Generalized Backprojection Operator: Fast Calculation

The inverse Radon transform and his straightforward implementation, known as filtered backprojection (also known as FBP), has become a powerful algorithm for solving a tomographic inverse problem. It has a wide range of applications, including geophysics, medicine and synchrotrons, and from kilo to centi to micro scale respectively. Such a classical inversion has a major computational disadvantage: increasing slowness proportionally to the data size. An ordinary implementation of this algorithm relies on a simple integral that has to be done pixelwise. Many accelerating techniques were proposed in the literature so as to make this part of the inversion as fast as possible. One the most promising strategies is converting the backprojection as a convolution operator (at log-polar coordinates). The generalized backprojector has many applications, for instance in the analytical inversion of single-photon emission tomography or x-ray fluorescence tomography. Our aim in this paper is to show how these ideas can be used for other inversion methods, the iterative ones; which deal much better with noise.