Geometric covariograms and their application to image analysis
Geometric covariograms and their application to image analysis - Context: geostatistical modeling of regionalized data Geostatistics is used to analyze and to model regionalized variables known through a set of sparse data. Regionalized variables are interpreted as realizations of random fields, the characteristics of which can be inferred from the available data. Object-based models. Random field models can be obtained by considering objects seeded at the points of a random point process in space and/or time. The shape, size and orientation of the objects may be deterministic or random, with the same distribution as a typical object, The value assigned to each object can be a constant or a function of the distance to the gravity center. When two or more objects overlap, the value at the spatial locations common to these objects can be defined by one of the following rules. alue of the first or last object: dead leaves model. sum of the object values: dilution model. maximum object value: Boolean model. Object-based models First example: a dead leaves model, with discs valued ,1 or 1. Second example: a dilution model, with discs valued ,1 or 1 (random coin model) Object-based models. Examples of application, grains in polycrystalline materials, fracture network in a rock mass, fluvial channels in reservoirs, fluvial channels in reservoirs, rocks on a conveyor belt, swarm of veinlets in an ore deposit. Random closed and compact sets. Deterministic compact set In practice, objects can be represented by compact sets, i.e., sets that are closed (i.e., that contain their boundary points) and bounded. In the following, we restrict ourselves to planar sets. Random closed set. A random closed set A in R2is a mapping from a probability space into the family F2 of closed sets equipped with the σ-algebra generated by the collections of closed sets of the form. Random compact set. A random closed set with almost surely compact realizations is called a random compact set. A random compact set that almost surely coincides with the closure of its interior is said to be regular. Random closed and compact sets - Characterization The distribution of a random closed set A is entirely described by its characteristic functionals, defined by. Geometric covariogram Definition The geometric covariogram of a random compact set A is: The homogeneous Poisson point process gives the impression of points distributed at random in space.
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