Point Lattices in Computer Graphics and Visualization

T. Möller, A. Entezari, J. Morey, K. Mueller, V. Ostromoukhov, D. Van De Ville

Tutorial, IEEE Visualization 2005 (VIS'05), Minneapolis MN, USA, October 23-28, 2005.

This course is motivated by the deep connections and applications of point lattice theory in the mathematics of computer graphics and the role it plays in multidimensional signal processing and tilings. Next to an introduction to the theory and history of point lattices and the related sampling and group theories, we present an in-depth survey from two different perspectives:

1. Signal Processing—Functional Analysis and Sampling Theory
   All computational fields in science and engineering have to deal with discrete representations of continuous phenomena. Clearly, sampling theory is crucial to provide the essential link between the discrete and the continuous domain. Digital signal processing algorithms can only act on the discrete data, but should not loose sight of the continuous-domain aspect of their operations. As we will show, many interesting practical problems are best approached from this theoretic framework. Therefore, we will review general sampling theory in arbitrary dimensions and focus on recent developments for optimal lattices. This part will contain many examples and good-practice in image processing, medical imaging, and volume rendering. We survey reconstruction filter designs, wavelet techniques, medical reconstruction, discretization and rendering aspects for 2D, 3D, and 4D lattices. At the end, the attendee will comprehend how to put a proper discrete/continuous model for his/her application.
2. Crystallography—Geometry and Group Theory
   The study of the formation and structure of crystals has been the interest of scientists for many centuries. Consequently, the symmetries and translation invariant properties of point lattices have been studied and investigated thoroughly in the field of crystallography and solid-state physics. Group theory brought mathematical rigor to these fields. We take the opportunity in this course to migrate the most interesting results from this domain to the computer graphics community. Besides intricate mathematical concepts, regular structures have a strong aesthetic impact and have been incorporated into artistic expressions from ancient ornamental structures to famous works of Escher and general tiling patterns. In this part, we introduce fundamental group theory related to point lattices; we also effectively demonstrate geometric tools for the visualization of tilings and patterns in 2D, 3D, and 4D.

Slides of the presentation (PDF, 6.5 Mb)

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