# Finite-Dimensional Lie Algebras for Fast Diffeomorphic Image Registration

@article{Zhang2015FiniteDimensionalLA, title={Finite-Dimensional Lie Algebras for Fast Diffeomorphic Image Registration}, author={Miaomiao Zhang and P. Thomas Fletcher}, journal={Information processing in medical imaging : proceedings of the ... conference}, year={2015}, volume={24}, pages={ 249-59 } }

This paper presents a fast geodesic shooting algorithm for diffeomorphic image registration. We first introduce a novel finite-dimensional Lie algebra structure on the space of bandlimited velocity fields. We then show that this space can effectively represent initial velocities for diffeomorphic image registration at much lower dimensions than typically used, with little to no loss in registration accuracy. We then leverage the fact that the geodesic evolution equations, as well as the adjoint…

## 55 Citations

Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

- Computer ScienceInternational Journal of Computer Vision
- 2018

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration that approximate the infinite-dimensional Lie algebra of smooth vector fields with a low-dimensional, bandlimited space.

Frequency Diffeomorphisms for Efficient Image Registration

- Computer Science, MedicineIPMI
- 2017

A novel finite dimensional Fourier representation of diffeomorphic deformations based on the key fact that the high frequency components of a diffeomorphism remain stationary throughout the integration process when computing the deformation associated with smooth velocity fields is introduced.

Data-Driven Model Order Reduction for Diffeomorphic Image Registration

- Computer ScienceIPMI
- 2019

A reduced order model (ROM) is introduced to substantially lower the overall computational cost while maintaining accurate alignment in the context of large deformation diffeomorphic metric mapping (LDDMM) and is demonstrated in neuroimaging applications of pairwise image registration and template estimation for population studies.

Low-dimensional shape analysis in the space of diffeomorphisms

- Mathematics
- 2020

Abstract Statistical shape analysis via diffeomorphic image registration is challenging due to the high-dimensional and nonlinear nature of the space of diffeomorphisms. In this chapter, we first…

Efficient diffeomorphic metric image registration via stationary velocity

- Computer ScienceJ. Comput. Sci.
- 2019

A new simplified model for optimal stationary velocity by representing temporal integration form of transformation variation via a single point model, e.g. the middle time point, according to mean value theorem and smoothness of transformation, is proposed.

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

- Mathematics, Computer ScienceGRAIL/MFCA/MICGen@MICCAI
- 2017

This paper uses the vector momenta formulation of LDDMM, which decouples the diffeomorphisms from the structures being transformed, e.g., images, point sets, etc, to lead to parallel transport expressed as a linear ODE in the Lie algebra.

Fast Geodesic Regression for Population-Based Image Analysis

- Computer Science, MedicineMICCAI
- 2017

A fast geodesic regression method that dramatically decreases the computational cost of the inference procedure while maintaining prediction accuracy and is dramatically faster than the state-of-the-art regression methods while producing equally good regression results on the large subject cohort.

PDE-constrained LDDMM via geodesic shooting and inexact Gauss-Newton-Krylov optimization using the incremental adjoint Jacobi equations

- Computer Science, MathematicsPhysics in medicine and biology
- 2019

This work proposes a method for PDE-constrained LDDMM parameterized in the space of initial velocity fields under the EPDiff equation that provides geodesics in the framework of PDE/LDDMM, and it shows performance competing with benchmark PDE’s large deformation L DDMM and EPDIFF-LDDMM methods.

Practical Implementation and Application of Geodesic Regression in Diffeomorphisms to Brain Image Time Series

- Mathematics
- 2016

Diffeomorphisms have received significant research focus in the medical image registration community over the past 15 years due in part to their desirable mathematical properties: the preservation of…

Stuttgart Efficient Algorithms for Geodesic Shooting in Diffeomorphic Image Registration

- 2019

Diffeomorphic image registration is a common problem in medical image analysis. Here, one searches for a diffeomorphic deformation that maps one image (the moving or template image) onto another…

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