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Analysis of Ancient Mosaic Images for Dedicated Applications
Published in Filippo Stanco, Sebastiano Battiato, Giovanni Gallo, Digital Imaging for Cultural Heritage Preservation, 2017
Filippo Stanco, Sebastiano Battiato, Giovanni Gallo
To study Islamic geometrical patterns, and all periodic patterns such as those encountered in textile patterns [41] or wallpapers, several works are based on the symmetry group theory. In [14], the authors first classify patterns into one of the three following categories: (1) pattern generated by translation along one dimension, (2) patterns which contain translational symmetries in two independent directions (refers to the seventeen wallpaper groups), and (3) “rosettes,” which describes patterns that begin at a central point and grow radially outward. For every pattern, authors extract the symmetry group and the fundamental region (i.e., a representative region in the image from which the whole image can be regenerated). Finally, they describe the fundamental region by a simple color histogram and build the feature vector, which is a combination of the symmetry feature and histogram information. The authors show promising experiments for either classification or indexing.
Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A group of symmetries of the plane that is doubly infinite is a wallpaper group, or crystallographic group. There are 17 types of such groups, corresponding to 17 essentially distinct ways to tile the plane in a doubly periodical pattern. (There are also 230 three‐dimensional crystallographic groups.)
A micropolar continuum model of diffusion creep
Published in Philosophical Magazine, 2021
The simplest hexagonal tiling is that of regular hexagons, with the wallpaper group p6m. With this hexagonal symmetry, Neumann's principle implies that the tensors and are isotropic, and the pseudo-tensor vanishes. Moreover, for the regular hexagons, the vector between centroids is in the same direction as the normal vector . It follows from (64) that there are the further minor symmetries , along with the Cauchy relation symmetry . As a consequence of these symmetries, the constitutive tensors reduce to