consequence, it seems promising to combine parametric and volumetric representations to effectively exploit both advantages. In this talk, a number of projects are presented and discussed in which such a combination leads to efficient and numerically stable algorithms for the solution of various geometry processing tasks. Applications include global error control for mesh decimation and smoothing, topology control for level-set surfaces, and shape modeling with unstructured point clouds.Item Geometric Modeling Based on Polygonal Meshes(The Eurographics Association, 2008) Botsch, Mario; Pauly, Mark; Kobbelt, Leif; Alliez, Pierre; Levy, Bruno; Maria Roussou and Jason LeighPolygonal meshes are nowadays intensively used in many different areas of computer graphics and geometry processing. In classical CAGD polygonal meshes developed into a valuable alternative to traditional spline surfaces, since their conceptual simplicity allows for more flexible and more efficient processing. Moreover, the consequent use of triangle meshes avoids error-prone conversions, e.g., the meshing of CAD surfaces for numerical simulations. Besides classical geometric modeling, other major areas frequently employing triangle meshes are computer games and movie production. In this context geometric models are often acquired by 3D scanning techniques and have to undergo post-processing and shape optimization before being actually used in production. The course starts with a comparison of different surface representations, motivating the use of polygonal meshes. We discuss the removal of geometric and topological degeneracies, and introduce quality measures for polygonal meshes, followed by their respective optimization, namely smoothing, decimation, and remeshing. We further discuss parametrization and present interactive shape editing, including a brief discussion on efficient numerical solvers. Since the course covers the whole mesh processing pipeline, it can give a full overview and point out interesting and important connections between the individual topics. For each topic we present the fundamental concepts and current state-of-the-art techniques. Frequent software demonstrations will give the participants a better understanding of the discussed algorithms. Moreover, these demo applications will be available from the course materials, both as binaries and in full source code, based on the popular mesh libraries OpenMesh and CGAL. This enables the participants to implement the discussed algorithms and reproduce the results published in the corresponding papers.