Structural analysis of regular textures using symmetry groups

Abstract

Symmetry group theory shows that every regular texture can be classified into one of seventeen symmetry groups. In this thesis, we propose an algorithm that classifies a noisy regular texture to identify the symmetry group for the texture and its fundamental region. A fundamental region is a smallest polygonal subregion of the texture that can reproduce the texture by applying the symmetries for the texture to the region. Our algorithm mainly consists of two parts: motif analysis and symmetry group identification. A motif is a smallest parallelogramed subregion of a texture that can regenerate the texture by repetition. To obtain the motif, we first propose a distance matching function to effectively compute equivalent points in the texture. The motif is then specified by determining two vectors which span the points dominantly. The motif contains all symmetries for a texture because it can reproduce the texture by repetition. This characteristic makes it possible to only consider symmetries embedded in the motif to determine the symmetry group for the texture. Since a symmetry in the motif is a combination of reflection, rotation and glide reflection, we present algorithms that identify each of the symmetries in a systematic way. Finally, we give the classification diagrams to identify all symmetry groups. Once we recognize the symmetry group, the fundamental region of the texture can be easily identified.