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

摘要

The Radon transform and its adjoint, the back-projection operator, can bothbe expressed as convolutions in log-polar coordinates. Hence, fast algorithmsfor the application of the operators can be constructed by using FFT, if datais resampled at log-polar coordinates. Radon data is typically measured on anequally spaced grid in polar coordinates, and reconstructions are represented(as images) in Cartesian coordinates. Therefore, in addition to FFT, severalsteps of interpolation have to be conducted in order to apply the Radontransform and the back-projection operator by means of convolutions. Both the interpolation and the FFT operations can be efficiently implementedon Graphical Processor Units (GPUs). For the interpolation, it is possible tomake use of the fact that linear interpolation is hard-wired on GPUs, meaningthat it has the same computational cost as direct memory access. Cubic orderinterpolation schemes can be constructed by combining linear interpolationsteps which provides important computation speedup. We provide details about how the Radon transform and the back-projection canbe implemented efficiently as convolution operators on GPUs. For large datasizes, speedups of about 10 times are obtained in relation to the computationaltimes of other software packages based on GPU implementations of the Radontransform and the back-projection operator. Moreover, speedups of more than a1000 times are obtained against the CPU-implementations provided in the MATLABimage processing toolbox.