Characterizing pixel and point patterns with a hyperuniformity disorder length

Comparison of Quantitative Morphology of Layered and Arbitrary Patterns: Contrary to Visual Perception, Binary Arbitrary Patterns Are Layered from a Structural Point of View

Patterns found among both living systems, such as fish scales, bones, and tree rings, and non-living systems, such as terrestrial and extraterrestrial dunes, microstructures of alloys, and geological seismic profiles, are comprised of anisotropic layers of different thicknesses and lengths. These layered patterns form a record of internal and external factors that regulate pattern formation in their various systems, making it potentially possible to recognize events in the formation history of these systems. In our previous work, we developed an empirical model (EM) of anisotropic layered patterns using an N-partite graph, denoted as G(N), and a Boolean function to formalize the layer structure. The concept of isotropic and anisotropic layers was presented and described in terms of the G(N) and Boolean function. The central element of the present work is the justification that arbitrary binary patterns are made up of such layers. It has been shown that within the frame of the proposed model, it is the isotropic and anisotropic layers themselves that are the building blocks of binary layered and arbitrary patterns; pixels play no role. This is why the EM can be used to describe the morphological characteristics of such patterns. We present the parameters disorder of layer structure, disorder of layer size, and pattern complexity to describe the degree of deviation of the structure and size of an arbitrary anisotropic pattern being studied from the structure and size of a layered isotropic analog. Experiments with arbitrary patterns, such as regular geometric figures, convex and concave polygons, contour maps, the shape of island coastlines, river meanders, historic texts, and artistic drawings are presented to illustrate the spectrum of problems that it may be possible to solve by applying the EM. The differences and similarities between the proposed and existing morphological characteristics of patterns has been discussed, as well as the pros and cons of the suggested method.

Quantifying the long-range structure of foams and other cellular patterns with hyperuniformity disorder length spectroscopy

We investigate the local and long-range structure of several space-filling cellular patterns: bubbles in a quasi-two-dimensional foam, and Voronoi constructions made around points that are uncorrelated (Poisson patterns), low discrepancy (Halton patterns), and displaced from a lattice by Gaussian noise (Einstein patterns). We study local structure with distributions of quantities including cell areas and side numbers. The former is the widest for the bubbles making foams the most locally disordered, while the latter show no major differences between the cellular patterns. To study long-range structure, we begin by representing the cellular systems as patterns of points, both unweighted and weighted by cell area. For this, foams are represented by their bubble centroids and the Voronoi constructions are represented by the centroids as well as the points from which they are created. Long-range structure is then quantified in two ways: by the spectral density, and by a real-space analog where the variance of density fluctuations for a set of measuring windows of diameter D is made more intuitive by conversion to the distance h(D) from the window boundary where these fluctuations effectively occur. The unweighted bubble centroids have h(D) that collapses for the different ages of the foam with random Poissonian fluctuations at long distances. The area-weighted bubble centroids and area-weighted Voronoi points all have constant h(D)=h_{e} for large D; the bubble centroids have the smallest value h_{e}=0.084sqrt[〈a〉], meaning they are the most uniform. Area-weighted Voronoi centroids exhibit collapse of h(D) to the same constant h_{e}=0.084sqrt[〈a〉] as for the bubble centroids. A similar analysis is performed on the edges of the cells and the spectra of h(D) for the foam edges show h(D)∼D^{1-ε} where ε=0.30±0.15.

Cloaking the underlying long-range order of randomly perturbed lattices

Random, uncorrelated displacements of particles on a lattice preserve the hyperuniformity of the original lattice, that is, normalized density fluctuations vanish in the limit of infinite wavelengths. In addition to a diffuse contribution, the scattering intensity from the the resulting point pattern typically inherits the Bragg peaks (long-range order) of the original lattice. Here we demonstrate how these Bragg peaks can be hidden in the effective diffraction pattern of independent and identically distributed perturbations. All Bragg peaks vanish if and only if the sum of all probability densities of the positions of the shifted lattice points is a constant at all positions. The underlying long-range order is then "cloaked" in the sense that it cannot be reconstructed from the pair correlation function alone. On the one hand, density fluctuations increase monotonically with the strength of perturbations a, as measured by the hyperuniformity order metric Λ[over ¯]. On the other hand, the disappearance and reemergence of long-range order, depending on whether the system is cloaked as the perturbation strength increases, is manifestly captured by the τ order metric. Therefore, while the perturbation strength a may seem to be a natural choice for an order metric of perturbed lattices, the τ order metric is a superior choice. It is noteworthy that cloaked perturbed lattices allow one to easily simulate very large samples (with at least 10^{6} particles) of disordered hyperuniform point patterns without Bragg peaks.

Methodology to construct large realizations of perfectly hyperuniform disordered packings

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space R^{d}. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles of general shape are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit, regardless of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R^{2} and R^{3}. Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-the-art methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials.

Universal hidden order in amorphous cellular geometries

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

Disordered hyperuniformity implies a hidden order on length scales that can be found in various amorphous materials. Klatt et al. analyse the evolution of random point patterns using Llyod’s algorithm and show that they converge to an effectively hyperuniform state regardless of the initial conditions.

Hyperuniformity disorder length spectroscopy for extended particles

The concept of a hyperuniformity disorder length h was recently introduced for analyzing volume fraction fluctuations for a set of measuring windows [Chieco et al., Phys. Rev. E 96, 032909 (2017).PLEEE81539-375510.1103/PhysRevE.96.032909]. This length permits a direct connection to the nature of disorder in the spatial configuration of the particles and provides a way to diagnose the degree of hyperuniformity in terms of the scaling of h and its value in comparison with established bounds. Here, this approach is generalized for extended particles, which are larger than the image resolution and can lie partially inside and partially outside the measuring windows. The starting point is an expression for the relative volume fraction variance in terms of four distinct volumes: that of the particle, the measuring window, the mean-squared overlap between particle and region, and the region over which particles have nonzero overlap with the measuring window. After establishing limiting behaviors for the relative variance, computational methods are developed for both continuum and pixelated particles. Exact results are presented for particles of special shape and for measuring windows of special shape, for which the equations are tractable. Comparison is made for other particle shapes, using simulated Poisson patterns. And the effects of polydispersity and image errors are discussed. For small measuring windows, both particle shape and spatial arrangement affect the form of the variance. For large regions, the variance scaling depends only on arrangement but particle shape sets the numerical proportionality. The combined understanding permit the measured variance to be translated to the spectrum of hyperuniformity lengths versus region size, as the quantifier of spatial arrangement. This program is demonstrated for a system of nonoverlapping particles at a series of increasing packing fractions as well as for an Einstein pattern of particles with several different extended shapes.