EUROGRAPHICS 2012/ M.-P. Cani, F. Ganovelli STAR – State of The Art Report
Symmetry in 3D Geometry: Extraction and Applications
Niloy J. Mitra Mark Pauly Michael Wand Duygu Ceylan
UCL EPFL MPII EPFL
Abstract
The concept of symmetry has received significant attention in computer graphics and computer vision research in recent years. Numerous methods have been proposed to find and extract geometric symmetries and exploit such high-level structural information for a wide variety of geometry processing tasks. This report surveys and classifies recent developments in symmetry detection. We focus on elucidating the similarities and differences between existing methods to gain a better understanding of a fundamental problem in digital geometry processing and shape understanding in general. We discuss a variety of applications in computer graphics and geometry that benefit from symmetry information for more effective processing. An analysis of the strengths and limitations of existing algorithms highlights the plenitude of opportunities for future research both in terms of theory and applications.
1. Introduction
The mathematical sciences particularly exhibit order, sym- metry, and limitation; and these are the greatest forms of the beautiful. - Aristotle Human brain Starfish Steam turbine Symmetry is a universal concept in nature, science, and art (see Figure 1). In the physical world, geometric symme- tries and structural regularity occur at all scales, from crystal lattices and carbon nano-structures to the human body, archi- tectural artifacts, and the formation of galaxies. Many bio- chemical processes are governed by symmetry and as a re- Taj Mahal Vitruvian Man Spiral galaxy sult we experience a wealth of biological structures that ex- hibit strong regularity patterns. This abundance of symmetry in the natural world has inspired mankind from its origins to incorporate symmetry in the design of tools, buildings, art- work, or even music. Besides aesthetic considerations, phys- ical optimality principles and manufacturing efficiency often Nautilus shell Simian virus Silicon nanostructures lead to symmetric designs in engineering and architecture.
Geometric data, acquired via scanning or modeled from scratch, is traditionally represented as a collection of low- level primitives, e.g., point clouds, polygon meshes, NURBS patches, etc., without explicit encoding of their high-level Insect eye Geodesic dome Persian carpet structure. Finding symmetries in such geometric data is thus an important problem in geometry processing that has re- Figure 1: Examples of symmetry in nature, engineering, ar- ceived significant attention in recent years. Numerous ap- chitecture, and art. plications immediately benefit from extracted symmetry in- formation, e.g., shape matching, segmentation, retrieval, ge-