Efficient Parallel Transport in the Group of Diffeomorphisms via Reduction to the Lie Algebra

Abstract

This paper presents an efficient, numerically stable algorithm for parallel transport of tangent vectors in the group of diffeomorphisms. Previous approaches to parallel transport in large deformation diffeomorphic metric mapping (LDDMM) of images represent a momenta field, the dual of a tangent vector to the diffeomorphism group, as a scalar field times the image gradient. This “scalar momenta” constraint couples tangent vectors with the images being deformed and leads to computationally costly horizontal lifts in parallel transport. This paper uses the vector momenta formulation of LDDMM, which decouples the diffeomorphisms from the structures being transformed, e.g., images, point sets, etc. This decoupling leads to parallel transport expressed as a linear ODE in the Lie algebra. Solving this ODE directly is numerically stable and significantly faster than other LDDMM parallel transport methods. Results on 2D synthetic data and 3D brain MRI demonstrate that our algorithm is fast and conserves the inner products of the transported tangent vectors.

Fast three‐dimensional image generation for healthy brain aging using diffeomorphic registration

“…For example, those group‐level methods aim at simulating spatiotemporal changes during aging across all subjects. Even though these methods capture the time‐varying changes of a population well, the way to leverage and extrapolate this to the target subject is still under development (Campbell & Fletcher, 2017 ). The work of Pathan and Hong ( 2018 ) has addressed this extrapolation problem by incorporating the regression model based on the framework of large deformation diffeomorphic metric mapping (LDDMM) (Pathan & Hong, 2018 ) with convolutional neural networks and recurrent neural networks.…”

Transporting Deformations of Face Emotions in the Shape Spaces: A Comparison of Different Approaches

“…The LS is an Euclidean version of the Levi Civita parallel transport recently proposed in [37]; the DT is a new Riemannian approach described by [49]; FS is a transport method introduced in [23,24], based on the parallel transport along geodesics of the LDDMM. Recently, many methods have been proposed for transporting deformations, based upon different techniques of deformations estimation: OPA [17,18,20,48], TPS [49], LDDMM [5,15,24,27,29,31,40], active contour [45,46], parametrized surfaces [50], stationary velocity field (SVF) [22]. The characteristics of most of these have been discussed in [49] and therein summarized in Table 1.…”