New method of averaging diffeomorphisms based on Jacobian determinant and curl vector

TL;DR

This paper introduces a novel method for averaging diffeomorphisms by averaging their Jacobian determinants and curl vectors, ensuring the averaged transformation remains a diffeomorphism, with promising numerical results.

Contribution

The paper proposes a new approach to diffeomorphism averaging that guarantees diffeomorphic properties by using Jacobian determinants and curl vectors, improving over simple Euclidean averaging.

Findings

Method produces valid diffeomorphisms after averaging.

Numerical examples demonstrate the effectiveness of the approach.

Averaged transformations retain key geometric properties.

Abstract

Averaging diffeomorphisms is a challenging problem, and it has great applications in areas like medical image atlases. The simple Euclidean average can neither guarantee the averaged transformation is a diffeomorphism, nor get reasonable result when there is a local rotation. The goal of this paper is to propose a new approach to averaging diffeomorphisms based on the Jacobian determinant and the curl vector of the diffeomorphisms. Instead of averaging the diffeomorphisms directly, we average the Jacobian determinants and the curl vectors, and then construct a diffeomorphism based on the averaged Jacobian determinant and averaged curl vector as the average of diffeomorphisms. Numerical examples with convincible results are presented to demonstrate the method.