Fast and exact Fourier rebinning using John´s equation

Three-dimensional (3D) positron emission tomography (PET) requires the inversion of the 3D X-ray transform. Fast reconstruction algorithms factor the 3D data into independent 2D data sets corresponding to the 2D Radon transforms of a stack of parallel slices. Each slice is then reconstructed using 2D FBP. These so-called rebinning methods are efficient but approximate. We describe a new exact approach to rebinning based on the fact that the 3D X-ray transform of a function is solution to the second order partial differential equation first studied by John. After introducing John´s equation for a pair of infinite plane detectors, we study the analogous equation for a multi-ring PET scanner. The new FORE-J algorithm based on this equation has been implemented and compared with the approximate Fourier rebinning algorithm FORE and with another exact rebinning algorithm, FOREX. Results with simulated data demonstrate a significant improvement in accuracy compared to FORE, while the reconstruction time is doubled. Compared to FOREX, the FORE-J algorithm is slightly less accurate but more than 3 times faster