For stationary fiber processes, the estimation of the directional distribution is an important task. We consider a stereological approach, assuming that the intersection points of the process with a finite number of test hyperplanes can be observed in a bounded window. The intensity of these intersection processes is proportional to the cosine transform of the directional distribution. We use the approximate inverse method to invert the cosine transform and analyze asymptotic properties of the estimator in growing windows for Poisson line processes. We show almost-sure convergence of the estimator and derive Berry–Esseen bounds, including formulae for the variance.