Growth processes based on 8-neighbor time delays

Abstract Unconstrained, uniform growth on a regular grid of cells, starting from a single cell, results in a simple (centrally symmetric) shape. On a square grid, for example, the resulting clusters of cells would be diamonds (rotated squares) using 4-neighbor growth, or squares using 8-neighbor growth. In order to model growth that is directed and focused in some directions and resisted in others, we formulate growth in terms of a kernel, which associates a time delay with each direction. On a square grid, a time delay kernel (TDK) consists of eight elements, one for each of the eight directions. We give a closed-form characterization of the clusters that result from TDK-based growth processes: specifically, we show that TDKs give rise to convex octagons. We also show that a much richer class of cluster shapes results from the composition of TDK-based growth processes, where activated cells synchronously cycle through sequences of TDKs. Lastly, we provide a “recovery” algorithm that outputs the sequence of TDKs that, when composed, can produce a given shape.

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