Modelisation of scattered objects as random closed sets Stefan Rolfes

Natural environments We consider natural, unstructured scenes as collection of marked closed sets: Observation : Objects that occur in natural scenes tend to form patches (alga, stone fields, …) The family of all closed sets The mark space, covering the possible types of objects 1

Application : Estimating amount of living and dead alga The mark space is defined as : Estimating the surface of dead and living alga in a large area Complete observation of Question : Can we estimate and based on partial observations? 2

Estimation based on partial observations Complete observationsPartial observations Covering trajectory : time and energy consuming, complete mapping in of the alga Sample trajectory : short path, but just partial mapping in of the alga Infer biomass in from partial observations 3

Capturing global characteristics of an environment by statistical models Distribution of the closed sets (number per unit area) Spatial and semantic relationship between them Distribution of basic morphological characteristics (size, shape, ….) Consider a natural scene as a realisation of a random closed set,defined by a ‘random closed set model’ describing just global characteristics (no need of detailed description): 4

Examples of random closed sets Uniform distribution Non isotropic distribution Cluster process Regular structures 5

Random closed sets Modelisation of the natural scene by a random closed set Observation of Estimation of RCSM (type and parameter) Estimate surface for the whole area Confidence of the estimate. Are more observations necessary ? 6

Parameter estimation of random closed set models The distribution of the compact sets and the intensity is determined by the hitting capactiy (Matheron 75) : Intersecting sets Non intersecting sets Objects can be counted, direct estimation of intensity and morphological characteristics No direct observation of number and morphological characteristics 7

The hitting capacity Analytical forms of can be found for some model types Hitting capacity of : Independent realisations of the RCS miss hit 8

Boolean model : simple model for random closed sets Frequently used to model random scenes, in that the objects are uniformly distributed (biological and physical contexts) The sequence of locations (germs) of the closed sets is a stationary Poisson process of intensity The sequence (grains) are i.i.d. realisations of random closed sets with distribution Boolean model : Goal : Estimate and such that is a typical realisation of 9

Hitting capacity for boolean models The hitting capacity for boolean models (Stoyan): Is the primary (typical) grain, often assumed to be characterizable by a parameter vector with distribution Examples : circles, ellipses, rectangles, … 10

Hitting capacity for isotropic cluster processes 1 The sequenceis Poisson distributed with intensity A cluster process is a union of clusters at locations Characterisation of the clusters can be done when the clusters are assumed to be isotropic 11

Hitting capacity for isotropic cluster processes 2 Intersection of a delimiting area with a boolean model The sequenceis Poisson distributed with intensity Hitting capacity for : Isotropy of the boolean model : 12

Results for isotropic simulated scenes (Boolean models) 1 Two hypothesis for primary grains, characterized by : (Circles of random radius) (Squares of random sidelength) Two hypothesis for the distribution : Estimate intensity and distribution parameter 13

Results for isotropic simulated scenes (Boolean models) 2 Observations Model Obtained by LSE criterium 14

RCSM for maerl mapping 1. Phase : Independend study of living and dead mearl The RCSM depends on environmental factors location ocean current, temperature,... Non isotropic model learned from data : sent video, diver information,... RCSM type and A priori information A posteriori probability (Bayes : a priori information + observations during survey) Confidence estimation 15

Conclusions and Future work Conclusions : Future work : Use Bayesian approach to the estimation of parameters Design and characterization of other appropriate RCSM’s Consideration of interdependence between living and dead mearl Modelisation of natural environments as Random Closed Sets Estimation of amount of living and dead mearl from partial observations Characterisation of simple isotropic RCSM (boolean, cluster model) 16