Large deformation diffeomorphic metric mapping

Diffeomorphic mapping 3-dimensional information across coordinate systems is central to high-resolution Medical imaging and the area of Neuroinformatics within the newly emerging field of bioinformatics. Diffeomorphic mapping 3-dimensional coordinate systems as measured via high resolution dense imagery has a long history in 3-D beginning with Computed Axial Tomography (CAT scanning) in the early 80's by the University of Pennsylvania group led by Ruzena Bajcsy,[24] and subsequently the Ulf Grenander school at Brown University with the HAND experiments.[25][26] In the 90's there were several solutions for image registration which were associated to linearizations of small deformation and non-linear elasticity.[27][28][29][30][31]

The central focus of the sub-field of Computational anatomy (CA) within medical imaging is mapping information across anatomical coordinate systems at the 1 millimeter morphome scale. In CA mapping of dense information measured within Magnetic resonance image (MRI) based coordinate systems such as in the brain has been solved via inexact matching of 3D MR images one onto the other. The earliest introduction of the use of diffeomorphic mapping via large deformation flows of diffeomorphisms for transformation of coordinate systems in image analysis and medical imaging was by Christensen, Rabbitt and Miller [17][32] and Trouve.[33] The introduction of flows, which are akin to the equations of motion used in fluid dynamics, exploit the notion that dense coordinates in image analysis follow the Lagrangian and Eulerian equations of motion. This model becomes more appropriate for cross-sectional studies in which brains and or hearts are not necessarily deformations of one to the other. Methods based on linear or non-linear elasticity energetics which grows with distance from the identity mapping of the template, is not appropriate for cross-sectional study. Rather, in models based on Lagrangian and Eulerian flows of diffeomorphisms, the constraint is associated to topological properties, such as open sets being preserved, coordinates not crossing implying uniqueness and existence of the inverse mapping, and connected sets remaining connected. The use of diffeomorphic methods grew quickly to dominate the field of mapping methods post Christensen's original paper, with fast and symmetric methods becoming available.[19][34]

Such methods are powerful in that they introduce notions of regularity of the solutions so that they can be differentiated and local inverses can be calculated. The disadvantages of these methods is that there was no associated global least-action property which could score the flows of minimum energy. This contrasts the geodesic motions which are central to the study of Rigid body kinematics and the many problems solved in Physics via Hamilton's principle of least action. In 1998, Dupuis, Grenander and Miller[35] established the conditions for guaranteeing the existence of solutions for dense image matching in the space of flows of diffeomorphisms. These conditions require an action penalizing kinetic energy measured via the Sobolev norm on spatial derivatives of the flow of vector fields.

The large deformation diffeomorphic metric mapping (LDDMM) code that Faisal Beg derived and implemented for his PhD at Johns Hopkins University[36] developed the earliest algorithmic code which solved for flows with fixed points satisfying the necessary conditions for the dense image matching problem subject to least-action. Computational anatomy now has many existing codes organized around diffeomorphic registration[17] including ANTS,[18] DARTEL,[19] DEMONS,[37] LDDMM,[2] StationaryLDDMM[21] as examples of actively used computational codes for constructing correspondences between coordinate systems based on dense images.

These large deformation methods have been extended to landmarks without registration via measure matching,[38] curves,[39] surfaces,[40] dense vector[41] and tensor [42] imagery, and varifolds removing orientation.[43]

Deformable shape in Computational Anatomy (CA)[44][45][46][47]is studied via the use of diffeomorphic mapping for establishing correspondences between anatomical coordinates in Medical Imaging. In this setting, three dimensional medical images are modelled as a random deformation of some exemplar, termed the template ${\displaystyle I_{temp}}$, with the set of observed images element in the random orbit model of CA for images ${\displaystyle I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in Diff_{V}\}}$. The template is mapped onto the target by defining a variational problem in which the template is transformed via the diffeomorphism used as a change of coordinate to minimize a squared-error matching condition between the transformed template and the target.

The diffeomorphisms are generated via smooth flows ${\displaystyle \phi _{t},t\in [0,1]}$, with ${\displaystyle \varphi \doteq \phi _{1}}$, satisfying the Lagrangian and Eulerian specification of the flow field associated to the ordinary differential equation,

${\displaystyle {\frac {d}{dt}}\phi _{t}=v_{t}\circ \phi _{t},\ \phi _{0}=id}$,

with ${\displaystyle v_{t},t\in [0,1]}$ the Eulerian vector fields determining the flow. The vector fields are guaranteed to be 1-time continuously differentiable ${\displaystyle v_{t}\in C^{1}}$ by modelling them to be in a smooth Hilbert space ${\displaystyle v\in V}$ supporting 1-continuous derivative.[48] The inverse ${\displaystyle \phi _{t}^{-1},t\in [0,1]}$ is defined by the Eulerian vector-field with flow given by

${\displaystyle {\frac {d}{dt}}\phi _{t}^{-1}=-(D\phi _{t}^{-1})v_{t},\ \phi _{0}^{-1}=id\ .}$

(Inverse Transport flow)

To ensure smooth flows of diffeomorphisms with inverse, the vector fields with components in ${\displaystyle {\mathbb {R} }^{3}}$ must be at least 1-time continuously differentiable in space[49][50] which are modelled as elements of the Hilbert space ${\displaystyle (V,\|\cdot \|_{V})}$ using the Sobolev embedding theorems so that each element ${\displaystyle v_{i}\in H_{0}^{3},i=1,2,3,}$ has 3-times square-integrable weak-derivatives. Thus ${\displaystyle (V,\|\cdot \|_{V})}$ embeds smoothly in 1-time continuously differentiable functions.[37][50] The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm

${\displaystyle Diff_{V}\doteq \{\varphi =\phi _{1}:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}=id,\int _{0}^{1}\|v_{t}\|_{V}dt<\infty \}\ .}$

(Diffeomorphism Group)

LDDMM matching based on the principal eigenvector of the diffusion tensor matrix takes the image ${\displaystyle I(x),x\in {\mathbb {R} }^{3}}$ as a unit vector field defined by the first eigenvector. [41] The group action becomes

${\displaystyle \varphi \cdot I={\begin{cases}{\frac {D_{\varphi ^{-1}}\varphi I\circ \varphi ^{-1}\|I\circ \varphi ^{-1}\|}{\|D_{\varphi ^{-1}}\varphi I\circ \varphi ^{-1}\|}}&I\circ \varphi \neq 0,\\0&{\text{otherwise.}}\end{cases}}}$

where ${\displaystyle \|\cdot \|}$ that denotes image squared-error norm.

LDDMM matching based on the entire tensor matrix [56] has group action ${\displaystyle \varphi \cdot M=(\lambda _{1}{\hat {e}}_{1}{\hat {e}}_{1}^{T}+\lambda _{2}{\hat {e}}_{2}{\hat {e}}_{2}^{T}+\lambda _{3}{\hat {e}}_{3}{\hat {e}}_{3}^{T})\circ \varphi ^{-1},}$ transformed eigenvectors

${\displaystyle {\begin{aligned}{\hat {e}}_{1}&={\frac {D\varphi e_{1}}{\|D\varphi e_{1}\|}}\ ,\ \ \ {\hat {e}}_{2}={\frac {D\varphi e_{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle {\hat {e}}_{1}}{\sqrt {\|D\varphi e_{2}\|^{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle ^{2}}}}\ ,\ \ \ {\hat {e}}_{3}={\hat {e}}_{1}\times {\hat {e}}_{2}\end{aligned}}}$.

High angular resolution diffusion imaging (HARDI) addresses the well-known limitation of DTI, that is, DTI can only reveal one dominant fiber orientation at each location. HARDI measures diffusion along ${\displaystyle n}$ uniformly distributed directions on the sphere and can characterize more complex fiber geometries by reconstructing an orientation distribution function (ODF) that characterizes the angular profile of the diffusion probability density function of water molecules. The ODF is a function defined on a unit sphere, ${\displaystyle {\mathbb {S} }^{2}}$.[57] Denote the square-root ODF (${\displaystyle {\sqrt {\text{ODF}}}}$) as ${\displaystyle \psi ({\bf {s}})}$, where ${\displaystyle \psi ({\bf {s}})}$ is non-negative to ensure uniqueness and ${\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi ^{2}({\bf {s}})d{\bf {s}}=1}$. The metric defines the distance between two ${\displaystyle {\sqrt {\text{ODF}}}}$ functions ${\displaystyle \psi _{1},\psi _{2}\in \Psi }$ as

${\displaystyle {\begin{aligned}\rho (\psi _{1},\psi _{2})=\|\log _{\psi _{1}}(\psi _{2})\|_{\psi _{1}}=\cos ^{-1}\langle \psi _{1},\psi _{2}\rangle =\cos ^{-1}\left(\int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi _{1}({\bf {s}})\psi _{2}({\bf {s}})d{\bf {s}}\right),\end{aligned}}}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the normal dot product between points in the sphere under the ${\displaystyle \mathrm {L} ^{2}}$ metric. The template and target are denoted ${\displaystyle \psi _{\mathrm {temp} }({\bf {s}},x)}$, ${\displaystyle \psi _{\mathrm {targ} }({\bf {s}},x)}$,${\displaystyle {\bf {s}}\in {{\mathbb {S} }^{2}}}$${\displaystyle x\in X}$ indexed across the unit sphere and the image domain, with the target indexed similarly.

Define the variational problem assuming that two ODF volumes can be generated from one to another via flows of diffeomorphisms ${\displaystyle \phi _{t}}$, which are solutions of ordinary differential equations ${\displaystyle {\dot {\phi }}_{t}=v_{t}(\phi _{t}),t\in [0,1],\phi _{0}={id}}$. The group action of the diffeomorphism on the template is given according to ${\displaystyle \phi _{1}\cdot \psi (x)\doteq (D\phi _{1})\psi \circ \phi _{1}^{-1}(x),x\in X}$, where ${\displaystyle (D\phi _{1})}$ is the Jacobian of the affined transformed ODF and is defined as

${\displaystyle {\begin{aligned}(D\phi _{1})\psi \circ \phi _{1}^{-1}(x)={\sqrt {\frac {\det {{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}}{\left\|{{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}{\bf {s}}\right\|^{3}}}}\quad \psi \left({\frac {(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}}{\|(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}\|}},\phi _{1}^{-1}(x)\right).\end{aligned}}}$

The LDDMM variational problem is defined as

${\displaystyle {\begin{aligned}\min _{v:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}={id}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dx\ dt+\lambda \int _{R^{3}}\|\log _{(D\phi _{1})\psi _{\mathrm {temp} }\circ \phi _{1}^{-1}(x)}(\psi _{\mathrm {targ} }(x))\|_{(D\phi _{1})\psi _{\mathrm {temp} }\circ \phi _{1}^{-1}(x)}^{2}dx\end{aligned}}}$.

Beg solved the early LDDMM algorithms by solving the variational matching taking variations with respect to the vector fields.[58] Another solution by Vialard,[59] reparameterizes the optimization problem in terms of the state ${\displaystyle q_{t}\doteq I\circ \phi _{t}^{-1},q_{0}=I}$, for image ${\displaystyle I(x),x\in X=R^{3}}$, with the dynamics equation controlling the state by the control given in terms of the advection equation according to ${\displaystyle {\dot {q}}_{t}=-\nabla q_{t}\cdot v_{t}}$. The endpoint matching term ${\displaystyle E(q_{1})\doteq {\frac {1}{2}}\|q_{1}-J\|^{2}}$ gives the variational problem:

${\displaystyle {\begin{matrix}&\ \ \ \ \ \min _{v:{\dot {q}}=v\circ q}C(v)\doteq {\frac {1}{2}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dxdt+{\frac {1}{2}}\int _{{\mathbb {R} }^{3}}|q_{1}(x)-J(x)|^{2}dx\end{matrix}}}$

(Advective-State-Image-Matching)

${\displaystyle {\begin{cases}{\text{Hamiltonian Dynamics}}&\ \ \ \ \ \ \ \ \ \ {\dot {q}}_{t}=-\nabla q_{t}\cdot v_{t}\\&\ \ \ \ \ \ \ \ \ \ {\dot {p}}_{t}=-{\text{div}}(p_{t}v_{t}),\ \ \ \ t\in [0,1]\\&\ \ \ \ \ \ \ \ \ \ Av_{t}=\mu _{t}=-p_{t}\nabla q_{t}\\{\text{Endpoint Condition}}&\ \ \ \ \ \ \ \ \ p_{1}=-{\frac {\partial E}{\partial q_{1}}}(q_{1})=-(q_{1}-J)=-(I\circ \phi _{1}^{-1}-J)\\&\ \ \ \ \ \ \ \ \ \ Av_{1}=\mu _{1}=(I\circ \phi _{1}^{-1}-J)\nabla (I\circ \phi _{1}^{-1})\ \ t=1\ .\\{\text{Conserved Dynamics}}&\ \ \ \ \ \ \ \ \ \ p_{t}=-(I\circ \phi _{t}^{-1}-J\circ \phi _{t1})|D\phi _{t1}|\ ,\ \ t\in [0,1]\ .\\\end{cases}}}$

(Hamiltonian Matching Condition)

Software suites containing a variety of diffeomorphic mapping algorithms include the following:

• Deformetrica[60]
• ANTS[18]
• DARTEL[61] Voxel-based morphometry(VBM)
• DEMONS[62]
• LDDMM[2]
• StationaryLDDMM[21]

• Computational anatomy § Dense image matching in computational anatomy
• Riemannian metric and Lie-bracket in computational anatomy
• Bayesian model of computational anatomy

• Ceritoglu, Can; Wang, Lei; Selemon, Lynn D.; Csernansky, John G.; Miller, Michael I.; Ratnanather, J. Tilak (2010-05-28). "Large Deformation Diffeomorphic Metric Mapping Registration of Reconstructed 3D Histological Section Images and in vivo MR Images". Frontiers in Human Neuroscience. 4: 43. doi:10.3389/fnhum.2010.00043. ISSN 1662-5161. PMC 2889720. PMID 20577633.