An axisymmetric object is reconstructed from its transaxial line-integral projection by the inverse Abel transform. An interesting variation of the Abel inversion problem is the finite-length line-spread function introduced by Dallas et al. [J. Opt. Soc. Am. A 4, 2039 (1987)], in which the path of integration does not extend completely across the object support, resulting in incomplete projections. We refer to this operation as the incomplete Abel transform and derive a space-domain inversion formula for it. It is shown that the kernel of the inverse transform consists of the usual Abel inversion kernel plus a number of correction terms that act to complete the projections. The space-domain inverse is shown to be equivalent to Dallas's frequency-domain inversion procedure. Finally, the space-domain inverse is demonstrated by numerical simulation.
© 1992 Optical Society of America
Eric W. Hansen, "Space-domain inversion of the incomplete Abel transform," J. Opt. Soc. Am. A 9, 2126-2137 (1992)