# 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…

## 56 Citations

Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

- MathematicsInternational 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 ScienceIPMI
- 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.

Efficient diffeomorphic metric image registration via stationary velocity

- EngineeringJ. Comput. Sci.
- 2019

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 ScienceMICCAI
- 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 SciencePhysics 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

- Computer Science
- 2019

It is demonstrated that low rank preconditioners are able to significantly reduce the number of iterations needed to solve the linear system in Newton-type optimizers, and different optimization methods are compared based on their overall performance.

## References

SHOWING 1-10 OF 16 REFERENCES

Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation

- Computer Science, MathematicsInternational Journal of Computer Vision
- 2011

The key contribution of this work is to provide an accurate estimation of the so-called initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics.

Diffeomorphic demons: Efficient non-parametric image registration

- Computer ScienceNeuroImage
- 2009

A Log-Euclidean Framework for Statistics on Diffeomorphisms

- Mathematics, Computer ScienceMICCAI
- 2006

This article focuses on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm, which is a simple 3D vector field and well-defined for diffe morphisms close enough to the identity.

A vector momenta formulation of diffeomorphisms for improved geodesic regression and atlas construction

- Mathematics2013 IEEE 10th International Symposium on Biomedical Imaging
- 2013

This paper presents a novel approach for diffeomorphic image regression and atlas estimation that results in improved convergence and numerical stability. We use a vector momenta representation of a…

Bayesian Estimation of Regularization and Atlas Building in Diffeomorphic Image Registration

- MathematicsIPMI
- 2013

An atlas estimation procedure that simultaneously estimates the parameters controlling the smoothness of the diffeomorphic transformations and a Monte Carlo Expectation Maximization algorithm, where the expectation step is approximated via Hamiltonian Monte Carlo sampling on the manifold of diffeomorphisms.

Geodesic Shooting for Computational Anatomy

- MathematicsJournal of Mathematical Imaging and Vision
- 2005

It is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint the authors introduce.

Diffeomorphic registration using geodesic shooting and Gauss–Newton optimisation

- BusinessNeuroImage
- 2011

Deformable templates using large deformation kinematics

- EngineeringIEEE Trans. Image Process.
- 1996

Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.