Statistical methods for the analysis of stochastic fibre processes

In this paper we describe some statistical methods related with stochastic geometries, and in particular with fibre processes, which can be applied to address a big set of biomedical problems, ranging from quantifying dose/effect ratios in medical treatments, to automatic diagnosis of pathologies related with the “shape” of the fibre process under study. One of the main mathematical instruments which is used to characterize the geometry of a fibre process is its mean density or intensity. Statistical methods for the estimate of the intensity are available in literature, but all of them are based either on the assumption of stationarity, or on the availability of a rather big i.i.d. sample of the fibre process under study. Unfortunately in many applications just one or a few images of the process are available, so that these estimators cannot be applied. In such situations an atlas-like division of the image into quasi-stationary subregions, where the techniques applicable in the stationary case can be locally used, is fundamental to be able to perform statistical inference. Here we describe two methods for the identification of quasi-stationary subregions, which exploit some prior knowledge of the structure of the fibre process under study, and we compare the results on simulated experiments. From the experimental results, the two methods look almost equivalent. Actually their robustness with respect to small random variations on the considered fibre process should still be investigated.