Morphological characterization of spatial patterns

Geometrical patterns are encountered in many different fields of science and technology, including porous media, biology, polymer chemistry, metallography, geology and astronomy. 1), 2) We characterize these patterns in terms of shape (geometry) and connectivity (topology) by means of morphological image analysis (MIA). This involves the calculation of the Minkowski functionals known from integral geometry. 3), 4) These functionals are related to familiar measures: In two (three) dimensions they correspond to the covered area, boundary length, and connectivity (volume, surface area, integral mean curvature and connectivity) of the pattern. For sufficiently smooth and regular objects some of these measures are related to quantities known from differential geometry. Integral geometry imposes no limitations on the properties of the patterns. Furthermore in integral geometry the calculation of the Minkowski functionals is relatively straightforward and requires little computational effort. 22) Given a set of patterns the first step in MIA is to compute the Minkowski functionals themselves. The second step is to analyze the behavior of the Minkowski functionals as a function of control parameters. This approach has proven to be instrumental to describe the morphology of porous media and complex fluids, the large-scale distribution of matter in the Universe, microemulsions, patterns in reaction diffusion systems, and spinodal decomposition kinetics. 5) In this paper we briefly discuss the underlying mathematics of the Minkowski