An alternative to inverse Abel transform and forward convolution methods is presented for extracting dynamical information from velocity-map ion images. Unlike most competing methods, that presented here does not require the probed three-dimensional distribution to possess cylindrical symmetry. The new method involves analysis of the Fourier moments of images measured in different experimental geometries, and allows speed distributions, angular differential cross sections, and angular momentum alignment and orientation to be determined from raw images of the products of photodissociation and photon-initiated bimolecular reactions. The methodology is developed within the semiclassical framework of Dixon's bipolar moment formalism [R. N. Dixon, J. Chem. Phys. 85, 1866 (1986)], although it is equally applicable to other common formulations of the product scattering distribution. To allow a comparison of the method with the Abel inversion, which requires that the velocity distribution of the probed product has an axis of cylindrical symmetry, the method is applied to newly acquired experimental images of atomic chlorine produced in the photolysis of NOCl. Extraction of product rotational alignment information is illustrated using newly acquired images of rotationally aligned NO formed by NO2 photolysis. Application of the Fourier moment methodology to studies of bimolecular reactions is also demonstrated, using simulated images for the reaction H+D-2-->HD(v=0,j=0,9)+D.