Forming Point Patterns by a Probabilistic Cellular Automata Rule

TL;DR

This paper develops a probabilistic cellular automata rule to generate 2D point patterns where points are maximally dense, separated, and cover the space efficiently, using a tiling approach with templates and noise injection.

Contribution

It introduces a novel probabilistic rule for cellular automata that optimizes point placement in 2D patterns through a tiling and template matching framework.

Findings

Maximal point density achieved with the proposed rule.

Effective separation of points with no overlaps of centers.

Pattern quality depends on parameter tuning.

Abstract

The objective is to find a Cellular Automata rule that can form a 2D point pattern with a maximum number of points (1-cells). Points are not allowed to touch each other, they have to be separated by 0-cells, and every 0-cell can find at least one point in its Moore-neighborhood. Probabilistic rules are designed that can solve this task with asynchronous updating and cyclic boundary condition. The task is considered as a tiling problem, where point tiles are used to cover the space with overlaps. A point tile consists of a center pixel (the kernel with value 1) and 8 surrounding pixels forming the hull with value 0. The term pixel is used to distinguish the cells of a tile from the cells of a cellular automaton. For each of the 9 tile pixels a so-called template is defined by a shift of the point tile. In the rule application, the 9 templates are tested at the actual cell position. If all template pixels (except the central reference pixel) of a template match with the corresponding neighbors of the actual cell under consideration, the cell's state is adjusted to the reference pixel's value. Otherwise the cell is set to the random value 0 or 1 with a certain probability. The hull pixels are allowed to overlap. In order to evolve a maximum of points, the overlap between tiles has to be maximized. To do that, the number of template hits is counted. Depending on the hit-number, additional noise is injected with certain probabilities. Thereby optimal patterns with the maximum number of points can be evolved. The behavior and performance of the designed rules is evaluated for different parameter settings.