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.
Ornament patterns have always been a source of curiosity and interest, not only in arts and crafts but also other fields including mathematics, computation, cultural studies etc. Early researchers mostly examined ornaments in cultural contexts, e.g. , @cite_39 , with a goal of revealing social structures and their interaction via dominant symmetries used in the ornament designs of individual cultures or geographical regions. In mathematics, the ornament patterns are studied in terms of the groups formed by the symmetry operations. Few examples include @cite_18 @cite_20 @cite_7 . The Dutch artist, Escher took a particular interest in patterns formed by repeating asymmetric shapes and discovered a local structure leading to the same wallpaper patterns; his work on symmetry is examined in @cite_23 . Regular repetitive patterns such as Wallpaper and Frieze groups are even utilized in quite practical problems; for example, to analyze human gait @cite_35 or to achieve automatic fabric defect detection in 2D patterned textures @cite_30 @cite_3 @cite_40 .
@cite_30 @cite_35 @cite_18 @cite_7 @cite_3 @cite_39 @cite_40 @cite_23 @cite_20
2056159211 2019654077 2080504245 2075487985
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. This paper proposes a generalized motif-based method for detecting defects in 16 out of 17 wallpaper groups in 2D patterned texture. It assumes that most patterned texture can be decomposed into lattices and their constituents-motifs. It then utilizes the symmetry property of motifs to calculate the energy of moving subtraction and its variance among different motifs. By learning the distribution of these values over a number of defect-free patterns, boundary conditions for discerning defective and defect-free patterns can be determined. This paper presents the theoretical foundation of the method, and defines the relations between motifs and lattice, from which a new concept called energy of moving subtraction is derived using norm metric measurement between a collection of circular shift matrices of motif and itself. It has been shown in this paper that the energy of moving subtraction amplifies the defect information of the defective motif. Together with its variance, an energy-variance space is further defined where decision boundaries are drawn for classifying defective and defect-free motifs. As the 16 wallpaper groups of patterned fabric can be transformed into three major groups, the proposed method is evaluated over these three major groups, from which 160 defect-free lattices samples are used for defining the decision boundaries, with 140 defect-free and 113 defective samples used for testing. An overall detection success rate of 93.32 is achieved for the proposed method. No other generalized approach can achieve this success rate has been reported before, and hence this result outperforms all other previously published approaches. Abstract An investigation of the Moorish ornaments from the Alhambra (in Granada, Spain) shows that their symmetry groups belong to 13 different crystallographic (wallpaper) classes; this corrects several earlier enumerations and claims. The four classes of wallpaper groups missing in Alhambra (pg, p2, pgg, p3ml) have not been found in other Moorish ornaments, either. But the classification of repeating patterns by their symmetry groups is in many cases not really appropriate—account should be taken of the coloring of the patterns, of their interlace characteristics, etc. This leads to a variety of “symmetry groups”, not all of which have been fully investigated. Moreovr, the “global” approach to repeating ornaments is only of limited applicability, since it does not correspond to the way of thinking of the artisans involved, and does not cover all the possibilities of “local” order. The proper mathematical tools for the study of such structures which are only “locally orderly” remain to be developed. New method to analyse mosaics based on the mathematical principles of Symmetry Groups.The method includes a higher level of knowledge based on objects.Extraction of objects and their main features of patterns with a Wallpaper Group (WG).Classification of objects according to their shape and obtaining their isometries.The extraction of the WG of the pattern using the relationships between objects. This article presents a new method for analysing mosaics based on the mathematical principles of Symmetry Groups. This method has been developed to get the understanding present in patterns by extracting the objects that form them, their lattice, and the Wallpaper Group. The main novelty of this method resides in the creation of a higher level of knowledge based on objects, which makes it possible to classify the objects, to extract their main features (Point Group, principal axes, etc.), and the relationships between them. In order to validate the method, several tests were carried out on a set of Islamic Geometric Patterns from different sources, for which the Wallpaper Group has been successfully obtained in 85 of the cases. This method can be applied to any kind of pattern that presents a Wallpaper Group. Possible applications of this computational method include pattern classification, cataloguing of ceramic coatings, creating databases of decorative patterns, creating pattern designs, pattern comparison between different cultures, tile cataloguing, and so on.