A "circularisation" method to repair deformations and determine the centre of velocity map images

A problem besetting the analysis of velocity map images, particularly those of photoelectrons, is the presence of distortions that cause the features in the image to deviate from circularity, leading to a loss of resolution in the spectrum extracted. A method is presented to repair such distortions based on fitting the angular behaviour of each of the ring structures to a trigonometric expansion. The repair function allows the intensity at any value of radius and angle to be mapped to a new position that removes the distortion and returns the features to circular. While the method relies on the analysis of the structure in an image, it could also be applied to determine the "repair function" using a calibration image (or series of images) for the experiment. Once the image has been circularised it can be processed by any of the approaches that have been developed for that purpose. The analysis also enables the image centre to be determined with high accuracy. The fitting method utilises an inverse Abel transformation of the image in polar coordinates as a means to reshape the image into a series of spectral features in order to determine the radial position of features at each angle. Although the velocity distribution is not in general spherically symmetric and so this is not a mathematically correct means to extract the velocity distribution, the feature positions are accurately reproduced in the resulting spectrum while the intensity and anisotropy parameters can be remarkably close to those obtained using the proper inverse Abel transformation of the image.