Models for Random Sets
Abstract:
The main question when defining random sets is : which questions are we allowed to ask about the properties of the set? In technical words: which measurements are random variables? Many different sigma-algebras have been proposed, however, no theory was able to put in a unified framework random sets, point processes, line processes, where the usual morphological transformations, like erosion, dilation, openings, closing and granulometries are measurable. In the seventies, Georges Matheron [1] proposed the the notion of random closed sets, based on the hit-or-miss topology. In this talk [2], we present these concepts, their relations with the notion of spatial law, continuity, a.s.o. and give some examples of important results in the inference of parameters in Boolean models and their derivatives.
[1] Random Sets and Integral Geometry, G. Matheron, Wiley, 1975
[2] Morphologie Mathématique, M. Schmitt and J. Mattioli, Chapter X, Presses des Mines, 2013
[1] Random Sets and Integral Geometry, G. Matheron, Wiley, 1975
[2] Morphologie Mathématique, M. Schmitt and J. Mattioli, Chapter X, Presses des Mines, 2013
Short Bio:
Michel Schmitt received an engineering degree from Ecole Polytechnique, Paris, in 1982 and the Ph.D. degree in Mathamatical Morphology from Mines ParisTech in 1989. He joined Thales Research as head of its «Perception» lab, working on applications in image processing and classification. Afterward, he came back to Mines ParisTech, as head of the «Geostatistics» lab, and then dean for research. In a near future, he will join PSL – Paris Sciences et Lettres Research University Paris – in charge of its digital policy. He is today at Institut Mines-Télécom. His research interests cover mathematical morphology – theoretical aspects, algorithmics and applications in image processing – and geostatistics – random sets theory and its application to earth sciences and environment. As a manager of research teams, he created two labs – Center for research on risks and crises (Mines ParisTech), Center for bioinformatics (Inserm, Institut Curie, Mines ParisTech) – and the Carnot Institute M.I.N.E.S.
