SUMMARY Within the concept of point processes, a review is presented of quantities which can be used in studies of three‐dimensional (3‐D) aggregates of particles. Suitable characteristics and estimators are given for both unmarked and marked point processes. To demonstrate the feasibility of such quantitative approaches, an application in histology, dealing with 3‐D arrangements of cell nuclei in rat liver, is described. Using a confocal scanning light microscope, 3‐D images are recorded and image analysis used to obtain the coordinates of the centroid, together with the volume and DNA content, of each cell nucleus. Examples of results are given, using both unmarked and marked point processes. In the latter case, cell type, nuclear volume and ploidy group are suitable marks.

The estimation of the fractal dimension in the case of concave log-log Richardson-Mandelbrot plots can be obtained by using asymptotic fractal equations. We demonstrate here, under asymptotic fractal conditions, that additional derivations making use of the Minkowski dilation in grey-scales lead to two asymptotes, one having a slope of 1 and the other a slope of DT-D + 1 (where DT is the topological dimension and D the fractal dimension). The resulting equation offers important advantages. It allows: (i) evaluation of scaling properties of a grey-scale image; (ii) estimation of D without any iteration and (iii) generation of texture and heterogeneity models. We concentrate here on the first two possibilities. Images from cultured cells in studies of cytoskeleton intermediate filaments and kinetic deformability of endothelial cells were used as examples.

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