M-lattice: from morphogenesis to image processing

Abstract

The paper is based on reaction-diffusion, a nonlinear mechanism first proposed by Turing in 1952 to account for morphogenesis, the formation of shape and pattern in nature. One of the key limitations of reaction-diffusion systems is that they are generally unbounded, making them awkward for digital image processing. In this paper we introduce the "M-lattice", a system that preserves the pattern-formation properties of reaction-diffusion and is bounded. On the theoretical front, we establish how the M-lattice is closely related to the analog Hopfield network and the cellular neural network, but has more flexibility in how its variables interact. Like many "neurally inspired" systems, the bounded M-lattice also enables computer or analog VLSI implementations to simulate a variety of partial and ordinary differential equations. On the practical front, we demonstrate two novel applications of reaction-diffusion formulated as the new M-lattice. These are adaptive filtering, applied to the restoration and enhancement of fingerprint images, and nonlinear programming, applied to image halftoning in both "faithful" and "special effects" styles.