In diesem Kapitel werden die wesentlichen Bausteine der 3D-Geometrie eingeführt. Dabei übertragen wir zunächst die 2D-Werkzeuge aus Kap. 2 und 3 in drei Dimensionen. Darüber hinaus werden wir einigen Konzepten begegnen, die wirklich 3D sind, d.h. die in zwei Dimensionen keine Entsprechung besitzen. Mit den geometrischen Werkzeugen dieses Kapitels werden wir in der Lage sein, einfache 3D-Objekte, wie sie in Abb. 10.1 dargestellt sind, zu erzeugen und zu analysieren.

We compare curvature combs and curvature plots.

WELCOME to the special section on Mathematics and Visualization. Mathematics is becoming an increasingly important tool in visualization, aiding the development and analysis of efficient algorithms and data structures. As visualization is beginning to evolve into a mature science and the underlying technology is becoming increasingly complex, the need for a rigorous framework is becoming obvious. Formal mathematical language provides a solid ground which allows us to embed visualization problems within an abstract setting and thereby to make well-established analytical tools applicable. For example, convergence issues and the numerical stability of visualization algorithms are essential for effective and reliable results in visualization. The editors of this special section have actively supported the convergence of mathematics and visualization as organizers of the international conference series Visualization and Mathematics since 1995, the film festival VideoMath at the International Congress of Mathematicians ICM 1998, as well as at various international workshops and summer schools. These activities helped to establish a small but worldwide community of researchers who perform work in this border area; they also gave rise to the new book series Mathematics + Visualization and the videoseries VideoMATH published by Springer Verlag. In particular, the international conference series Visualization and Mathematics in Berlin intensified the interaction of mathematicians and scientists from computer graphics and visualization. The current special section is comprised of a set of five carefully selected articles which are elaborated publications of the latest conference in 2002. In order to emphasize the interdisciplinary character of the young scientific field of mathematical visualization, the articles in this special section were selected to cover a broad range of topics from theory to praxis. The papers include topics from discrete and differential geometry, combinatorial and differential topology, graph theory, vector field analysis, geometry compression, image processing, signal processing, feature detection, image segmentation, numerical methods, and partial differential equations. The articles were selected by the editors of this special section and each carefully reviewed by at least three referees. In the following, we summarize the articles. “Applications of Forman’s Discrete Morse Theory to Topology Visualization and Mesh Compression” by Thomas Lewiner, Héio Lopes and Geovan Tavares: Many advances in science arise from the combination of different fields. Lewiner et al. relate Forman’s discrete geometry, a purely mathematical construction, with popular mesh compression schemes, such as Edgebreaker. This link unveils a new topological view on compression algorithms. The introduction of Forman’s discrete Morse theory and vector fields provides a new tool for topology visualization and the understanding of visualization algorithms. “Visualizing a Sphere Eversion” by George Francis and John M. Sullivan: Finding a process to turn a sphere inside out has been an active research area for geometers and topologists for more than 50 years. Mathematical visualization—using computer graphics tools, numerical algorithms, and geometric invention—provides an adequate collection of techniques to calculate and visualize the complex deformations of the eversion process. Here, mathematical visualization provides insight into abstract constructions and explains mathematics to a broader audience. The paper by Francis and Sullivan provides a glimpse of the exciting history of the sphere eversion problem and presents the underlying techniques of the author’s awarded computer graphics animation “Optiverse.” “Robust Feature Detection and Local Classification for Surfaces Based on Moment Analysis” by Ulrich Clarenz, Martin Rumpf, and Alex Telea: Geometry analysis and processing has become a major topic in computer graphics, as well as feature detection in visualization. The authors present a surface classification method for triangulated surfaces, with applications in surface fairing, mesh decimation, surface segmentation, and surface matching. The method uses local zero and first order moments as an alternative to curvature and utilizes structure tensor-like concepts. The method is well-founded, explained in detail, and shows convincing performance under various conditions. It provides an indicator for the smoothness of a given discrete surface and comes together with a built-in multiscale. Interesting aspects are the difference between the computation of moments in the smooth and nonsmooth surface as well as the analysis of how the moments scale with the scale parameter for smooth and nonsmooth areas. “Fast Evolution of Image Manifolds and Application to Filtering and Segmentation in 3D Medical Images” by Thomas Deschamps, Ravi Malladi, and Igor Ravve: Image analysis techniques are interesting in visualization as they represent “feature detection” methods for one major data type, namely, images. Additionally, many of IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 10, NO. 5, SEPTEMBER/OCTOBER 2004 497

This chapter introduces the practical side of linear algebra used in many of its real-life applications. It commences with a discussion on linear systems with m equations and n unknowns in detail. The chapter also introduces row echelon forms, row, column and null spaces, and rank and nullity of matrices. Linear independence, basis vectors, linear transformations and their representations are among the other topics discussed in the chapter. Finally, the chapter introduces some of the frequently used numerical methods of linear algebra, where partial pivoting, LU-factorization, interpolation, and numerical methods of finding solutions are among the topics covered. Gauss's method involves certain basic operations that leads to an equivalent system, which has the same solution as the original system, but where the solution can be obtained easily by a method called back-substitution.

We discuss G1 smoothness conditions for rectangular and triangular Gregory patches. We then incorporate these G1 conditions into a surface fitting algorithm. Knowledge of the patch type is inconsequential to the formulation of the G1 conditions, hence the term agnostic G1 Gregory surfaces.

The video game graphics pipeline has traditionally rendered the scene using a polygonal approach. Advances in modern graphics hardware now allow the rendering of parametric methods. This thesis explores various smooth surface rendering methods that can be integrated into the video game graphics engine. Moving over to parametric or smooth surfaces from the polygonal domain has its share of issues and there is an inherent need to address various rendering bottlenecks that could hamper such a move. The game engine needs to choose an appropriate method based on in-game characteristics of the objects; character and animated objects need more sophisticated methods whereas static objects could use simpler techniques. Scaling the polygon count over various hardware platforms becomes an important factor. Much control is needed over the tessellation levels, either imposed by the hardware limitations or by the application, to be able to adaptively render the mesh without significant loss in performance. This thesis explores several methods that would help game engine developers in making correct design choices by optimally balancing the trade-offs while rendering the scene using smooth surfaces. It proposes a novel technique for adaptive tessellation of triangular meshes that vastly improves speed and tessellation count. It develops an approximate method for rendering Loop subdivision surfaces on tessellation enabled hardware. A taxonomy and evaluation of the methods is provided and a unified rendering system that provides automatic level of detail by switching between the methods is proposed.

A method for calculating the product of two B-spline functions is presented. The product is computed by solving a linear system. The coefficient matrix of the system is a Gramian, which guarantees that the system has a unique solution. Every element of the coefficient matrix and the righthand vector of the system is an inner product of B-splines. The inner product can be computed accurately by making use of numerical methods.

The eccentricity of rational quadratic Bézier curves is formulated directly in terms of their control-points and weights. Based on this expression, we analyze the range and extreme values of the eccentricity of conic sections expressed in this form. We also provide an explicit expression for the eccentricity of the osculating conic of a rational Bézier curve of high degree.

The Double Insertion, Nonuniform, Stationary subdivision surface (DINUS) generalizes both the nonuniform, bicubic spline surface and the Catmull-Clark subdivision surface. DINUS allows arbitrary knot intervals on the edges, allows incorporation of special features, and provides limit point as well as limit normal rules. It is the first subdivision scheme that gives the user all this flexibility and at the same time all essential limit information, which is important for applications in modeling and adaptive rendering. DINUS is also amenable to analysis techniques for stationary schemes. We implemented DINUS as an Autodesk Maya plugin to show several modeling and rendering examples.

We compare a variety of triangle shape measures using concepts such as smoothness and convexity. We show that one of these measures, the elongation measure, lends itself to an intuitive geometric interpretation.

Among locally supported scattered data schemes, natural neighbor interpolation has some unique features that makes it interesting for a range of applications. However, its restriction to the convex hull of the data sites is a limitation that has not yet been satisfyingly overcome. We use this setting to discuss some aspects of scattered data extrapolation in general, compare existing methods, and propose a framework for the extrapolation of natural neighbor interpolants on the basis of dynamic ghost points.

In this paper we extend the method of inter-modality image registration using the maximization of normalized mutual information (NMI) for the registration of [F-18]-2-fluoro-deoxy-D-glucose (FDG)-positron emission tomography (PET) with T1-weighted magnetic resonance (MR) Volumes. We investigate the impact on the NMI maximization with respect to using coarse-to-fine grained B-spline bases and to the number of bills required for the voxel intensity histograms of each volume. Our results demonstrate that the efficiency and accuracy of elastic, as well as rigid body, registration is improved both through the use of a reduced number of bins in the PET and MR histograms, and of a limited coarse-to-fine grain interpolation of the volume data. To determine the appropriate number of bins prior to registration, we consider the NMI between the two volumes, the mutual information content of the two volumes, as a function of the binning of each volume. Simulated data sets are used for validation and the registration improves that obtained with a standard approach based oil the Statistical Parametric Mapping software. Copyright (C) 2008 John Wiley & Sons, Ltd.

The 7th Dagstuhl seminar on Geometric Modeling was held in May 2008 at the Leibniz-Zentrum für Informatik at Schloss Dagstuhl in Germany.The participants came from 4 continents and 19 countries.There were a total of 46 technical presentations.The seminar was also the setting for the John Gregory Award.The award, named after a pioneer of the field, honored the long-term innovative contributions to field of Hartmut Prautzsch, Helmut Pottmann and Tom Sederberg.The seminar succeeded in bringing together leading researchers to present and discuss radically different approaches to the challenge of modeling complex geometric phenomena on the computer.Acquisition, representation and analysis of 3-dimensional geometry call for the combination of technically complex and often

The 7th Dagstuhl seminar on Geometric Modeling was held in May 2008 at the Leibniz-Zentrum für Informatik at Schloss Dagstuhl in Germany. The participants came from 4 continents and 19 countries. There were a total of 46 technical presentations. The seminar was also the setting for the John Gregory Award. The award, named after a pioneer of the field, honored the long-term innovative contributions to field of Hartmut Prautzsch, Helmut Pottmann and Tom Sederberg. The seminar succeeded in bringing together leading researchers to present and discuss radically different approaches to the challenge of modeling complex geometric phenomena on the computer. Acquisition, representation and analysis of 3-dimensional geometry call for the combination of technically complex and often The 7th Dagstuhl seminar on Geometric Modeling was held in May 2008 at the Leibniz-Zentrum für Informatik at Schloss Dagstuhl in Germany. The participants came from 4 continents and 19 countries. There were a total of 46 technical presentations. The seminar was also the setting for the John Gregory Award. The award, named after a pioneer of the field, honored the long-term innovative contributions to field of Hartmut Prautzsch, Helmut Pottmann and Tom Sederberg. The seminar succeeded in bringing together leading researchers to present and discuss radically different approaches to the challenge of modeling complex geometric phenomena on the computer. Acquisition, representation and analysis of 3-dimensional geometry call for the combination of technically complex and often The 7th Dagstuhl seminar on Geometric Modeling was held in May …