  Accurate interpolation and approximation techniques for functions with
discontinuities are key tools in many applications as, for instance, medical
imaging. In this paper, we study an RBF type method for scattered data
interpolation that incorporates discontinuities via a variable scaling
function. For the construction of the discontinuous basis of kernel functions,
information on the edges of the interpolated function is necessary. We
characterize the native space spanned by these kernel functions and study error
bounds in terms of the fill distance of the node set. To extract the location
of the discontinuities, we use a segmentation method based on a classification
algorithm from machine learning. The conducted numerical experiments confirm
the theoretically derived convergence rates in case that the discontinuities
are a priori known. Further, an application to interpolation in magnetic
particle imaging shows that the presented method is very promising.
