Discretization of the radon transform and of its inverse by spline convolutions

We present an explicit formula for B-spline convolution kernels; these are defined as the convolution of several B-splines of variable widths hi and degrees ni. We apply our results to derive spline-convolution-based algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the least-squares sense. We then consider the reverse problem and introduce a new spline-convolution version of the filtered back-projection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-degree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.