Positive definite kernel functions are popular tools for multivariate scattered data approximation.
In particular, the utility of kernel-based reconstructions from generalized Hermite-Birkhoff data has been demonstrated in many applications. The approximation of images from scattered Radon data is only one relevant example. As we show, however, standard kernel-based reconstruction methods fail to work for this particular application. Therefore, we first explain limitations of radial kernels, before we propose weighted positive definite kernels, which are symmetric but not radially symmetric. We discuss the characterization and construction of weighted positive definite kernels in general, before we provide concrete examples.
This leads us to a larger class of flexible kernel-based approximation schemes, which work for image reconstruction from scattered Radon data and other relevant applications.