Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

MIRZA FAISAL BEGMICHAEL I. MILLERALAIN TROUVÉLAURENT YOUNES

article2005IJCV1,958 citations

Develops the algorithmic and mathematical foundation for large deformation diffeomorphic metric mapping by deriving Euler-Lagrange equations and implementing a semi-Lagrangian particle flow method to compute shortest-path geodesic transformations between anatomical images.

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Modern medical imaging techniques generate intricate anatomical datasets, yet significant natural variability across individuals makes comparing and standardizing these images challenging. Traditional registration approaches frequently fail during large deformations because they allow coordinate grids to fold or tear, disrupting essential anatomical structures. To overcome these limitations, the article develops, implements, and validates the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. This framework aims to compute smooth, invertible coordinate transformations between biological images while establishing a mathematically rigorous metric distance to quantify anatomical differences.

The authors formulated the registration task as an optimal control variational problem solved via Euler-Lagrange equations. Instead of evaluating displacement fields greedily or locally in time, the method computes geodesic shortest paths across the full time-dependent flow within a space of smooth velocity fields. The numerical framework pairs a Sobolev Hilbert gradient descent algorithm with a semi-Lagrangian particle flow scheme to integrate velocities without numerical dissipation. The performance of this framework was evaluated across synthetic tests, two-dimensional biological datasets (including canine hearts, macaque cortexes, and hippocampi from Alzheimer's and schizophrenia patients), three-dimensional brain volumes, and segmented electron micrographs of mitochondria.

The findings show that LDDMM successfully yields smooth, folding-free, and invertible transformations across extensive structural deformations while reducing image mismatch errors to approximately 2% to 7% in two-dimensional benchmarks. Furthermore, the length of the estimated geodesic path forms a valid metric distance that accurately mirrors intuitive morphological differences across varying shapes. Comparisons against the established Christensen viscous-fluid registration algorithm revealed that while both achieve comparable matching accuracy, LDDMM yields shorter paths on the deformation manifold with velocity fields that remain smooth across both space and time. Computationally, the Hilbert gradient proved stable by suppressing high-frequency noise that typically destabilizes standard L2 gradients, with 2D runtimes taking a few minutes on single processors and 3D volumes requiring up to a few hours on parallel architectures.

These results establish that rigorous metric mapping can transform qualitative visual comparisons into reliable quantitative measurements of anatomical shape. Establishing statistical distributions of anatomical distances relative to a common atlas offers significant potential to enhance clinical diagnostic baselines and monitor disease progression in conditions such as Alzheimer's disease and schizophrenia. The primary trade-off involves computational cost; the global space-time optimization requires substantially more processing time and memory than greedy, locally optimal approaches.

Moving forward, organizations and researchers should apply this metric framework to map large population cohorts to standardize morphological baselines for clinical diagnosis. The algorithm is also directly suitable for extension to multi-channel modalities, including color and diffusion tensor imaging. Future work should refine operator smoothing parameters using empirical population statistics, implement accelerated solver architectures, and establish confidence boundaries on broader, unaligned clinical datasets.

ENS Paris-Saclay Johns Hopkins University Simon Fraser University elastic deformation large deformation metric mappings geodesic flows of diffeomorphisms diffeomorphic transformations

BEG et al (2005).pdf

No sufficiently relevant recommendations were found.

Paper: VoxelMorph: A Learning Framework for Deformable Medical Image Registration, Guha Balakrishnan et al. (2018). After seeing LDDMM’s accurate but costly pairwise optimization, VoxelMorph shows how learning can amortize registration into a fast model for new image pairs.

Cover for Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

Abstract

This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I0, I1 are given and connected via the diffeomorphic change of coordinates I0 ∘ φ−1 = I1 where φ = ϕ1 is the end point at t = 1 of curve ϕt, t ∈ [0, 1] satisfying ϕ̇t = vt(ϕt), t ∈ [0, 1] with ϕ0 = id. The variational problem takes the form

argmin v:ϕ̇t=vt(ϕt) ( ∫0^1 ‖vt‖_V^2 dt + ‖I0 ∘ ϕ_1^−1 − I1‖_L^2^2 ),

where ‖vt‖_V is an appropriate Sobolev norm on the velocity field vt(·), and the second term enforces matching of the images with ‖·‖_L^2 representing the squared-error norm.

In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt, t ∈ [0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-Lagrangian method of computing particle flows and show the solutions for various examples. We also compute the metric distance on several anatomical configurations as measured by ∫0^1 ‖vt‖_V dt on the geodesic shortest paths.

Topics

elastic deformation

Keywords

large deformation metric mappings geodesic flows of diffeomorphisms diffeomorphic transformations computational anatomy smooth velocity vector fields elastic matching deformable template Eulerian frame Lagrangian schemes Christensen algorithm

1. Introduction
2. Euler Equations for the Variational Minimization on Vector Fields
3. Numerical Implementation of LDDMM Algorithm
3.1. Gradient Descent Scheme Based Optimization
3.2. Choice of Operator L and Evaluation of L † L, ( L † L ) - 1
3.3. Integration of Velocity Field to Generate Maps
3.4. Constant Speed Reparametrization of Velocity Vector Field
3.5. The Large Deformation Metric-Matching Algorithm
3.6. ∫ 1 0 ‖ v t ‖ V d t Defines a Metric on I
4. Numerical Results
5. The 'Christensen Algorithm' for Computing Large Deformation Diffeomorphisms
5.1. Comparison of the LDDMM Algorithm with the GEC Algorithm
6. Discussion
6.1. Hilbert Gradient Versus L 2 Gradient
6.2. The Metric Structure on I
Acknowledgments
References

Knowls

Knowl 1 — Large Deformation Diffeomorphic Metric Mapping Variational Formulation
model/method
Large Deformation Diffeomorphic Metric Mapping (LDDMM) solves the inexact dense matching problem between a template image $I_0: \Omega \to \mathbb{R}^d$ and a target image $I_1: \Omega \to \mathbb{R}^d$ on a spatial domain $\Omega \subset \mathbb{R}^n$ ( $n \in \{2, 3\}$ ) by optimizing over a time-dependent velocity vector field $v \in L^2([0, 1], V)$ .

The transformation of coordinates is governed by the ordinary differential equation flow:

$\frac{d}{dt}\phi_t^v(x) = v_t(\phi_t^v(x)), \quad \phi_0^v(x) = x$

where $V$ is a Hilbert space of smooth, compactly supported vector fields on $\Omega$ equipped with inner product $\langle f, g \rangle_V = \langle L f, L g \rangle_{L^2} = \langle L^\dagger L f, g \rangle_{L^2}$ for an admissible linear differential operator $L$ . The inverse flow mapping from time $s$ to time $t$ is denoted $\phi_{s,t}^v = \phi_t^v \circ (\phi_s^v)^{-1}$ , such that $\phi_{1,0}^v = (\phi_1^v)^{-1}$ .

The optimal velocity field $\hat{v}$ minimizes the global space-time energy functional:

$\hat{v} = \arg\min_{v \in L^2([0, 1], V)} E(v) = \int_0^1 \|v_t\|_V^2 \, dt + \frac{1}{\sigma^2} \|I_0 \circ \phi_{1,0}^v - I_1\|_{L^2}^2$

where $\sigma^2 > 0$ balances the kinetic regularization energy against the squared $L^2$ image mismatch.
Knowl 2 — Euler-Lagrange Hilbert Gradient for Dense LDDMM Matching
theoretical result
For the LDDMM inexact image matching energy functional $E(v) = \int_0^1 \|v_t\|_V^2 \, dt + \frac{1}{\sigma^2} \|I_0 \circ \phi_{1,0}^v - I_1\|_{L^2}^2$ defined on $v \in L^2([0, 1], V)$ , where $V$ has inner product $\langle f, g \rangle_V = \langle L^\dagger L f, g \rangle_{L^2}$ and Green's operator $K = (L^\dagger L)^{-1}$ , the Fréchet gradient $(\nabla_v E_t)_V \in V$ with respect to the Hilbert inner product on $V$ at time $t \in [0, 1]$ is:

$(\nabla_v E_t)_V = 2 v_t - \frac{2}{\sigma^2} K \left( |D\phi_{t,1}^v| \nabla J_t^0 (J_t^0 - J_t^1) \right)$

where:
- $J_t^0 = I_0 \circ \phi_{t,0}^v$ is the template image deformed forward to time $t$ ;
- $J_t^1 = I_1 \circ \phi_{t,1}^v$ is the target image deformed backward to time $t$ ;
- $\nabla J_t^0$ is the spatial gradient of $J_t^0$ ;
- $|D\phi_{t,1}^v| = \det(D\phi_{t,1}^v)$ is the determinant of the spatial Jacobian matrix of $\phi_{t,1}^v = \phi_1^v \circ (\phi_t^v)^{-1}$ .
The optimizing velocity field $\hat{v}$ satisfies the Euler-Lagrange equation $(\nabla_v E_t)_V = 0$ for all $t \in [0, 1]$ .
Knowl 3 — Variation of Flow Mappings under Velocity Perturbations
theoretical result
Let $\phi_{s,t}^v = \phi_t^v \circ (\phi_s^v)^{-1}$ denote the position at time $t$ of a particle located at position $y$ at time $s$ under the flow generated by $\frac{d}{dt}\phi_t^v = v_t \circ \phi_t^v$ for a velocity field $v \in L^2([0, 1], V)$ on a domain $\Omega \subset \mathbb{R}^n$ .

When the velocity field $v$ is perturbed along a direction $h \in L^2([0, 1], V)$ , the Gateaux variation of the mapping $\phi_{s,t}^v$ is given by:

$\partial_h \phi_{s,t}^v = \lim_{\epsilon \to 0} \frac{\phi_{s,t}^{v + \epsilon h} - \phi_{s,t}^v}{\epsilon} = D\phi_{s,t}^v \int_s^t (D\phi_{s,u}^v)^{-1} (h_u \circ \phi_{s,u}^v) \, du$

where $D\phi_{s,t}^v$ is the spatial Jacobian matrix of $\phi_{s,t}^v$ , and $(D\phi_{s,u}^v)^{-1}$ is its matrix inverse.
Knowl 4 — Large Deformation Diffeomorphic Metric Mapping Algorithm
algorithm
The LDDMM algorithm optimizes the time-dependent velocity vector field $v \in L^2([0, T], V)$ to match a template image $I_0$ to a target image $I_1$ via gradient descent in the Hilbert space $V$ .
Input: Template image $I_0$ , target image $I_1$ , number of timesteps $N$ , total time $T$ , step size $\epsilon$ , smoothing operator $K = (L^\dagger L)^{-1}$ , parameter $\sigma^2$
Output: Optimal velocity field $\hat{v}$ , matching diffeomorphism $\phi_{T, 0}^{\hat{v}}$ , geodesic metric distance $\rho$
$\delta t \leftarrow T / N$
Initialize $v_{t_j}^{(0)} \leftarrow 0$ for all $j \in \{0, \dots, N-1\}$
$k \leftarrow 0$
repeat
    for $j = 0$ to $N-1$ do
         $v_{t_j}^{(k+1)} \leftarrow v_{t_j}^{(k)} - \epsilon (\nabla_v E_{t_j})_V^{(k)}$
    end for
    if $k \pmod{10} == 0$ then
         $v^{(k+1)} \leftarrow \text{ConstantSpeedReparameterize}(v^{(k+1)})$
    end if
    Compute backward maps $\phi_{t_j, T}^{(k+1)}$ for $j = N-1$ down to 0 using semi-Lagrangian backward tracking
    Compute forward maps $\phi_{t_j, 0}^{(k+1)}$ for $j = 0$ to $N-1$ using semi-Lagrangian forward tracking
    for $j = 0$ to $N-1$ do
         $J_{t_j}^0 \leftarrow I_0 \circ \phi_{t_j, 0}^{(k+1)}$
         $J_{t_j}^1 \leftarrow I_1 \circ \phi_{t_j, T}^{(k+1)}$
        Compute spatial gradient $\nabla J_{t_j}^0$ and Jacobian determinant $|D\phi_{t_j, T}^{(k+1)}|$
         $(\nabla_v E_{t_j})_V^{(k+1)} \leftarrow 2 v_{t_j}^{(k+1)} - \frac{2}{\sigma^2} K\left(|D\phi_{t_j, T}^{(k+1)}| \nabla J_{t_j}^0 (J_{t_j}^0 - J_{t_j}^1)\right)$
    end for
     $E^{(k+1)} \leftarrow \sum_{j=0}^{N-1} \|v_{t_j}^{(k+1)}\|_V^2 \delta t + \frac{1}{\sigma^2 |\Omega|} \sum_{y \in \Omega} |J_T^0(y) - I_1(y)|^2$
     $k \leftarrow k + 1$
until $\|\nabla_v E\|_V < \text{threshold}$ or $k \ge K_{\max}$
$\hat{v} \leftarrow v^{(k)}$
$\rho \leftarrow \sum_{j=0}^{N-1} \|\hat{v}_{t_j}\|_V \delta t$
return $\hat{v}$ , $\phi_{T, 0}^{\hat{v}}$ , $\rho$
Knowl 5 — Semi-Lagrangian Integration of Forward and Backward Flow Maps
algorithm
The inverse diffeomorphic mapping $\phi_{t,0}^v = (\phi_t^v)^{-1}$ satisfies the Eulerian transport equation $\frac{\partial \phi_{t,0}^v}{\partial t}(y) + D\phi_{t,0}^v(y) v_t(y) = 0$ . To integrate the forward and backward maps stably without grid distortion or Courant-Friedrichs-Lewy (CFL) time step restrictions, a two-step semi-Lagrangian scheme tracks streamlines backward to land exactly on Cartesian grid points at the next timestep.
Input: Velocity field sequence $v_{t_j}$ on regular grid for $j \in \{0, \dots, N-1\}$ , timestep $\delta t$
Output: Forward maps $\phi_{t_j, 0}^v$ and backward maps $\phi_{t_j, T}^v$ for all $j \in \{0, \dots, N\}$
$\phi_{0,0}^v(y) \leftarrow y$ for all $y \in \Omega$
$\phi_{T,T}^v(y) \leftarrow y$ for all $y \in \Omega$
for $j = 1$ to $N$ do
    for each grid point $y \in \Omega$ do
         $\alpha \leftarrow \delta t \cdot v_{t_j}(y)$
        repeat
             $\alpha \leftarrow \delta t \cdot v_{t_j}\left(y - \frac{\alpha}{2}\right)$
        until convergence of $\alpha$
         $\phi_{t_j, 0}^v(y) \leftarrow \phi_{t_{j-1}, 0}^v(y - \alpha)$ evaluated via multilinear interpolation
    end for
end for
for $j = N-1$ down to 0 do
    for each grid point $y \in \Omega$ do
         $\alpha \leftarrow \delta t \cdot v_{t_j}(y)$
        repeat
             $\alpha \leftarrow \delta t \cdot v_{t_j}\left(y - \frac{\alpha}{2}\right)$
        until convergence of $\alpha$
         $\phi_{t_j, T}^v(y) \leftarrow \phi_{t_{j+1}, T}^v(y + \alpha)$ evaluated via multilinear interpolation
    end for
end for
return $\{\phi_{t_j, 0}^v\}_{j=0}^N$ , $\{\phi_{t_j, T}^v\}_{j=0}^N$
Knowl 6 — Constant-Speed Reparameterization of Velocity Fields
algorithm
Because geodesic paths on the diffeomorphism group have constant velocity norm $\|v_t\|_V = \text{constant}$ , enforcing constant speed along the trajectory monotonically decreases the kinetic energy $\int_0^T \|v_t\|_V^2 \, dt$ while preserving the endpoint transformation $\phi_T^v$ .
Input: Time-dependent velocity field $v_t$ for $t \in [0, T]$ , operator norm $\|\cdot\|_V$
Output: Reparameterized velocity field $\tilde{v}_t$ with constant norm $\|\tilde{v}_t\|_V = \text{Length}/T$
$\text{Length} \leftarrow \int_0^T \|v_t\|_V \, dt$
for $t \in [0, T]$ do
     $s(t) \leftarrow \frac{T}{\text{Length}} \int_0^t \|v_\tau\|_V \, d\tau$
end for
Compute inverse function $h: [0, T] \to [0, T]$ such that $s(h(t)) = t$
for $t \in [0, T]$ do
     $\dot{h}(t) \leftarrow \frac{\text{Length}}{T \|v_{h(t)}\|_V}$
     $\tilde{v}_t \leftarrow \dot{h}(t) v_{h(t)}$
end for
return $\tilde{v}_t$
By the Cauchy-Schwarz inequality, $\int_0^T \|v_t\|_V^2 \, dt \ge \frac{\text{Length}^2}{T} = \int_0^T \|\tilde{v}_t\|_V^2 \, dt$ , guaranteeing that reparameterization strictly decreases or preserves total energy.
Knowl 7 — Cauchy-Navier Regularization Operator and Fourier Domain Inversion
model/method
To ensure that generated vector fields yield smooth diffeomorphisms, the differential operator $L$ on domain $\Omega \subset \mathbb{R}^n$ is defined as the Cauchy-Navier operator:

$L = (-\alpha \nabla^2 + \gamma I) I_{n \times n}$

where $\alpha > 0$ controls regularity, $\gamma > 0$ prevents singularity, and $\nabla^2 = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$ is the Laplacian operator. Under periodic boundary conditions, $L$ is self-adjoint ( $L = L^\dagger$ ) and $L^\dagger L = L^2$ .

Applying the smoothing Green's operator $f = K g = (L^\dagger L)^{-1} g$ on a discrete grid with spacings $\Delta x_1, \dots, \Delta x_n$ is carried out in the discrete Fourier domain by:

$F(k) = \frac{G(k)}{A(k)^2}$

where $F(k)$ and $G(k)$ are the discrete Fourier transforms of $f$ and $g$ at spatial frequency $k = (k_1, \dots, k_n)$ , and the discrete symbol $A(k)$ is:

$A(k) = \gamma + 2\alpha \sum_{i=1}^n \frac{1 - \cos(2\pi \Delta x_i k_i)}{\Delta x_i^2}$

The smoothed velocity field $f$ is obtained via the inverse discrete Fourier transform of $F$ .
Knowl 8 — Metric Distance Structure on Diffeomorphisms and Image Orbits
theoretical result
Let $\mathcal{G} = \{ \phi_1^v \mid v \in L^1([0, 1], V) \}$ be the group of diffeomorphisms generated by flows in a Hilbert space $V$ . The Riemannian distance between any two diffeomorphisms $\phi_0, \phi_1 \in \mathcal{G}$ is:

$\rho_{\mathcal{G}}(\phi_0, \phi_1) = \inf_{v \in L^2([0, 1], V)} \left\{ \int_0^1 \|v_t\|_V \, dt \ \middle|\ \phi_1 = \phi_1^v \circ \phi_0 \right\}$

The metric $\rho_{\mathcal{G}}$ is complete on $\mathcal{G}$ and right-invariant: $\rho_{\mathcal{G}}(\phi_0 \circ \psi, \phi_1 \circ \psi) = \rho_{\mathcal{G}}(\phi_0, \phi_1)$ for all $\psi \in \mathcal{G}$ .

For the anatomical orbit $\mathcal{I} = \{ I_0 \circ \phi^{-1} \mid \phi \in \mathcal{G} \}$ , the induced distance function $\rho_{\mathcal{I}}: \mathcal{I} \times \mathcal{I} \to \mathbb{R}_+$ defined by:

$\rho_{\mathcal{I}}(I_0, I_1) = \inf_{\phi \in \mathcal{G}} \{ \rho_{\mathcal{G}}(\text{Id}, \phi) \mid I_1 = I_0 \circ \phi^{-1} \}$

is positive, symmetric, satisfies the triangle inequality, and satisfies $\rho_{\mathcal{I}}(I_0, I_1) = 0 \iff I_0 = I_1$ , making $\rho_{\mathcal{I}}$ a genuine metric on the image orbit $\mathcal{I}$ . For constant speed paths, $\rho_{\mathcal{G}}^2(\phi_0, \phi_1) = \inf_{v: \|v_t\|_V = \text{const}} \int_0^1 \|v_t\|_V^2 \, dt$ .
Knowl 9 — Numerical Stability of Sobolev (Hilbert) Gradient versus L2 Gradient
theoretical result
Let $L^\dagger L$ be the self-adjoint operator on $L^2(\Omega, \mathbb{R}^n)$ with eigenvalues $\lambda_i \to \infty$ as $i \to \infty$ on an orthonormal basis $\{w_i\}_{i \in \mathbb{N}}$ , and let $K = (L^\dagger L)^{-1}$ be its compact inverse. Let $b_t = -\frac{2}{\sigma^2} |D\phi_{t,1}^v| \nabla J_t^0 (J_t^0 - J_t^1)$ .

The gradient of the matching functional expressed in the standard $L^2$ inner product versus the Sobolev inner product on $V$ expand in the eigenbasis as:

$(\nabla_v E_t)_{L^2} = 2(L^\dagger L) v_t + b_t = \sum_{i \in \mathbb{N}} \left( \lambda_i \langle 2v_t, w_i \rangle_{L^2} + \langle b_t, w_i \rangle_{L^2} \right) w_i$

$(\nabla_v E_t)_V = 2 v_t + K b_t = \sum_{i \in \mathbb{N}} \left( \langle 2v_t, w_i \rangle_{L^2} + \frac{\langle b_t, w_i \rangle_{L^2}}{\lambda_i} \right) w_i$

The $L^2$ gradient multiplies high-frequency components by $\lambda_i$ , leading to severe high-frequency noise amplification and numerical instability. In contrast, the Hilbert gradient $(\nabla_v E_t)_V = K (\nabla_v E_t)_{L^2}$ divides high-frequency components of $b_t$ by $\lambda_i$ , smoothing high-frequency noise and ensuring stable gradient descent optimization.
Knowl 10 — Theoretical and Kinetic Comparison of LDDMM and Christensen Greedy Fluid Matching
theoretical result
The Christensen viscous-fluid deformable template method computes velocity fields $v_t$ step-by-step by solving $L^\dagger L v_t + b_t = 0$ with $b_t = -\frac{2}{\sigma^2} (J_t^0 - I_1) \nabla J_t^0$ , which is equivalent to instantaneous Riemannian steepest descent on the image mismatch cost $E_2(\phi_t) = \frac{1}{\sigma^2} \|I_0 \circ \phi_{t,0} - I_1\|_{L^2}^2$ on the diffeomorphism group $\mathcal{G}$ :

$v_t = -\nabla_{\mathcal{G}} E_2(\phi_t) \circ \phi_t^{-1} = -K(b_t)$

While computationally fast due to time-decoupled updates, this approach is greedy and only locally optimal in time. Consequently:
1. The integrated path length $\int_0^1 \|v_t\|_V \, dt$ from the Christensen method is an upper bound on the true geodesic distance $\rho_{\mathcal{G}}(\text{Id}, \phi_1)$ and does not yield a valid metric on image orbits.
2. The velocity field generated by the Christensen algorithm fluctuates from one timestep to the next, whereas LDDMM optimizes globally over $L^2([0, 1], V)$ to produce a velocity field that is smooth in both space and time and traces the shortest geodesic path.

Knowl 11 — 2D Anatomical Image Matching Performance and Metric Distances

data/table

Dense inexact matching experiments were conducted on five 2D image pairs with $N = 20$ timesteps (step size $\delta t = 0.1$ ) on a single processor using Cauchy-Navier operator $L = -\alpha \nabla^2 + \gamma I$ . All image pairs except the synthetic translation were rigidly pre-aligned.

Experiment (image size)	$L$	Image error (%)	Est. metric	Iter.	Exec. time
“Parallel Translation” ( $32 \times 32$ )	$-0.01\nabla^2 + 0.1I$	1.97	0.769	70	0.2 min
“Heart Mapping” ( $80 \times 80$ )	$-0.01\nabla^2 + I$	7.12	6.122	159	3.7 min
“Schiz.” ( $64 \times 64$ )	$-0.01\nabla^2 + I$	3.01	4.891	277	4.2 min
“Alzh.” ( $64 \times 64$ )	$-0.01\nabla^2 + I$	2.69	5.592	255	3.9 min
“Macaq.” ( $80 \times 80$ )	$-0.01\nabla^2 + I$	3.64	6.869	134	3.2 min

Image error measures residual $L^2$ intensity mismatch after registration as a percentage of the initial unregistered mismatch. The estimated metric is the geodesic length $\sum_{j=0}^{N-1} \|v_{t_j}\|_V \delta t$ on the diffeomorphism manifold. Execution times on a single processor ranged between 0.2 and 4.2 minutes depending on image resolution and iteration count.

Knowl 12 — Comparison of Manifold Path Length: LDDMM versus Greedy Christensen Algorithm

data/table

The globally optimal LDDMM algorithm and the greedy Christensen (GEC) fluid algorithm were compared across three matching experiments. Optimization was stopped when both algorithms reached comparable image matching errors.

	Final image error (%)		Distance on manifold		Time (s)
Experiment	LDDMM	GEC	LDDMM	GEC	LDDMM / GEC
“Parallel Translation”	5.648	5.65	0.7223	0.722757	12 / 1
“Heart Mapping”	11.558	11.799	4.631	6.7145	120 / 2.4
“Macaque Cortex Mapping”	9.55	9.53	4.636	5.45	98 / 1.2

At equivalent levels of residual image mismatch error, LDDMM consistently achieved shorter path lengths on the manifold of diffeomorphisms compared to GEC (e.g., 4.631 versus 6.7145 for Heart Mapping), demonstrating that GEC follows a sub-optimal, non-geodesic trajectory.

Coverage note — Qualitative metric distance evaluations on 2D mitochondria shapes and 3D hippocampal volumes in Alzheimer's and Schizophrenia populations were omitted as standalone knowls because their numerical setups and behavior are fully represented by the comprehensive 2D and comparative benchmark knowls.

References

1.Amit, Y. 1994. A nonlinear variational problem for image matching. SIAM Journal on Scientific Computing, 15(1):207–224.
2.Bajcsy, R. and Broit, C. 1982. Matching of deformed images. In Proc. 6th Int. Joint Conf. Patt. Recog., pp. 351–353.
3.Bajcsy, R., Lieberson, R., and Reivich, M. 1983. A computerized system for the elastic matching of deformed radiographic images to idealized atlas images. Journal of Computer Assisted Tomography, 7(4):618–625.
4.Broit, C. 1981. Optimal registration of deformed images. PhD thesis, University of Pennsylvania.
5.Do Carmo, M.P. 1976. Differential geometry of curves and surfaces. Prentice-Hall Engineering/Science/Mathematics.
6.Do Carmo, M.P. 1993. Riemannian Geometry. Birkhauser.
7.Christensen, G.E., Rabbitt, R.D., and Miller, M.I. 1996. Deformable templates using large deformation kinematics. IEEE Transactions on Image Processing, 5(10):1435–1447.
8.Christensen, G. 1994. Deformable shape models for anatomy. PhD Thesis, Dept. of Electrical Engineering, Sever Institute of Technology, Washington Univ., St. Louis, MO.
9.Dupuis, P., Grenander, U., and Miller, M.I. 1998. Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, LVI:587–600.
10.Grenander, U. and Miller, M.I. 1998. Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, 56:617–694.
11.Miller, M.I., Trouvé, A., and Younes, L. 2002. On the metrics and Euler-Lagrange equations of computational anatomy. Annual Review of Biomedical Engineering, 4:375–405.
12.Miller, M.I. and Younes, L. 2001. Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision, 41:61–84.
13.Morton, K.W. and Mayers, D.F. 1996. Numerical Solution of Partial Differential Equations. Cambridge University Press, University of Cambridge.
14.Robb, R.A. 1999. Biomedical Imaging, Vizualization and Analysis. John Wiley and Sons, Inc., New York, NY.
15.Staniforth, A. and Côté, J. 1991. Semi-lagrangian integration schemes for atmospheric models-a review. Monthly Weather Review, 119:2206–2223.
16.Thirion, J.-P. 1998. Image matching as a diffusion process: An analogy with maxwell’s demons. Medical Image Analysis, 2(3):243–260.
17.Trouvé, A. 1995. An infinite dimensional group approach for physics based models in patterns recognition. Preprint.
18.Trouvé, A. 1998. Diffeomorphic groups and pattern matching in image analysis. Int. J. Computer Vision, 28:213–221.
19.Younes, L. 1999. Optimal matching between shapes via elastic deformations. Image and Vision Computing, 17:381–389.

Citation

MLA

Beg, M. F., et al. “Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms”. International Journal of Computer Vision, vol. 61, no. 2, 2005, pp. 139–57, https://doi.org/10.1023/B:VISI.0000043755.93987.aa.

APA

Beg, M. F., Miller, M. I., Trouvé, A., & Younes, L. (2005). Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. International Journal of Computer Vision, 61(2), 139–157. https://doi.org/10.1023/B:VISI.0000043755.93987.aa

Chicago

Beg, M. F., M. I. Miller, A. Trouvé, and L. Younes. 2005. “Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms”. International Journal of Computer Vision 61 (2): 139–57. https://doi.org/10.1023/B:VISI.0000043755.93987.aa.

Harvard

Beg, M.F. et al. (2005) “Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms”, International Journal of Computer Vision, 61(2), pp. 139–157. Available at: https://doi.org/10.1023/B:VISI.0000043755.93987.aa.

Vancouver

1. Beg MF, Miller MI, Trouvé A, Younes L (2005) Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. International Journal of Computer Vision 61:139–157

BibTeX

@article{Beg_2005, title={Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms}, volume={61}, ISSN={1573-1405}, url={http://dx.doi.org/10.1023/B:VISI.0000043755.93987.aa}, DOI={10.1023/b:visi.0000043755.93987.aa}, number={2}, journal={International Journal of Computer Vision}, publisher={Springer Science and Business Media LLC}, author={Beg, M. Faisal and Miller, Michael I. and Trouvé, Alain and Younes, Laurent}, year={2005}, month=Feb, pages={139–157} }

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Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

Abstract

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Table of Contents

Knowls

Knowl 1 — Large Deformation Diffeomorphic Metric Mapping Variational Formulation

Knowl 2 — Euler-Lagrange Hilbert Gradient for Dense LDDMM Matching

Knowl 3 — Variation of Flow Mappings under Velocity Perturbations

Knowl 4 — Large Deformation Diffeomorphic Metric Mapping Algorithm

Knowl 5 — Semi-Lagrangian Integration of Forward and Backward Flow Maps

Knowl 6 — Constant-Speed Reparameterization of Velocity Fields

Knowl 7 — Cauchy-Navier Regularization Operator and Fourier Domain Inversion

Knowl 8 — Metric Distance Structure on Diffeomorphisms and Image Orbits

Knowl 9 — Numerical Stability of Sobolev (Hilbert) Gradient versus L2 Gradient