[en] In this thesis, new models and methodologies are introduced for the analysis of dynamic processes characterized by image sequences with spatial temporal overlapping. The spatial temporal overlapping exists in many natural phenomena and should be addressed properly in several Science disciplines such as Microscopy, Material Sciences, Biology, Geostatistics or Communication Networks. This work is related to the Point Process and Random Closed Set theories, within Stochastic Geometry. The proposed models are an extension of Boolean Models in R2 by adding a temporal dimension. The study has been motivated for its application in a multidisciplinary project that combined Statistics, Computer Sciences, Biology and Microscopy, with the aim of analysing the cell exocytosis and endocytosis. Exocytosis is the process by which cells secrete vesicles outside the plasma membrane and endocytosis is the opposite mechanism. Our data were image sequences obtained by Electron Microscopy and Total Internal Reflection Fluorescence Microscopy. Fluorescent tagged-proteins are observed as overlapped clusters with random shape, area and duration. They can be modelled as realizations of a stationary and isotropic stochastic process. The methodology herein proposed could be used to analyze similar phenomena in other Fields of Science. First, the temporal Boolean model is introduced and some estimation methods for the parameters of the model are presented. Second, we proposed a method for the estimation of the event duration distribution function of a univariate temporal Boolean model based on spatial temporal covariance. A simulation study is performed with several duration probability density functions, and an application to the cell endocytosis is realized. Third, we introduce the bivariate temporal Boolean model to study interactions between two overlapped spatial temporal processes and to quantify their overlapping and dependencies. We propose a non-parametric approach based on a generalization of the Ripley K-function, the spatial-temporal covariance and the pair correlation functions for a bivariate temporal random closed set. A Monte Carlo test was performed to test the independence hypothesis. This methodology is not only a test procedure but also allows us to quantify the degree and spatial temporal interval of the interaction. No parametric assumption is needed. A simulation study has been conducted and an application to the study of proteins that mediate in cell endocytosis has been performed. Fourth, from high spatial resolution EM images, we model the distribution of exocytic vesicles (granules) within the cell cytoplasm as a realization of a finite point process (a point pattern), and the point patterns of several cell groups are considered replicates of different point processes. Our aim was to study differences between two treatment groups in terms of granule location. We characterize the spatial distribution of granules with respect to the plasma membrane by means of several functional descriptors, that allowed us to detect significant differences between the two cell groups that would not be observed by a classical approach. To perform image segmentation, we developed an automatic granule detection tool with similar performance to that of the manual one-by-one marking. Finally, we have implemented a software toolbox for the simulation and analysis of temporal Boolean models (available at http : ==www:uv:es=tracs=), so scientists and technicians of any discipline can apply the proposed methods. In summary, the spatial temporal stochastic models proposed allow modelling of dynamic processes from image sequences where several forms of random shape, size and duration overlap. It is the first time these tools are applied to the study of cell exo and endocytosis, and they would contribute to improve their understanding. Our methodologies will help future research in Cell Biology, e.g. in the study of diseases related to secretion dysfunctions, such as diabetes.

[es] En esta tesis presentamos nuevos modelos y metodolog as para el an alisis de pro- cesos din amicos a partir de secuencias de im agenes, con solapamiento espacial y tem- poral de los objetos de an alisis, un fen omeno habitual en la naturaleza. El trabajo realizado se enmarca en la teor a de Procesos Puntuales y Conjuntos Aleatorios Ce- rrados (RACS), dentro de la Geometr a Estoc astica. Los modelos propuestos son una extensi on de la teor a de modelos booleanos en R2 incorporando una componente temporal. La motivaci on del trabajo fue su aplicaci on a un proyecto multidisciplinar donde analizamos la exocitosis y la endocitosis celular, procesos en que la c elula segrega o absorbe sustancias a trav es de la membrana citoplasm atica, respectivamente. El es- tudio se realiz o utilizando secuencias de im agenes obtenidas con microscop a TIRFM, donde se observan las prote nas como agrupaciones uorescentes superpuestas. Mo- delizamos las im agenes como realizaciones de un proceso estoc astico estacionario e isotr opico. Esta metodolog a permite analizar fen omenos reales en otros campos de la Ciencia con superposici on espacio-temporal de objetos con formas y duraciones aleatorias, como Geolog a, Qu mica, Comunicaciones, etc. Primero, introducimos el modelo booleano temporal. Presentamos un m etodo de estimaci on de la funci on de distribuci on de la duraci on basado en la covarianza espacio-temporal, y el estudio de simulaci on realizado. Segundo, estudiamos la in- terrelaci on entre dos procesos espacio-temporales mediante la K-funci on de Ripley, la covarianza espacio-temporal y la funci on de correlaci on para conjuntos aleatorios bivariados. Realizamos un estudio de simulaci on y una aplicaci on a la endocitosis celular. Tercero, modelizamos la distribuci on de ves culas exoc ticas (gr anulos) en el cito- plasma celular como un proceso puntual nito. Caracterizamos su distribuci on espa- cial respecto a la membrana mediante varios descriptores funcionales. Para segmentar las im agenes, desarrollamos una herramienta autom atica de detecci on de gr anulos. Hemos desarrollado una herramienta de software completa para la simulaci on y es- timaci on de modelos booleanos temporales (disponible en http : ==www:uv:es=tracs=).