Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration

TL;DR

This paper analyzes the topology preservation properties of various Radial Basis Functions, including Matérn functions, in landmark-based image registration, comparing their effectiveness in maintaining topology during deformation modeling.

Contribution

It provides a detailed analysis of topology preservation for RBFs, especially Matérn functions, in landmark-based image registration, with comparative numerical results.

Findings

Matérn functions can preserve topology in landmark registration

Comparison shows differences among RBFs in topology preservation

Numerical results highlight the effectiveness of specific RBFs

Abstract

In image registration, a proper transformation should be topology preserving. Especially for landmark-based image registration, if the displacement of one landmark is larger enough than those of neighbourhood landmarks, topology violation will be occurred. This paper aim to analyse the topology preservation of some Radial Basis Functions (RBFs) which are used to model deformations in image registration. Mat\'{e}rn functions are quite common in the statistic literature (see, e.g. \cite{Matern86,Stein99}). In this paper, we use them to solve the landmark-based image registration problem. We present the topology preservation properties of RBFs in one landmark and four landmarks model respectively. Numerical results of three kinds of Mat\'{e}rn transformations are compared with results of Gaussian, Wendland's, and Wu's functions.