Scattered data interpolation of Radon data

Linear combinations of translates of a given basis function have long been successfully used to solve scattered data interpolation and approximation problems. We demonstrate how the classical basis function approach can be transferred to the projective space ℙ ^d−1. To be precise, we use concepts from harmonic analysis to identify positive definite and strictly positive definite zonal functions on ℙ ^d−1. These can then be applied to solve problems arising in tomography since the data given there consists of integrals over lines. Here, enhancing known reconstruction techniques with the use of a scattered data interpolant in the “space of lines”, naturally leads to reconstruction algorithms well suited to limited angle and limited range tomography. In the medical setting algorithms for such incomplete data problems are desirable as using them can limit radiation dosage.