Integral-geometry morphological image analysis

This paper reviews a general method to characterize the morphology of two- and three-dimensional patterns in terms of geometrical and topological descriptors. Based on concepts of integral geometry, it involves the calculation of the Minkowski functionals of black-and-white images representing the patterns. The result of this approach is an objective, numerical characterization of a given pattern. We briefly review the basic elements of morphological image processing, a technique to transform images to patterns that are amenable to further morphological image analysis. The image processing technique is applied to electron microscope images of nano-ceramic particles and metal-oxide precipitates. The emphasis of this review is on the practical aspects of the integral-geometry-based morphological image analysis but we discuss its mathematical foundations as well. Applications to simple lattice structures, triply periodic minimal surfaces, and the Klein bottle serve to illustrate the basic steps of the approach. More advanced applications include random point sets, percolation and complex structures found in block copolymers.