Topology-preserving image registration with novel multi-dimensional Beltrami regularization
Chongfei Huang, Ke Chen, Meixiang Huang, De-Xing Kong, Jing Yun Yuan · Applied Mathematical Modelling · 2023
Beltrami coefficient provides a proper measure to quantify local distortions for a topology-preserving mapping, a.k.a general quasi-conformal mapping, in 2D. Recent studies managed to extend Beltrami coefficient from 2D to nD, and introduced its corresponding Beltrami-like measures to encode local deformation information in challenging nD topology-preserving image registration tasks. However, all these nD measures are not consistent with the classical 2D Beltrami coefficient. We propose a novel nD measure for evaluating local distortions of an nD diffeomorphic mapping in this work. We show its key mathematical properties , and prove that the new measure coincides with the classical 2D Beltrami coefficient. We further introduce a new nD topology-preserving image registration model by means of regularizing such proposed Beltrami-consistent coefficient, and show the existence of its solution. Meanwhile, a generalized Gauss-Newton scheme is applied to establish a fast multi-level numerical solver. Extensive experiments over different 3D medical image registration tasks, including the challenging multi-modal medical image registration, illustrate that the proposed approach by regularizing the introduced Beltrami-consistent coefficient outperforms the other state-of-the-art topology-preserving methods in both efficiency and accuracy.