Fast Algorithms and Efficient GPU Implementations for the Radon Transform and the Back-Projection Operator Represented as Convolution Operators

Abstract

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 the operators can be constructed by using 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. It 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 which 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, speedups of about 10 times are obtained in relation to the computational times of other software packages based on GPU implementations of the Radon transform and the back-projection operator. Moreover, speedups of more than a 1000 times are obtained against the CPU-implementations provided in the MATLAB image processing toolbox.

Dynamic in-situ imaging of methane hydrate formation and self-preservation in porous media

“…The constructed pipeline for studying gas-hydrate formation processes includes plugins for flat field correction, ring removal, and phase-retrieval filter. We also prepared our own plugin for tomography data reconstruction by using the log-polar-based imaging algorithm optimized for GPUs (Andersson et al, 2016).…”

The MAX IV imaging concept

“…One being a Log-polar fast Radon transform approach developed at the Mathematics Department at Lund University [20], the second being algebraic methods such as SIRT and the third group being discrete methods such as DART and an energy minimization method [21]. The latter are more computationally intensive and therefore slower.…”

“…Contrary, the first method is fast and qualifies for on-the-fly data analysis during experiment making it particularly valuable for processing time resolved acquisition. Performance of all the methods in terms of speed is evaluated in Table 1, taken from [20]. …”

“…The data are taken from [20] where the authors compared among others the following four methods: LP Log-Polar Radon transform and backprojection developed by the authors, FBP as implemented in the Astra toolbox [19], IRT Image Reconstruction Toolbox [22]Each algorithm is evaluated for the image size of 1024 and 2048 pixels (values represent computational time in seconds)…”

Fast hyperbolic Radon transform represented as convolutions in log-polar coordinates

“…A fast method for the standard Radon transform was proposed in [1] by expressing the Radon transform and its adjoint in terms of convolutions in log-polar coordinates. Computationally efficient algorithms for GPUs were presented for this approach in [2]. In this paper we propose to use the same approach and construct algorithms with complexity O(N 2 log N ) for evaluation of the hyperbolic Radon transform.…”