Image translation and rotation on hexagonal structure

Image translation and rotation are becoming essential operations in many application areas such as image processing, computer graphics and pattern recognition. Conventional translation moves image from pixels to pixels and conventional rotation usually comprises of computation-intensive CORDIC operations. Traditionally, images are represented on a square pixel structure. In this paper, we perform reversible and fast image translation and rotation based on a hexagonal structure. An image represented on the hexagonal structure is a collection of hexagonal pixels of equal size. The hexagonal structure provides a more flexible and efficient way to perform image translation and rotation without losing image information. As there is not yet any available hardware for capturing image and for displaying image on a hexagonal structure, we apply a newly developed virtual hexagonal structure. The virtual hexagonal structure retains image resolution during the process of image transformations, and almost does not introduce distortion. Furthermore, images can be smoothly and easily transferred between the traditional square structure and the hexagonal structure. © 2006 IEEE

Basic transformations on virtual hexagonal structure

Hexagonal structure is different from the traditional square structure for image representation. The geometrical arrangement of pixels on hexagonal structure can be described in terms of a hexagonal grid. Hexagonal structure provides an easy way for image translation and rotation transformations. However, all the existing hardware for capturing image and for displaying image are produced based on square architecture. It has become a serious problem affecting the advanced research based on hexagonal structure. In this paper, we introduce a new virtual hexagonal structure. Based on this virtual structure, a more flexible and powerful image translation and rotation are performed. The virtual hexagonal structure retains image resolution during the process of image transformations, and does not introduce distortion. Furthermore, images can be smoothly and easily transferred between the traditional square structure and the hexagonal structure. © 2006 IEEE

An aperiodic hexagonal tile

We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three--dimensional prototile.Comment: 32 pages, 24 figures; submitted to Journal of Combinatorial Theory Series A. Version 2 is a major revision. Parts of Version 1 have been expanded and parts have been moved to a separate article (arXiv:1003.4279

Forcing nonperiodicity with a single tile

An aperiodic prototile is a shape for which infinitely many copies can be arranged to fill Euclidean space completely with no overlaps, but not in a periodic pattern. Tiling theorists refer to such a prototile as an "einstein" (a German pun on "one stone"). The possible existence of an einstein has been pondered ever since Berger's discovery of large set of prototiles that in combination can tile the plane only in a nonperiodic way. In this article we review and clarify some features of a prototile we recently introduced that is an einstein according to a reasonable definition. [This abstract does not appear in the published article.]Comment: 18 pages, 10 figures. This article has been substantially revised and accepted for publication in the Mathematical Intelligencer and is scheduled to appear in Vol 33. Citations to and quotations from this work should reference that publication. If you cite this work, please check that the published form contains precisely the material to which you intend to refe

Spatial period-multiplying instabilities of hexagonal Faraday waves

A recent Faraday wave experiment with two-frequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called `superlattice-II') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2sqrt{3} from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (sqrt{3} larger than the original scale) and apparently has 60 degree rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-II pattern presented in and the subharmonic nature of the secondary instability, we show (a) that the superlattice-II pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [1]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60 degree rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude

A comprehensive analysis of the (R13xR13)R13.9{\deg} type II structure of silicene on Ag(111)

In this paper, using the same geometrical approach than for the (2R3x2R3) R30{\deg} structure (H. Jamgotchian et al., 2015, Journal of Physics. Condensed Matter 27 395002), for the (R13xR13)R13.9{\deg} type II structure, we propose an atomic model of the silicene layer based on a periodic relaxation of the strain epitaxy. This relaxation creates periodic arrangements of perfect areas of (R13xR13)R13.9{\deg} type II structure surrounded by defect areas. A detailed analysis of the main published experimental results, obtained by Scanning Tunneling Microscopy and by Low Energy Electron Diffraction, shows a good agreement with the geometrical model.Comment: 20 pages, 9 figure

Crystal image analysis using 2D2D synchrosqueezed transforms

We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of non-overlapping segments of 2D general intrinsic mode type functions, which are superpositions of non-linear wave-like components. In particular, crystal defects are interpreted as the irregularity of local energy; crystal rotations are described as the angle deviation of local wave vectors from their references; the gradient of a crystal elastic deformation can be obtained by a linear system generated by local wave vectors. Several numerical examples of synthetic and real crystal images are provided to illustrate the efficiency, robustness, and reliability of our methods.Comment: 27 pages, 17 figure

Symmetry-guided nonrigid registration: the case for distortion correction in multidimensional photoemission spectroscopy

Image symmetrization is an effective strategy to correct symmetry distortion in experimental data for which symmetry is essential in the subsequent analysis. In the process, a coordinate transform, the symmetrization transform, is required to undo the distortion. The transform may be determined by image registration (i.e. alignment) with symmetry constraints imposed in the registration target and in the iterative parameter tuning, which we call symmetry-guided registration. An example use case of image symmetrization is found in electronic band structure mapping by multidimensional photoemission spectroscopy, which employs a 3D time-of-flight detector to measure electrons sorted into the momentum ($k_x$, $k_y$) and energy ($E$) coordinates. In reality, imperfect instrument design, sample geometry and experimental settings cause distortion of the photoelectron trajectories and, therefore, the symmetry in the measured band structure, which hinders the full understanding and use of the volumetric datasets. We demonstrate that symmetry-guided registration can correct the symmetry distortion in the momentum-resolved photoemission patterns. Using proposed symmetry metrics, we show quantitatively that the iterative approach to symmetrization outperforms its non-iterative counterpart in the restored symmetry of the outcome while preserving the average shape of the photoemission pattern. Our approach is generalizable to distortion corrections in different types of symmetries and should also find applications in other experimental methods that produce images with similar features