MELEXIR: maximum entropy Legendre expanded image reconstruction. A fast and efficient method for the analysis of velocity map imaging or photoelectron imaging data

A fast and efficient method is presented that recovers a three-dimensional cylindrically symmetric probability distribution from its two-dimensional projection onto a plane parallel to the cylinder axis. This problem arises regularly in the analysis of data from velocity map imaging or photoelectron imaging experiments. The data can be considered the (numerically stable) Abel transform of the unknown probability distribution. The inverse Abel transform belongs to the class of ill-posed problems. In 2014 I presented two methods which solve this problem, termed maximum entropy velocity image reconstruction (MEVIR) and maximum entropy velocity Legendre reconstruction (MEVELER) (Phys. Chem. Chem. Phys., 2014, 16, 570). The maximum entropy concept finds the most probable solution that agrees with the data for a Gaussian or Poissonian particle statistics. The new method presented here also uses the maximum entropy concept and finds a solution in terms of an expansion in Legendre polynomials like MEVELER. The new method dramatically reduces the size of the numerical problem by using an expansion in terms of Legendre polynomials also for the image data. The new method performs at least as well as MEVELER (i.e. down to intensities of ca. 0.01 counts per pixel), but requires only a small fraction of the CPU time and core memory. It should hence be applicable for on-the-fly data analysis during measurements. It can analyze distributions containing higher-order and odd-order Legendre polynomials, whereas MEVELER performed well only for Legendre polynomials of order l = 0 and l = 2. Higher-order and odd-order Legendre polynomials are required in the analysis of multiphoton dissociation reactions, photoelectrons from higher harmonics experiments, or involving circular dichroism.