Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI

TL;DR

This paper introduces a novel RBF-based interpolation method that effectively handles functions with discontinuities by incorporating edge detection and variable scaling, with proven error bounds and promising applications in medical imaging.

Contribution

It develops a new discontinuous kernel interpolation framework that integrates edge detection, providing theoretical error analysis and demonstrating effectiveness in magnetic particle imaging.

Findings

Error bounds depend on node distribution and discontinuity location

Numerical experiments confirm convergence rates when edges are known

Application to magnetic particle imaging shows practical potential

Abstract

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.