Numerical inversion of the sperical Radon transform: estimation of the directional distribution of fibres of stationary anisotropic cylinder processes
Final Report Abstract
The spherical Radon and the cosine transform appear frequently both in applications and in theoretical work, for example in tomography, stereology and geometry. Our investigations in this project were motivated by a special inverse problem appearing in stereology, namely the estimation of the directional distribution of fibres of stationary anisotropic porous media, e.g. cylinder processes. For example materials with a filament structure or containing metal pins can be modelled by this class, and it is known that the directional distribution has a crucial influence on the stability and other properties of a material. We suppose that it is impossible to observe the whole fibre process, but only the numbers of intersections of the process with finitely many test hyperplanes can be determined. This has applications for example when a solid material is examined which cannot be penetrated by x-rays or in confocal microscopy. It turns out that the intensity of the observed intersection process (called the rose of intersections) is the cosine transform of the directional distribution of the fibre process. Thus, our idea for the estimation of the directional distribution is a two-step procedure: 1. Estimate the rose of intersections based on the numbers of intersections of the process with finitely many test hyperplanes. 2. Numerically approximately invert the cosine transform to obtain an estimation for the directional distribution. While the first step is a simple counting procedure, the second step is quite involved: We use the method of the approximate inverse to solve this inverse problem. One crucial point for the use of this method is the analytical inversion of the cosine transform for a certain function, a so-called mollifier which approximates the delta distribution. The result of the inversion is called the reconstruction kernel. For the spherical Radon transform, we were able to determine the corresponding reconstruction kernels analytically (using an inversion formula for rotation invariant functions) for two classes of mollifiers, namely a polynomial class and one based on the Gaussian density. Since both transforms are closely related, this formula could be used to come up with a reconstruction kernel for the cosine transform, too. Note, that it is rather difficult to determine the reconstruction kernel for the cosine transform directly for dimensions d ≥ 3. (For d = 2 this can be done by an easy calculation.) By this, we have accomplished to invert the spherical Radon and the cosine transform in a numerically stable way and also managed to estimate the directional distribution of a fibre process. We also want to point out that our algorithm is very fast because the reconstruction kernel can be calculated in advance and the evaluation of the estimator at a certain point coincides with the calculation of the inner product of two known functions, the given data and the reconstruction kernel. To verify the applicability of our approach, we have first analysed the results of the inversion method more closely. While the results in numerical experiments looked very good for both transforms, we were also interested in the theoretical properties. Thus, we have derived sufficient conditions on the mollifier such that the inversion method leads to a regularisation and verified them for our two examples in arbitrary dimensions d ≥ 2. Furthermore, we have considered the stochastic asymptotic properties of the resulting estimator as the number of test hyperplanes and the size of the observation window tend to infinity. We have shown that under certain (mild) conditions it converges to the directional distribution density in the supremum norm almost surely. Further, for the value of the density at a given point we have derived a central limit theorem with Berry-Esseen bounds and large deviation results. Finally, a χ2-test for the directional distribution was obtained. To see how well our method works in comparison to other known approaches, we have teamed up with four well-known researchers in this field to extensively analyse the quality of the reconstruction for simulated 3D data sets which have been voxelised as well as real data from microscopic images. While the results are not completely finished, we can already say our output looks at least comparably good, especially for sparse processes. We do not directly compare the speed of the methods, but given the computational time our co-workers have reported on their systems, our approach seems to be the fastest.
Publications
- Inversion algorithms for the spherical Radon and cosine transforms. Oberwolfach Report, 18:64–66, 2010
M. Riplinger and M. Spiess
- Anisotropic Poisson processes of cylinders. Methodology and Computing in Applied Probability, 13(4):801–819, 2011
M. Spiess and E. Spodarev
- Inversion algorithms for the spherical Radon and cosine transform. Inverse Problems 27, 035015 (25pp)
A. K. Louis, M. Riplinger, M. Spiess and E. Spodarev
