A review on anisotropy analysis of spatial point patterns

A spatial point pattern is called anisotropic if its spatial structure depends on direction. Several methods for anisotropy analysis have been introduced in the literature. In this paper, we give an overview of nonparametric methods for anisotropy analysis of (stationary) point patterns in $\mathbf{R}^2$ and $\mathbf{R}^3$. We discuss methods based on nearest neighbour and second order summary statistics as well as spectral and wavelet analysis. All techniques are illustrated on both a clustered and a regular example. Finally, we discuss methods for testing for isotropy as well as for estimating preferred directions in a point pattern.Comment: Submitted to Spatial Statistics -journal's special issue of the Spatial Statistics 2017 conferenc

An Analytical Representation of the 2d Generalized Balanced Power Diagram

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram

Detecting anisotropy in spatial point patterns - a comparison of statistical indices

Isotropy of a point process, defined as invariance of the distribution\ua0under rotation, is often assumed in spatial statistics. Formal\ua0tests for the hypothesis of isotropy can be created by comparing\ua0directional summary statistics in different directions. In this paper,\ua0the statistical powers of tests based on a variety of summary\ua0statistics and several choices of deviance measures are compared\ua0in a simulation study. Four models for anisotropic point processes\ua0are considered covering both regular and clustered cases.\ua0We discuss the robustness of the results to changes of the tuning\ua0parameters, and highlight the strengths and limitations of the\ua0methods

ON THE DILATED FACETS OF A POISSON-VORONOI TESSELLATION

In this paper, the parallel set ΞR of the facets ((d−1)-faces) of a stationary Poisson-Voronoi tessellation in ℝ2 and ℝ3 is investigated. An analytical formula for the spherical contact distribution function of the tessellation allows for the derivation of formulae for the volume density and the specific surface area of ΞR. The densities of the remaining intrinsic volumes are studied by simulation. The results are used for fitting a dilated Poisson-Voronoi tessellation to the microstructure of a closed-cell foam

Crack modeling via minimum-weight surfaces in 3d Voronoi diagrams

Abstract As the number one building material, concrete is of fundamental importance in civil engineering. Understanding its failure mechanisms is essential for designing sustainable buildings and infrastructure. Micro-computed tomography (μCT) is a well-established tool for virtually assessing crack initiation and propagation in concrete. The reconstructed 3d images can be examined via techniques from the fields of classical image processing and machine learning. Ground truths are a prerequisite for an objective evaluation of crack segmentation methods. Furthermore, they are necessary for training machine learning models. However, manual annotation of large 3d concrete images is not feasible. To tackle the problem of data scarcity, the image pairs of cracked concrete and corresponding ground truth can be synthesized. In this work we propose a novel approach to stochastically model crack structures via Voronoi diagrams. The method is based on minimum-weight surfaces, an extension of shortest paths to 3d. Within a dedicated image processing pipeline, the surfaces are then discretized and embedded into real μCT images of concrete. The method is flexible and fast, such that a variety of different crack structures can be generated in a short amount of time