A Birth-and-Growth Process for Random Closed Sets Based on Minkowski Sum

In literature,birth-and-growth processes are the composition of a marked point process together with a real process that represents a local isotropic dilatation.They describe those evolution phenomena that are typically associated to crystals which border is supposed regular enough.The aim of thesis is to redefine the framework to avoid regularity hypothesis for the border.In order to do this,a geometrical point of view is used to present a particular family of set-valued continuous time stochastic processes:the growth process is described by a bounded closed set-valued process {G_t,t>0},at the same time the nucleation is described by a non-decreasing closed set-valued process {H_t,t>0}.The process is defined as a suitable combination of those processes.In particular,the discrete time process is derived by a maximality property of Minkowski sum,whilst the continuous one is defined thanks to a construction that is analogous to the definition of the Riemann integral.The proposed setting allows us to infer the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth.As a logical consequence,different consistent set-valued estimators are studied for growth process.Moreover,the nucleation process is studied via the Choquet capacity,and some consistent estimators are derived.In order to make image analysis and to test the obtained results in R^2 (a simple case),are implemented some codes.They are tested on benchmark.