IJPAM: Volume 19, No. 2 (2005)

GEOMETRIC AND STOCHASTIC ANALYSIS
OF REACTION-DIFFUSION PATTERNS

Jianhong Shen$^1$, Yoon Mo Jung$^2$
$^{1,2}$School of Mathematics
University of Minnesota
206 Church Street, S.E., Minneapolis, MN 55455, USA
$^1$e-mail: jhshen@math.umn.edu
$^2$e-mail: ymjung@math.umn.edu


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 (in both chemistry and biology), pattern growing in complex laboratory environments, and novel applications in computer graphics. A fundamental question that remains unanswered in the literature is what one precisely means by (reaction-diffusion) patterns. Most patterns have only been discovered, identified, or explained by human vision and human intelligence. Inspired by the recent advancement in Mathematical Image and Vision Analysis (MIVA), the current paper develops both geometric and stochastic 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, or various analytical tools for the automatic characterization of generic complex patterns by artificial intelligence.

Received: January 3, 2005

AMS Subject Classification: 68T45, 92C15, 62H35

Key Words and Phrases: reaction-diffusion, Gray-Scott model, turing instabilities, entropy, maturity, skewness, kurtosis, geometric measures, total curvatures, skeleton curves, singularities

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 19
Issue: 2