110 research outputs found
Image Sampling with Quasicrystals
We investigate the use of quasicrystals in image sampling. Quasicrystals
produce space-filling, non-periodic point sets that are uniformly discrete and
relatively dense, thereby ensuring the sample sites are evenly spread out
throughout the sampled image. Their self-similar structure can be attractive
for creating sampling patterns endowed with a decorative symmetry. We present a
brief general overview of the algebraic theory of cut-and-project quasicrystals
based on the geometry of the golden ratio. To assess the practical utility of
quasicrystal sampling, we evaluate the visual effects of a variety of
non-adaptive image sampling strategies on photorealistic image reconstruction
and non-photorealistic image rendering used in multiresolution image
representations. For computer visualization of point sets used in image
sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary
materials, please visit at
http://www.Eyemaginary.com/Portfolio/Publications.htm
Enumeration of octagonal tilings
Random tilings are interesting as idealizations of atomistic models of
quasicrystals and for their connection to problems in combinatorics and
algorithms. Of particular interest is the tiling entropy density, which
measures the relation of the number of distinct tilings to the number of
constituent tiles. Tilings by squares and 45 degree rhombi receive special
attention as presumably the simplest model that has not yet been solved exactly
in the thermodynamic limit. However, an exact enumeration formula can be
evaluated for tilings in finite regions with fixed boundaries. We implement
this algorithm in an efficient manner, enabling the investigation of larger
regions of parameter space than previously were possible. Our new results
appear to yield monotone increasing and decreasing lower and upper bounds on
the fixed boundary entropy density that converge toward S = 0.36021(3)
Translational tilings by a polytope, with multiplicity
We study the problem of covering R^d by overlapping translates of a convex
body P, such that almost every point of R^d is covered exactly k times. Such a
covering of Euclidean space by translations is called a k-tiling. The
investigation of tilings (i.e. 1-tilings in this context) by translations began
with the work of Fedorov and Minkowski. Here we extend the investigations of
Minkowski to k-tilings by proving that if a convex body k-tiles R^d by
translations, then it is centrally symmetric, and its facets are also centrally
symmetric. These are the analogues of Minkowski's conditions for 1-tiling
polytopes. Conversely, in the case that P is a rational polytope, we also prove
that if P is centrally symmetric and has centrally symmetric facets, then P
must k-tile R^d for some positive integer k
- …