On the foundations of vision modeling III. Pattern-theoretic analysis of Hopf and Turing's reaction-diffusion patterns

Abstract

After Turing's ingenious work on the chemical basis of morphogenesis fifty years ago, reaction-diffusion patterns have been extensively studied in terms of modelling and analysis of pattern formations (both in chemistry and biology), pattern growing in complex laboratory environments, and novel applications in computer graphics. But one of the most fundamental elements has still been missing in the literature. That is, what do we mean exactly by (reaction-diffusion) {\em patterns}? When presented to human vision and visual system, the patterns usually look deceptively simple and are often tagged by household names like {\em spots} or {\em stripes}. But are such split-second pattern identification and classification equally simple for a computer vision system? The answer does not seem to be confirmative, just as in the case of face recognition, one of the greatest challenges in contemporary A.I. and computer vision research. Inspired and fuelled by the recent advancement in mathematical image and vision analysis (Miva), as well as modern {\em pattern theory}, the current paper develops both statistical and geometrical tools and frameworks for identifying, classifying, and characterizing common reaction-diffusion patterns and pattern formations. In essence, it presents a data mining theory for the scientific simulations of reaction-diffusion patterns.