A new mathematical algorithm is reported for the accurate and efficient analysis of pore properties of nanoporous anodic alumina (NAA) membranes using scanning electron microscope (SEM) images. NAA membranes of the desired pore size were fabricated using a two-step anodic oxidation process. Surface morphology of the NAA membranes with different pore properties was studied using SEM images along with computerized image processing and analysis. The main objective was to analyze the SEM images of NAA membranes quantitatively, systematically, and quickly. The method uses a regularized shock filter for contrast enhancement, mathematical morphological operators, and a segmentation process for efficient determination of pore properties. The algorithm is executed using MATLAB, which generates a statistical report on the morphology of NAA membrane surfaces and performs accurate quantification of the parameters such as average pore-size distribution, porous area fraction, and average interpore distances. A good comparison between the pore property measurements was obtained using our algorithm and ImageJ software. This algorithm, with little manual intervention, is useful for optimizing the experimental process parameters during the fabrication of such nanostructures. Further, the algorithm is capable of analyzing SEM images of similar or asymmetrically porous nanostructures where sample and background have distinguishable contrast.

In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem in mathematical morphology: to represent an image in such a way that maxima and minima can be computationally dealt with simultaneously. We prove the finiteness of the tree when the image is the result of applying any extrema killer (a classical denoising filter in image processing). The shape tree also yields an easy mathematical definition of adaptive image quantization.

A new and practically tractable formula for estimating the intensity of the underlying Poisson process of a Boolean model is given, assuming only almost sure boundedness of the primary grain.

A large class of statistics of planar and spatial data is closely connected with empirical distributions, which estimate ‘ergodic’ distributions of stationary random sets. The main result is a functional limit theorem concerning the deviation of the empirical distribution from the ‘true’ one. Examples in mathematical morphology are given.

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.