M-lattice: from morphogenesis to image processing

This work 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. Reaction-diffusion systems have been researched primarily by biologists working on theories of natural pattern formation and by chemists modeling dynamics of oscillating reactions. Although these nonlinear systems can potentially model a wealth of image patterns, they have only recently been applied to problems in image analysis and synthesis. 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 Mlattice 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.