Fernand Meyer - Böcker

Space, structure, and randomness: these are the three key concepts underlying Georges Matherons scientific work. He first encountered them at the beginning of his career when working as a mining engineer, and then they resurfaced in fields ranging from meteorology to microscopy. What could there be in common between such radically different types of applications? First, in each one only a single realisation of the phenomenon is available for study, but its features repeat themselves in space; second, the sampling pattern is rarely regular, and finally there are problems of change of scale.This volume is divided in three sections on random sets, geostatistics and mathematical morphology. They reflect his professional interests and his search for underlying unity. Some readers may be surprised to find theoretical chapters mixed with applied ones. We have done this deliberately. GM always considered that the distinction between the theory and practice was purely academic.When GM tackled practical problems, he used his skill as a physicist to extract the salient features and to select variables which could be measured meaningfully and whose values could be estimated from the available data.Then he used his outstanding ability as a mathematician to solve the problems neatly and efficiently. It was his capacity to combine a physicists intuition with a mathematicians analytical skills that allowed him to produce new and innovative solutions to difficult problems. The book should appeal to graduate students and researchers working in mathematics, probabilities, statistics, physics, spatial data analysis, and image analysis. In addition it will be of interest to those who enjoy discovering links between scientific disciplines that seem unrelated at first glance. In writing the book the contributors have tried to put GMs ideas into perspective. During his working life, GM was a genuinely creative scientist. He developed innovative concepts whose usefulness goes far beyond the confines of the discipline for which they were originally designed. This is why his work remains as pertinent today as it was when it was first written.

Mathematical morphology has developed a powerful methodology for segmenting images, based on connected filters and watersheds. We have chosen the abstract framework of node- or edge-weighted graphs for an extensive mathematical and algorithmic description of these tools. Volume 1 is devoted to watersheds. The topography of a graph appears by observing the evolution of a drop of water moving from node to node on a weighted graph, along flowing paths, until it reaches regional minima. The upstream nodes of a regional minimum constitute its catchment zone. The catchment zones may be constructed independently of each other and locally, in contrast with the traditional approach where the catchment basins have to be constructed all at the same time. Catchment zones may overlap, and thus, a new segmentation paradigm is proposed in which catchment zones cover each other according to a priority order. The resulting partition may then be corrected, by local and parallel treatments, in order to achieve the desired precision.

Mathematical morphology has developed a powerful methodology for segmenting images, based on connected filters and watersheds. We have chosen the abstract framework of node- or edge-weighted graphs for an extensive mathematical and algorithmic description of these tools.Volume 2 proposes two physical models for describing valid flooding on a node- or edge-weighted graph, and establishes how to pass from one to another. Many new flooding algorithms are derived, allowing parallel and local flooding of graphs. Watersheds and flooding are then combined for solving real problems. Their ability to model a real hydrographic basin represented by its digital elevation model constitutes a good validity check of the underlying physical models. The last part of Volume 2 explains why so many different watershed partitions exist for the same graph. Marker-based segmentation is the method of choice for curbing this proliferation. This book proposes new algorithms combining the advantages of the previous methods which treated node- and edge-weighted graphs differently.