Intensity estimation of stationary fibre processes from digital images with a learned detector

Stationary fibre processes are processes of curves in a higher dimensional space, whose distribution is translation invariant. In practical applications, they can be used to model several real objects, such as roots, vascular networks and fibres of materials. Often it is required to compare processes showing similar shape, thus a quantitative approach to describe their stochastic geometry is necessary. One of the basic geometric characteristics of these processes is the intensity (i.e., mean total length per unit area or volume). Here, a general computational-statistical approach is proposed for the estimation of this quantity from digital images of the process, thus only planar fibre processes or projections of processes onto a plane are considered. Differently from approaches based on segmentation, it does not depend on the particular application. The statistical estimator of the intensity is proportional to the number of intersections between the process under study and an independent motion invariant test fibre process. The intersections are detected on the real digital image by a learned detector, easily trained by the user. Under rather mild regularity conditions on the fibre process under study, the method also allows to estimate approximate confidence intervals for the intensity, which is useful especially for comparison purposes.