Macro X-ray Fluorescence (MA-XRF) scanning is an increasingly widely used technique for analytical imaging of paintings and other artworks. The datasets acquired must be processed to produce maps showing the distribution of the chemical elements that are present in the painting. Existing approaches require varying degrees of expert user intervention, in particular to select a list of target elements against which to fit the data. In this paper, we propose a novel approach that can automatically extract and identify chemical elements and their distributions from MA-XRF datasets. The proposed approach consists of three parts: 1) pre-processing steps, 2) pulse detection and model order selection based on Finite Rate of Innovation theory, and 3) chemical element estimation based on Cramér-Rao bounding techniques. The performance of our approach is assessed using MA-XRF datasets acquired from paintings in the collection of the National Gallery, London. The results presented show the ability of our approach to detect elements with weak X-ray fluorescence intensity and from noisy XRF spectra, to separate overlapping elemental signals and, excitingly, to aid visualisation of hidden underdrawing in a masterpiece by Leonardo da Vinci.