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    Freeform Shape Representations for Efficient Geometry Processing
    (Blackwell Publishers, Inc and the Eurographics Association, 2003) Kobbelt, Leif
    The 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.
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    Multiresolution Surface Representation Based on Displacement Volumes
    (Blackwell Publishers, Inc and the Eurographics Association, 2003) Botsch, Mario; Kobbelt, Leif
    We propose a new representation for multiresolution models which uses volume elements enclosed between thedifferent resolution levels to encode the detail information. Keeping these displacement volumes locally constantduring a deformation of the base surface leads to a natural behaviour of the detail features. The correspondingreconstruction operator can be implemented efficiently by a hierarchical iterative relaxation scheme, providingclose to interactive response times for moderately complex models.Based on this representation we implement a multiresolution editing tool for irregular polygon meshes that allowsthe designer to freely edit the base surface of a multiresolution model without having to care about self-intersectionsin the respective detailed surface. We demonstrate the effectiveness and robustness of the reconstructionby several examples with real-world data.
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    Sub-Voxel Topology Control for Level-Set Surfaces
    (Blackwell Publishers, Inc and the Eurographics Association, 2003) Bischoff, Stephan; Kobbelt, Leif
    Active contour models are an efficient, accurate, and robust tool for the segmentation of 2D and 3D image data.In particular, geometric deformable models (GDM) that represent an active contour as the level set of an implicitfunction have proven to be very effective. GDMs, however, do not provide any topology control, i.e. contours maymerge or split arbitrarily and hence change the genus of the reconstructed surface. This behavior is inadequate insettings like the segmentation of organic tissue or other objects whose genus is known beforehand. In this paperwe describe a novel method to overcome this limitation while still preserving the favorable properties of the GDMsetup. We achieve this by adding (sparse) topological information to the volume representation at locations whereit is necessary to locally resolve topological ambiguities. Since the sparse topology information is attached to theedges of the voxel grid, we can reconstruct the interfaces where the deformable surface touches itself at sub-voxelaccuracy. We also demonstrate the efficiency and robustness of our method.