Automatic sorting of point pattern sets using Minkowski functionals

Abstract

Point pattern sets arise in many different areas of physical, biological, and applied research, representing many random realizations of underlying pattern formation mechanisms. These pattern sets can be heterogeneous with respect to underlying spatial processes, which may not be visually distiguishable. This heterogeneity can be elucidated by looking at statistical measures of the patterns sets and using these measures to divide the pattern set into distinct groups representing like spatial processes. We introduce here a numerical procedure for sorting point pattern sets into spatially homogenous groups using Functional Principal Component Analysis (FPCA) applied to the approximated Minkowski functionals of each pattern. We demonstrate that this procedure correctly sorts pattern sets into similar groups both when the patterns are drawn from similar processes and when the 2nd-order characteristics of the pattern are identical. We highlight this routine for distinguishing the molecular patterning of fluorescently labeled cell membrane proteins, a subject of much interest in studies investigating complex spatial signaling patterns involved in the human immune response.

InterCells: A Generic Monte-Carlo Simulation of Intercellular Interfaces Captures Nanoscale Patterning at the Immune Synapse

“…We provide multiple analysis tools, including univariate and bivariate PCFs ( 36 ), clustering algorithms and the topological analysis demonstrated here (Figure 9 ). Additional analyses of relevance may include Minkowski functionals ( 37 ), conditional second order PCFs ( 13 ), and more. Our simulation enable batch runs for scanning systematically values of parameters of choice.…”

“…The densities of the other molecules (namely, red or blue points in our example) can now be calculated on this perimeter. The consecutive operation of these steps with growing radii from the molecules yields the Minkowski perimeter functional ( 37 ). The conditional densities of the molecules are then calculated for the growing perimeters, as shown in Figures 9B–E .…”

Characterization of Self-Assembled 2D Patterns with Voronoi Entropy

“…Several methods can be used to quantify the orderliness of 2D patterns: The Minkowski functionals [45], Fourier analysis [44], and correlation functions [47]. An alternative method is the calculation of the entropy of the Voronoi diagram, which is the 2D analogy of 3D Wigner-Seitz partition [59][60].…”

Characterization of Self-Assembled 2D Patterns with Voronoi Entropy

“…Martin et al studied pattern formation during 2D nanoparticle self-assembly controlled by direct modification of solvent dewetting dynamics [ 43 ]. The authors compared three different techniques for the study of ordering in the resulting patterns: the Voronoi diagrams, two-dimensional fast Fourier transform analysis of the images [ 44 ], and the Minkowski functional method [ 45 , 46 ]. The Minkowski functionals of point patterns are calculated by centering a disk on each point and analyzing the topology of this secondary patterns of overlapping disks as a function of the radius [ 45 ].…”

“…The second Minkowski measure, the total perimeter of the pattern, is the perimeter of all of the shapes, which is reduced from the perimeter of the individual disks because of overlaps. The Euler number χ , supplied by Equation (1) is the final Minkowski measure, defined as the total number of distinct shapes or components in the window (created by the overlapping disks) minus the number of holes [ 45 ]. Mathematically, the three functionals do completely classify a pattern [ 45 ].…”