  The Abel transform is a mathematical operation that transforms a
cylindrically symmetric three-dimensional (3D) object into its two-dimensional
(2D) projection. The inverse Abel transform reconstructs the 3D object from the
2D projection. Abel transforms have wide application across numerous fields of
science, especially chemical physics, astronomy, and the study of laser-plasma
plumes. Consequently, many numerical methods for the Abel transform have been
developed, which makes it challenging to select the ideal method for a specific
application. In this work eight transform methods have been incorporated into a
single, open-source Python software package (PyAbel) to provide a direct
comparison of the capabilities, advantages, and relative computational
efficiency of each transform method. Most of the tested methods provide
similar, high-quality results. However, the computational efficiency varies
across several orders of magnitude. By optimizing the algorithms, we find that
some transform methods are sufficiently fast to transform 1-megapixel images at
more than 100 frames per second on a desktop personal computer. In addition, we
demonstrate the transform of gigapixel images.
