Inverse projection of axisymmetric orientation distributions

Abstract

Abstract We show that the projection of an axisymmetric three-dimensional orientation distribution to two dimensions can be cast into an Abel transform. Based on this correspondence, we derive an exact integral inverse, which allows for the quantification of three-dimensional uniaxial alignment of rod-like units from two-dimensional sliced images, thus providing an alternative to X-ray or tomographic analysis. A matrix representation of the projection and its inverse is derived, providing a direct relationship between two- and three-dimensional order parameters for both polar and non-polar systems.