Stephan Bischoff

Geometric modeling based on polygonal meshes Video files associated with this course are available from the citation page

In 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.

Automatic restoration of polygon models

We present a fully automatic technique which converts an inconsistent input mesh into an output mesh that is guaranteed to be a clean and consistent mesh representing the closed manifold surface of a solid object. The algorithm removes all typical mesh artifacts such as degenerate triangles, incompatible face orientation, non-manifold vertices and edges, overlapping and penetrating polygons, internal redundant geometry, as well as gaps and holes up to a user-defined maximum size ρ. Moreover, the output mesh always stays within a prescribed tolerance ϵ to the input mesh. Due to the effective use of a hierarchical octree data structure, the algorithm achieves high voxel resolution (up to 40963 on a 2GB PC) and processing times of just a few minutes for moderately complex objects. We demonstrate our technique on various architectural CAD models to show its robustness and reliability.

Structure Preserving CAD Model Repair

There are two major approaches for converting a tessellated CAD model that contains inconsistencies like cracks or intersections into a manifold and closed triangle mesh. Surface oriented algorithms try to x the inconsistencies by perturbing the input only slightly, but they often cannot handle special cases. Volumetric algorithms on the other hand produce guaranteed manifold meshes but mostly destroy the structure of the input tessellation due to global resampling. In this paper we combine the advantages of both approaches: We exploit the topological simplicity of a voxel grid to reconstruct a cleaned up surface in the vicinity of intersections and cracks, but keep the input tessellation in regions that are away from these inconsistencies. We are thus able to preserve any characteristic structure (i.e. iso-parameter or curvature lines) that might be present in the input tessellation. Our algorithm closes gaps up to a user-dened maximum diameter, resolves intersections, handles incompatible patch orientations and produces a feature-sensitive, manifold output that stays within a prescribed error-tolerance to the input model.

Geometric modeling based on triangle meshes

Course Summary This course is designed to cover the entire geometry processing pipeline based on triangle meshes. We will present the latest concepts for mesh generation and mesh repair, for geometry and topology optimizations like mesh smoothing, decimation, and remeshing, for parametrization, segmentation, and shape editing. In addition to describing and discussing the related algorithms, we will also give valuable implementation hints and provide source code for most of the covered topics. The course assumes only very basic knowledge on geometric concepts in general, but does not require specific knowledge on polygonal meshes and how to discretize the respective problems for those. It is intended for computer graphics researchers, software developers and engineers from CAGD, computer games, or the movie industry, who are interested in geometry processing algorithms based on triangle meshes. Stephan Bischoff graduated in 1999 with a masters in computer science from the University of Karlsruhe, Germany. He then worked at the graphics group of the Max-Planck-Institute for Computer Science in Saarbrücken, Germany. In 2001 he joined the Computer Graphics Group at the Aachen University of Technology, Germany, where he is working as a research associate with Prof. Dr. Leif Kobbelt and is currently pursuing his PhD. He is an experienced speaker and presented courses at Eurographics and Shape Modeling International. His research interests focus on freeform shape representations for efficient geometry processing, topology control techniques for level-set surfaces, reconstruction of medical data sets and the restoration and healing of CAD models. of which he is paper co-chair this year. He is an experienced speaker and presented courses at Eurographics and Shape Modeling International. His research interests include geometry processing in general, and mesh generation, mesh optimization, and multiresolution shape editing in particular. His research interests include all areas of Computer Graphics and Geometry Processing with a focus on multiresolution and freeform modeling, 3D model optimization, as well as the efficient handling of polygonal mesh data. during the last years resulted in numerous publications in top scientific journals and international conferences. He is invited regularly to give keynote presentations and tutorial lectures. For his contributions he received several scientific awards. He has ongoing collaborations with colleagues in Europe, North America, and Asia, and frequently serves on international program committees. He organized and co-chaired several workshops and conferences. he was a postdoctoral scholar at Stanford University , where he also held a position as visiting assistant professor during the …

Towards hardware implementation of loop subdivision

We present a novel algorithm to evaluate and render Loop subdivision surfaces. The algorithm exploits the fact that Loop subdivision surfaces are piecewise polynomial and uses the forward difference technique for efficiently computing uniform samples on the limit surface. The main advantage of our algorithm is that it only requires a small and constant amount of memory that does not depend on the subdivision depth. The simple structure of the algorithm enables a scalable degree of hardware implementation. By low-level parallelization of the computations, we can reduce the critical computations costs to a theoretical minimum of about one float [3]-operation per triangle.

Isosurface reconstruction with topology control

Extracting isosurfaces from volumetric datasets is an essential step for indirect volume rendering algorithms. For physically measured data, e.g. in medical imaging applications, one often introduces topological errors such as small handles that stem from measurement inaccuracy and cavities that are generated by tight folds of an organ. During isosurface extraction these measurement errors result in a surface whose genus is much higher than that of the actual surface. In many cases, however, the topological type of the object under consideration is known beforehand, e.g., the cortex of a human brain is always homeomorphic to a sphere. By using topology preserving morphological operators we can exploit this knowledge to gradually dilate an initial set of voxels with correct topology until it fits the target isosurface. This approach avoids the formation of handles and cavities and guarantees a topologically correct reconstruction of the objects surface.

Towards robust broadcasting of geometry data

Abstract We present new algorithms for the robust transmission of geometric datasets, i.e. transmission which allows the receiver to recover (an approximation of) the original geometric object even if parts of the data get lost on the way. These algorithms can be considered as hinted point cloud triangulation schemes since the general manifold reconstruction problem is simplified by adding tags to the vertices and by providing a coarse base mesh which determines the global surface topology. Robust transmission techniques exploit the geometric coherence of the data and do not require redundant transmission protocols on lower software layers. As an example application scenario, we describe the teletext-like broadcasting of three-dimensional (3D) models.

Ellipsoid decomposition of 3D-models

In this paper we present a simple technique to approximate the volume enclosed by a given triangle mesh with a set of overlapping ellipsoids. This type of geometry representation allows us to approximately reconstruct 3D-shapes from a very small amount of information being transmitted. The two central questions that we address are: how can we compute optimal fitting ellipsoids that lie in the interior of a given triangle mesh and how do we select the most significant (least redundant) subset from a huge number of candidate ellipsoids. Our major motivation for computing ellipsoid decompositions is the robust transmission of geometric objects where the receiver can reconstruct the 3D-shape even if part of the data gets lost during transmission.

Sub-Voxel Topology Control for Level-Set Surfaces

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.

Snakes on Triangle Meshes

In this work we introduce a new method for representing and evolving snakes that are constrained to lie on a prescribed surface (triangle mesh). The new representation allows to automatically adapt the snake resolution to the surface tesselation and does not need any (unstable) back-projection operations. Furthermore, it enables efficient and robust collision detection and gives us complete control on the topological behaviour of the snakes, i.e. snakes may split or merge depending on the intended task. Possible applications include enhanced mesh scissoring operations and the detection of constrictions of a surface.