Algebraic Medical Image Reconstruction from Scattered Radon Data by Positive Definite Kernels

Computerized tomography requires customized numerical methods for the approximation of a bivariate target function f from a finite set of discrete Radon data, each of whose data samples represents one line integral of f . In standard reconstruction methods, specific assumptions concerning the distribution of the sample lines are usually made, e.g. by parallel line geometry. In relevant applications of medical image reconstruction, however, such assumptions are often too restrictive. In this case, one would rather prefer to work with reconstruction methods allowing for arbitrary distributions of scattered sample lines. In this paper, we propose a novel kernel-based algebraic reconstruction method for medical image reconstruction from scattered Radon data. Our reconstruction relies on generalized Hermite-Birkhoff interpolation by positive definite kernel functions in combination with a suitable regularization of the Radon transform. This leads to a very flexible reconstruction method for medical images, whose good performance is supported by numerical examples and comparisons with classical Fourier-based methods relying on the filtered back projection formula.