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.
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. We analyze walking people using a gait sequence representation that bypasses the need for frame-to-frame tracking of body parts. The gait representation maps a video sequence of silhouettes into a pair of two-dimensional spatio-temporal patterns that are near-periodic along the time axis. Mathematically, such patterns are called "frieze" patterns and associated symmetry groups "frieze groups". With the help of a walking humanoid avatar, we explore variation in gait frieze patterns with respect to viewing angle, and find that the frieze groups of the gait patterns and their canonical tiles enable us to estimate viewing direction of human walking videos. In addition, analysis of periodic patterns allows us to determine the dynamic time warping and affine scaling that aligns two gait sequences from similar viewpoints. We also show how gait alignment can be used to perform human identification and model-based body part segmentation. 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.  In this paper, we propose a method of automatic detection of texture-periodicity using superposition of distance matching functions (DMFs) followed by computation of their forward differences. The method has been specifically devised for automatically identifying row and column periodicities and thereby the size of periodic units from textile fabrics belonging to any of the 17 wallpaper groups and is a part of automatic fabric defect detection scheme being developed by us that needs periodicities along row and column directions. Overall row-DMF (or overall column-DMF) is obtained based on superposition of DMF of all rows (or columns) from the input image and its second forward difference is computed to get the overall maximum which is a direct measure of periodicity along row (or column) direction. Results from experiments on various near-regular textures demonstrate the capability of the proposed method for automatic periodicity extraction without the need of human intervention. AcknowledgmentsIntroduction1. History and Theory of Plane Pattern Analysis2. Mathematical Principles and Terminology3. Color Symmetry4. One-Dimensional Patterns5. Two-Dimensional Patterns6. Finite Designs7. Problems in ClassificationConclusionAppendixes--1. The Four Rigid Motions of the Plane--2. The Seven Classes of One-Dimensional Design--3. Comparative Notation for the Two-Color, Two-Dimensional PatternsBibliographyIndexes This paper presents a study of using ellipsoidal decision regions for motif-based patterned fabric defect detection, the result of which is found to improve the original detection success using max-min decision region of the energy-variance values. In our previous research, max-min decision region was found to be effective in distinct cases but ill detect the ambiguous false-positive and false-negative cases. To alleviate this problem, we first assume that the energy-variance values can be described by a Gaussian mixture model. Second, we apply k-means clustering to roughly identify the various clusters that make up the entire data population. Third, convex hull of each cluster is employed as a basis for fitting an ellipsoidal decision region over it. Defect detection is then based on these ellipsoidal regions. To validate the method, three wallpaper groups are evaluated using the new ellipsoidal regions, and compared with those results obtained using the max-min decision region. For the p2 group, success rate improves from 93.43 to 100 . For the pmm group, success rate improves from 95.9 to 96.72 , while the p4m group records the same success rate at 90.77 . This demonstrates the superiority of using ellipsoidal decision regions in motif-based defect detection.  