Pattern Design Using Cellular Automata and an Iterative Relocation System

We introduce a two-step method for generating decorative images. In the first step, a special kind of 2D Cellular Automata (CA) is used as an engine for generating various patterns in pixel format. In the second step, an iterative relocation system is applied to convert the pixel based pattern into a mesh based system. Example patterns show the power of the combined method. Also we introduce another way of using the mesh based system: a concept of magic canvas that maps simple figure into complex decorative patterns. Introduction The application of cellular automata as a tool for a computerized pattern design system is promising because CA can be interpreted as working on the smallest element of patterns, i.e. the pixel. We know famous patterns, such as snowflakes or mollusk skin patterns, for example, generated from CA. Many of these CA patterns were found by chance, not through a systematic application of the CA. However, such chance discoveries were enough to show the diversity of patterns that CA can generate. The authors are trying to develop a generative art system based on a special class of CA, called a one-rule firing CA [1]. While a traditional CA fires and updates all the applicable cells in the system, only a subset of the applicable cells fires in the one-rule firing CA. A pattern generated from the one-rule system will be called an F-pattern, for convenience. One of the merits of the F-patterns is that the pattern can be completely defined by a series of numbers, called its F-code. For example, F=1 defines a Packard snow crystal or a line, depending on the cell type and neighborhood definition. The pattern can be decoded without a rule table. Patterns from One-Rule Firing Cellular Automata We can generate thousands of F-patterns in a couple of hours, using typical PCs today. We can control the category of patterns to be generated, simply by defining the number of symmetries and the number of colors used for the patterns. Figure 1 shows single F-patterns generated under different categories. Fcodes for each of the patterns are quoted here simply to have an idea how an F-code looks like. In actuality, an F-code can be considered as a convenient way of storing rule table for a pattern. A complete rule table, for example, for a 6-colored CA with Moore neighborhood, will be composed of 6=1,679,616 individual rules. However, the number of rules actually used for a typical pattern will be less than 100. The F-code stores the small fraction of the rule table actually used for a pattern. The number of digits in an F-code corresponds to the number of rules actually used for the pattern. While there are almost an infinite number of F-patterns, most of them are monotonous. For a generative art system, we need a method to create interesting patterns in a systematic way, not by simple trial and error. An F-pattern generally changes dramatically even at a single change of digit in its F-code, making it difficult to control the F-pattern through its F-code. Figure 1: Example of F-patterns from one-rule firing CA. (left) F=1(square cell) (middle) F=20212020 (right) F=652040430032650204605024105 To make a system more fruitful, a method of mixing multiple F-codes for a pattern was previously proposed [2]. Two different ways of mixing F-patterns were proposed. The first way is to mix several F-patterns time-step-wise. Starting from an initial seed, a pattern is grown up to time-step 100, for example, using pattern F1. From time step 101 to 201, another pattern F2 is applied, and so on. Second way of mixing is a hierarchical one. In a hierarchical mixing, an F-pattern F1 is grown first to a certain time-step. And next, the canvas is expanded with respect to the center of canvas to get a sparsely populated canvas of isolated pixels. Then a second F-pattern F2 is grown on each of these isolated pixels. Figure 2 shows various patterns generated through hierarchical mixing. Figure 2: CA patterns obtained by a rule-mixing technique. Iterative Relocation System While CA based patterns can be mixed for a more diverse patterns, all the patterns are given in pixel format and have several limitations. For an increased scalability and for a more diverse application of the F-pattern, we introduce a method of iterative relocation. For the method, we start from pixel-based images (see Fig. 3(a)). A pixel-based image is interpreted as a grid system composed of nodes and edges. Initially, the nodes are located on each of the pixels and edges connect neighboring nodes. Thus the initial grid system constitutes a system of uniform net composed of unit square patches (Fig. 3(b)). On each of the nodes is associated a mass corresponding to the color codes of the pixels. From the initial uniform net, the nodes iteratively relocate their positions under an interaction with the neighboring nodes. When the position of i-th node at time-step t is denoted as , its position at time-step t+1 will be defined as the following equation. In the equation above, Bi denotes index set of neighbors of node i . The weight is defined in terms of the color codes of the set Bi. Throughout the present study, Moore neighborhood is used and a node has 8 neighbors except the boundary nodes. At a given time step, a node relocates its position, for example, to the center of mass of the neighboring nodes. The update is simultaneous throughout all the nodes and boundary conditions are applied along the edges of the canvas. When the nodes move, the shape of the patches changes, in general, from unit square to quadrilateral. It is because the amount of move is not identical in general. As the relocation process continues, the initially straight grid lines turns into natural looking curved lines as shown in Fig. 3(c).