Use of box splines in computer tomography
Numerical computerized tomography tackles the problem of constructing an approximation \(f^*\) of \(f\) from a finite set of measurements of \(Rf\). The approximation \(f^*\) of \(f\) belongs to the space of box splines. Box splines are attractive multivariate approximations, since they possess good approximation power and can be evaluated very efficiently. The author wants to give an idea of how their qualities can be made to come into play in the field of image reconstruction in computerized tomography. To keep the exposition simple, he concentrates on a special situation: His tomograph is characterized by the bivariate standard scanning geometry and his reconstructions always lie in scales of the linear space spanned by the integer translates of a fixed piecewise quadratic box 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. The author presents another reconstruction procedure based on quasi-interpolation, which compares to filtered backprojection in computational complexity. Along with his exposition, he gives a generalization of a certain theorem due to \textit{Y. Nievergelt} [SIAM Rev. 28, 79-84 (1986; Zbl 0623.65139)] which may be of interest of practical applications.
scientific article; zbMATH DE number 1220841
scientific article