Revisiting the inverse Abel integral for reconstructing velocity-map images

The velocity-map imaging (VMI) technique is used near ubiquitously throughout the study of gas-phase photophysics and chemical dynamics. Many VMI experiments rely on numerical reconstruction techniques to recover the full three-dimensional (3D) velocity distribution of photoproducts from the two-dimensional (2D) geometric projection - the Abel transform of the distribution - that is recorded in a typical experiment. The simplest mathematical approach for this reconstruction procedure is through use of the inverse Abel integral transform. Historically, though, this approach has performed poorly on real experimental data, and so the VMI community has devoted much effort into the development of alternative inversion strategies that avoid direct use of the integral. In this article, we challenge this firmly held belief, and show instead what advantages can be realised through this approach. Unlike many other competing approaches, the reconstruction technique presented here, which we refer to as the modified Abel integral transform (MAIT), does not require the lengthy pre-computation time for a large basis set or any manually adjustable regularisation parameters. Examples involving simulated and real experimental data are used to demonstrate the efficacy of our new approach. This method is shown to perform similarly to the most popular alternative strategies for extracting photoproduct angular distributions, and have a significant advantage over them when handling data with high levels of background noise, in particular.