In [CITE], Kronrod proves that the connected components of isolevelsets of a continuous function can be endowed with a treestructure. Obviously, the connected components of upper level sets are aninclusion tree, and the same is true for connected components of lower levelsets. We prove that in the case of semicontinuous functions, those trees canbe merged into a single one, which, following its use in image processing, wecall “tree of shapes”. This permits us to solve a classical representationproblem in mathematical morphology: to represent an image in such a way thatmaxima and minima can be computationally dealt with simultaneously. We provethe finiteness of the tree when the image is the result of applying anyextrema killer (a classical denoising filter in image processing). The shapetree also yields an easy mathematical definition of adaptive imagequantization.

This paper describes a compact perceptual image model intended formorphologicalrepresentation of the visual information contained innatural images. We explain why the total variation can be a criterionto split the information between the two main visual structures, whichare the sketch and the microtextures. We deduce a morphological decompositionscheme, based on a segmentation where the borders of the regions correspondto the location of the topological singularities of a topographic map.This leads to propose a new and morphological definition of edges. The sketch is computed by approximating the image with a piecewise smoothnon-oscillating function, using a Lipshitz interpolant given as the solutionof a PDE. The data needed to reconstruct the sketch image are very compact,so that an immediate outcome of this image model is the design of a progressive, and artifact-free, image compression scheme.

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.