Planar ornaments, a.k.a. wallpapers, are regular repetitive patterns which exhibit translational symmetry in two independent directions. There are exactly 17 distinct planar symmetry groups. We present a fully automatic method for complete analysis of planar ornaments in 13 of these groups, specifically, the groups called p6, p6m, p4g, p4m, p4, p31m, p3m, p3, cmm, pgg, pg, p2 and p1. Given the image of an ornament fragment, we present a method to simultaneously classify the input into one of the 13 groups and extract the so-called fundamental domain, the minimum region that is sufficient to reconstruct the entire ornament. A nice feature of our method is that even when the given ornament image is a small portion such that it does not contain multiple translational units, the symmetry group as well as the fundamental domain can still be defined. This is because, in contrast to common approach, we do not attempt to first identify a global translational repetition lattice. Though the presented constructions work for quite a wide range of ornament patterns, a key assumption we make is that the perceivable motifs (shapes that repeat) alone do not provide clues for the underlying symmetries of the ornament. In this sense, our main target is the planar arrangements of asymmetric interlocking shapes, as in the symmetry art of Escher.
It is also possible to perform a continuous characterization of the ornament by comparing ornament images. This is for example encountered in @cite_29 , where ornament images are classified according to a symmetry feature vector calculated based on a prior lattice extraction and yes no questions; for lattice detection, they used method in @cite_32 . In @cite_36 , ornament images are directly compared in a transformed domain after applying a global transformation. Note that among the works addressing planar patterns, there are also several interesting works on pattern synthesis, including how to generate an ornament in a certain symmetry group, how to use a given motif to tile the plane in a certain style, or how to map a given wallpaper pattern to a curved surface @cite_4 @cite_5 .
@cite_4 @cite_36 @cite_29 @cite_32 @cite_5
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From art to science, ornaments constructed by repeating a base motif (tiling) have been a part of human culture. These ornaments exhibit various kinds of symmetries depending on the construction process as well as the symmetries of the base motif. The scientific study of the ornaments is the study of symmetry, i.e., the repetition structure. There is, however, an artistic side of the problem too: intriguing color permutations, clever choices of asymmetric interlocking forms, several symmetry breaking ideas, all that come with the artistic freedom. In this paper, in the context of Escher's Euclidean ornaments, we study ornaments without reference to fixed symmetry groups. We search for emergent categorical relations among a collection of tiles. We explore how these relations are affected when new tiles are inserted to the collection. We ask and answer whether it is possible to code symmetry group information implicitly without explicitly extracting the repetition structure, grids and motifs. The interest shown recently in the algorithmic treatment of symmetries, also known as Computational Symmetry, covers several application areas among which textile and tiles design are some of the most significant. Designers make new creations based on the symmetric repetition of motifs that they find in databases of wallpaper images. The existing methods for dealing with these images have several drawbacks because the use of heuristics achieves low recovery rates when images exhibit imperfections due to the fabrication or the handmade process. A system and apparatus for visually cuing a human subject wherein an incandescent lamp or lamps (40, 41) are energizable to provide a source of visible illumination. These incandescent lamps have a predetermined rise and fall characteristic. An observation surface (12) is positioned for confronting illumination from the lamps (40, 41) including a pattern through which filtered light of a first intensity may pass, which pattern is configured for visually conveying information to the subject. The pattern is surrounded by a region (54) opaque to illumination, that region extending to a periphery (55). A peripheral surface arrangement (14, 16) is provided extending from the observation surface (12) periphery and is positioned for confronting and transmitting illumination from the lamps (40, 41) at a second intensity selected as greater than the first intensity. Lamps (40, 41) are illuminated to effect the transmission of illumination through the peripheral surface and observation surface in an intermittent fashion at a frequency selected to provide visual stimuli of predetermined temporar pause, p, to evoke a gamma effect with respect to human visual perception of the pattern. In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.