USE OF BOX SPLINES IN COMPUTER-TOMOGRAPHY

Box splines are attractive for practical multivariate approximation, s ince they possess good approximation power and can be evaluated very e fficiently. We want to give an idea of how their qualities can be made to come into play in the field of image reconstruction in computerize d tomography (CT). To keep the exposition simple, we will concentrate on a special situation: our tomograph will be characterized by the biv ariate standard scanning geometry anal our reconstructions will always lie in scales of the linear space spanned by the integer translates o f a fixed piecewise quadratic box spline. On the other hand we give de tails of an algorithm based on Fourier reconstruction, which produces approximations of optimal order for the box splines used, whilst the a mount of computational work required is of no higher order than for cl assical Fourier reconstruction. We present another reconstruction proc edure based on quasi-interpolation, which compares to filtered backpro jection in computational complexity. Along with our exposition, pre gi ve a generalization of a certain Theorem due to Nievergelt which may b e of interest for practical applications.