Mathematics and Visualization

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