Use of box splines in computer tomography

Abstract

Box splines are attractive for practical multivariate approximation, since they possess good approximation power and can be evaluated very efficiently. We want to give an idea of how their qualities can be made to come into play in the field of image reconstruction in computerized tomography (CT). To keep the exposition simple, we will concentrate on a special situation: our tomograph will be characterized by the bivariate standard scanning geometry and our reconstructions will always lie in scales of the linear space spanned by the integer translates of a fixed piecewise quadratic box spline. On the other hand we give details of an algorithm based on Fourier reconstruction, which produces approximations of optimal order for the box splines used, whilst the amount of computational work required is of no higher order than for classical Fourier reconstruction. We present another reconstruction procedure based on quasi-interpolation, which compares to filtered backprojection in computational complexity. Along with our exposition, we give a generalization of a certain Theorem due to Nievergelt which may be of interest for practical applications.