Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators

The Radon transform and its adjoint, the back-projection operator, can both be expressed as convolutions in log-polar coordinates. Hence, fast algorithms for the application of these operators can be constructed by using the FFT, if data is resampled at log-polar coordinates. Radon data is typically measured on an equally spaced grid in polar coordinates, and reconstructions are represented (as images) in Cartesian coordinates. Therefore, in addition to FFT, several steps of interpolation have to be conducted in order to apply the Radon transform and the back-projection operator by means of convolutions. However, in comparison to the interpolation conducted in Fourier-based gridding methods, the interpolation performed in the Radon and image domains will typically deal with functions that are substantially less oscillatory. Reasonable reconstruction results can thus be expected using interpolation schemes of moderate order. The approach also provides better control over the artifacts that can appear due to measurement errors. Both the interpolation and the FFT operations can be efficiently implemented on graphical processor units (GPUs). For the interpolation, it is possible to make use of the fact that linear interpolation is hard-wired on GPUs, meaning that it has the same computational cost as direct memory access. Cubic order interpolation schemes can be constructed by combining linear interpolation steps, and this provides important computation speedup. We provide details about how the Radon transform and the back-projection can be implemented efficiently as convolution operators on GPUs. For large data sizes, these algorithms are several times faster than those of other software packages based on GPU implementations of the Radon transform and the back-projection operator. Moreover, the gain in computational speed is substantially higher when comparing against other CPU-based algorithms.