Reconstruction of Abel-transformable images: The Gaussian basis-set expansion Abel transform method

TLDR

The authors introduce a novel method for reconstructing three‑dimensional images with cylindrical symmetry from two‑dimensional projections. The method expands the projection in a Gaussian‑like basis of analytically projectable functions, reconstructs the 3D image as a linear combination with identical coefficients, and applies regularization to stabilize noisy data. The approach is efficient, computationally inexpensive, requires minimal user input, and reliably reconstructs images—especially when the conventional Fourier–Hankel Abel transform fails—making it well suited for photoion and photoelectron imaging experiments.

Abstract

In this article we present a new method for reconstructing three-dimensional (3D) images with cylindrical symmetry from their two-dimensional projections. The method is based on expanding the projection in a basis set of functions that are analytical projections of known well-behaved functions. The original 3D image can then be reconstructed as a linear combination of these well-behaved functions, which have a Gaussian-like shape, with the same expansion coefficients as the projection. In the process of finding the expansion coefficients, regularization is used to achieve a more reliable reconstruction of noisy projections. The method is efficient and computationally cheap and is particularly well suited for transforming projections obtained in photoion and photoelectron imaging experiments. It can be used for any image with cylindrical symmetry, requires minimal user’s input, and provides a reliable reconstruction in certain cases when the commonly used Fourier–Hankel Abel transform method fails.