Inversion of the noisy Radon transform on SO(3) by Gabor frames and sparse recovery principles

In this paper we present algorithms to calculate the fast Fourier synthesis and its adjoint on the rotation group SO(3) for arbitrary sampling sets. They are based on the fast Fourier transform for nonequispaced nodes on the three-dimensional torus. Our algorithms evaluate the SO(3) Fourier synthesis and its adjoint, respectively, of B-bandlimited functions at M arbitrary input nodes in \(\mathcal O(M+B^4)\) or even \(\mathcal O(M + B^3 \log^2 B)\) flops instead of \(\mathcal O(MB^3)\). Numerical results will be presented establishing the algorithm’s numerical stability and time requirements.

/pdf/a-fast-algorithm-for-nonequispaced-fourier-transforms-on-the-3dybmtg8kz.pdf

A fast algorithm for nonequispaced Fourier transforms on the rotation group

https://www-user.tu-chemnitz.de/~rahi/paper/kdeSO3.pdf

Kernel density estimation on the rotation group and its application to crystallographic texture analysis

Neutron, gamma-ray, and x-ray imaging are important diagnostic tools at the National Ignition Facility (NIF) for measuring the two-dimensional (2D) size and shape of the neutron producing region, for probing the remaining ablator and measuring the extent of the DT plasmas during the stagnation phase of Inertial Confinement Fusion implosions. Due to the difficulty and expense of building these imagers, at most only a few two-dimensional projections images will be available to reconstruct the three-dimensional (3D) sources. In this paper, we present a technique that has been developed for the 3D reconstruction of neutron, gamma-ray, and x-ray sources from a minimal number of 2D projections using spherical harmonics decomposition. We present the detailed algorithms used for this characterization and the results of reconstructed sources from experimental neutron and x-ray data collected at OMEGA and NIF.

Three-dimensional reconstruction of neutron, gamma-ray, and x-ray sources using spherical harmonic decomposition

Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross correlation on the sphere.

The inversion of the one-dimensional Radon transform on the rotation group SO(3) is an ill-posed inverse problem which applies to x-ray tomography with polycrystalline materials. This paper presents a novel approach to the numerical inversion of the one-dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Radon transform of a function sampled on the product space of two two-dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests.

/pdf/the-radon-transform-on-so-3-a-fourier-slice-theorem-and-4oc788spc2.pdf

M Schmalz

Papers

The Radon transform on SO(3): a Fourier slice theorem and numerical inversion *