A Computational Model for Periodic Pattern Perception Based on Frieze and Wallpaper Groups

Abstract

Humans have an innate ability to perceive symmetry, yet it is not obvious how to automate this powerful insight. This paper presents a computational model for periodic pattern perception based on the mathematical theory of crystallographic groups. Despite an infinite variety of periodic patterns, only a small set of symmetry groups is needed to characterize pattern structure. In 2D space, there are seven frieze groups describing monochrome patterns that repeat along one direction, and seventeen wallpaper groups for patterns that repeat along two linearly independent directions to tile the plane. The novelty of this work is to "understand" a periodic pattern by automatically finding its underlying lattice and identifying its symmetry group and representative motifs. We also explore the use of this computational model for characterizing patterns that are not exactly periodic, using geometric AIC. Applications of this work include pattern indexing, texture synthesis, image compression, and gait analysis.