We consider two integral transforms which are frequently used in integral geometry and related fields, namely the cosine and the spherical Radon transform. Fast algorithms are developed which invert the respective transforms in a numerically stable way. So far, only theoretical inversion formulas or algorithms for atomic measures have been derived, which are not so important for applications. We focus on the two and threedimensional case, where we also show that our method leads to a regularization. Numerical results are presented and show the validity of the resulting algorithms. First, we use synthetic data for the inversion of the Radon transform. Then we apply the algorithm for the inversion of the cosine transform to reconstruct the directional distribution of line processes from finitely many intersections of their lines with test lines (2D) or planes (3D), respectively. Finally we apply our method to analyze a series of microscopic two-and three-dimensional images of a fibre system.

The focus of this paper is on the numerical inversion of two integral transforms, namely the spherical Radon and the cosine transform. Both transforms are frequently used in integral geometry and related fields, and their numerical inversion is needed in several applications. To derive fast regularization schemes, the method of the approximate inverse is utilized. We introduce a new family of mollifiers and calculate the corresponding reconstruction kernels analytically for dimension d D 3 and even dimensions d 4. Numerical results for the three-dimensional case are presented showing that the new class of mollifiers clearly improves the quality of the reconstruction in comparison to the Gaussian mollifier. Moreover, the regularization theory for the method is extended to a framework for arbitrary dimension d 3 (the special case d D 3 was already considered in [21]).

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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.

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