Topology-preserving image registration with novel multi-dimensional Beltrami regularization

Abstract

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.