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ABSTRACT
Stochastic geometry deals with models and stochastic properties of random geometric sets, mainly in Euclidean spaces. This article presents a synthesis of the basic concepts and notions of stochastic geometry in n?dimensional Euclidean spaces. Some basics of set theory, topology and measure theory are first given. Random closed sets and their properties are then introduced. Their main functional properties, covariance functions, and main spatial statistics are then presented. The following random models are then described: point fields, particle fields, curve fields, and surface fields. Some information is provided on statistical estimation of numerical spatial characteristics and on stochasticity hypothesis tests.
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Jean-Charles PINOLI: Professor - École Nationale Supérieure des Mines de Saint-Étienne, Saint-Étienne, France - To Andrée-Aimée Toucas for her bibliographical support. - To Professor Frédéric Gruy for his scientific interest.
INTRODUCTION
Stochastic Geometry is a branch of mathematics whose name appeared in the 1960s .
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Geometry and probability
Stochastic geometry applies the concepts and tools of probability theory (Probability Theory) to geometry, generally Euclidean. It is concerned with the study of the spatial distributions of random geometric objects (e.g. points, curves, surfaces or bodies in dimensions less than or equal to 3). From a geometric point of view, stochastic geometry is the heir to Geometric Probability , where random events are no longer "counted", but "measured". It is largely based on Convex Geometry , Integral Geometry and Geometric Measure Theory . Spatial Point Process Theory is also fundamental.
Stochastic geometry is essentially based on two general notions: random sets
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KEYWORDS
estimators | set functionals | random closed sets | random point fields | random geometric object fields
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Stochastic geometry
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