Large deformation diffeomorphic metric mapping

The diffeomorphism orbit model in computational anatomy

Deformable shape in computational anatomy (CA)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 ${\displaystyle I\in \mathcal{I}\doteq \{I=I_{\text{temp}}\circ \phi ,\phi \in {\mathrm{Diff}}_V\}}$. 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 ${\displaystyle \phi \doteq {\phi }_1}$, satisfying the Lagrangian and Eulerian specification of the flow field associated to the ordinary differential equation,

${\displaystyle \frac{d}{dt}{\phi }_t=v_t\circ {\phi }_t,{\phi }_0=\mathrm{i}\mathrm{d},}$

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. The inverse ${\displaystyle {\phi }_t^{-1},t\in [0,1]}$ is defined by the Eulerian vector-field with flow given by

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 which are modelled as elements of the Hilbert space ${\displaystyle (V,\Vert \cdot {\Vert }_V)}$ 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 ${\displaystyle (V,\Vert \cdot {\Vert }_V)}$ embeds smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm

The variational problem of dense image matching and sparse landmark matching

LDDMM algorithm for dense image matching

In CA the space of vector fields ${\displaystyle (V,\Vert \cdot {\Vert }_V)}$ are modelled as a reproducing Kernel Hilbert space (RKHS) defined by a 1-1, differential operator${\displaystyle A:V\to V^{\ast }}$ determining the norm ${\displaystyle \Vert v{\Vert }_V^2\doteq {\int }_{R^3}Av\cdot vdx,v\in V,}$ where the integral is calculated by integration by parts when ${\displaystyle Av}$ is a generalized function in the dual space ${\displaystyle V^{\ast }}$. The differential operator is selected so that the Green's kernel, the inverse of the operator, is continuously differentiable in each variable implying that the vector fields support 1-continuous derivative; see for the necessary conditions on the norm for existence of solutions.

The original large deformation diffeomorphic metric mapping (LDDMM) algorithms of Beg, Miller, Trouve, Younes was derived taking variations with respect to the vector field parameterization of the group, since ${\displaystyle v=\dot{\varphi }\circ {\varphi }^{-1}}$ are in a vector spaces. Beg solved the dense image matching minimizing the action integral of kinetic energy of diffeomorphic flow while minimizing endpoint matching term according to

• Beg's Iterative Algorithm for Dense Image Matching

Update until convergence, ${\displaystyle {\varphi }_t^{old}\leftarrow {\varphi }_t^{new}}$ each iteration, with ${\displaystyle {\varphi }_{t1}\doteq {\varphi }_1\circ {\varphi }_t^{-1}}$:

This implies that the fixed point at ${\displaystyle t=0}$ satisfies

${\displaystyle {\mu }_0^{\ast }=A{v}_0^{\ast }=(I-J\circ {\varphi }_1^{\ast })\mathrm{\nabla }I|D{\varphi }_1^{\ast }|}$,

which in turn implies it satisfies the Conservation equation given by the Endpoint Matching Condition according to

${\displaystyle A{v}_t^{\ast }=(D{\varphi }_t^{\ast -1}{)}^{T}A{v}_0^{\ast }\circ {\varphi }_t^{\ast -1}|D{\varphi }_t^{\ast -1}|}$

LDDMM registered landmark matching

The landmark matching problem has a pointwise correspondence defining the endpoint condition with geodesics given by the following minimum:

${\displaystyle \underset{v:{\dot{\varphi }}_t=v_t\circ {\varphi }_t}{min}C(v)\doteq \frac{1}{2}{\int }_0^1{\int }_{R^3}A{v}_t\cdot v_{t}dxdt+\frac{1}{2}\sum _i({\varphi }_1(x_i)-y_i)\cdot ({\varphi }_1(x_i)-y_i)}$;

• Iterative Algorithm for Landmark Matching

Joshi originally defined the registered landmark matching probleme,. Update until convergence, ${\displaystyle {\varphi }_t^{old}\leftarrow {\varphi }_t^{new}}$ each iteration, with ${\displaystyle {\varphi }_{t1}\doteq {\varphi }_1\circ {\varphi }_t^{-1}}$:

This implies that the fixed point satisfy

${\displaystyle A{v}_0=-\sum _i(D{\varphi }_1)(x_i{)}^T(y_i-{\varphi }_1(x_i)){\delta }_{x_i}}$

with

${\displaystyle A{v}_t=-\sum _i(D{\varphi }_{t1}{)}^T{|}_{{\varphi }_t(x_i)}(y_i-{\varphi }_1(x_i)){\delta }_{{\varphi }_t(x_i)}}$.

Variations for LDDMM dense image and landmark matching

The Calculus of variations was used in Beg^[49] to derive the iterative algorithm as a solution which when it converges satisfies the necessary maximizer conditions given by the necessary conditions for a first order variation requiring the variation of the endpoint with respect to a first order variation of the vector field. The directional derivative calculates the Gateaux derivative as calculated in Beg's original paper^[49] and.

LDDMM Diffusion Tensor Image Matching

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. The group action becomes

${\displaystyle \phi \cdot I=\{\begin{array}{ll}\frac{D_{{\phi }^{-1}}\phi I\circ {\phi }^{-1}\Vert I\circ {\phi }^{-1}\Vert }{\Vert D_{{\phi }^{-1}}\phi I\circ {\phi }^{-1}\Vert }& I\circ \phi \ne 0,\\ 0& \text{otherwise.}\end{array}}$

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

LDDMM matching based on the entire tensor matrix has group action ${\displaystyle \phi \cdot M=({\lambda }_1{\hat{e}}_1{\hat{e}}_1^T+{\lambda }_2{\hat{e}}_2{\hat{e}}_2^T+{\lambda }_3{\hat{e}}_3{\hat{e}}_3^T)\circ {\phi }^{-1},}$ transformed eigenvectors

${\displaystyle \begin{array}{cc}{\hat{e}}_1& =\frac{D\phi e_1}{\Vert D\phi e_1\Vert },{\hat{e}}_2=\frac{D\phi e_2-⟨{\hat{e}}_1,D\phi e_2⟩{\hat{e}}_1}{\sqrt{\Vert D\phi e_2{\Vert }^2-⟨{\hat{e}}_1,D\phi e_2{⟩}^2}},{\hat{e}}_3={\hat{e}}_1\times {\hat{e}}_2\end{array}}$.

Dense matching problem onto principle eigenvector of DTI

The variational problem matching onto vector image ${\displaystyle I^{\mathrm{\prime }}(x),x\in {\mathbb{R}}^3}$with endpoint

${\displaystyle E({\varphi }_1)\doteq \alpha {\int }_{{\mathbb{R}}^3}\Vert {\varphi }_1\cdot I-I^{\mathrm{\prime }}{\Vert }^{2}dx+\beta {\int }_{{\mathbb{R}}^3}(\Vert {\varphi }_1\cdot I\Vert -\Vert I^{\mathrm{\prime }}\Vert {)}^{2}dx).}$

becomes

${\displaystyle \underset{v:\dot{\varphi }\circ {\varphi }^{-1}}{min}\frac{1}{2}{\int }_0^1{\int }_{R^3}A{v}_t\cdot v_{t}dxdt+\alpha {\int }_{{\mathbb{R}}^3}\Vert {\varphi }_1\cdot I-I^{\mathrm{\prime }}{\Vert }^{2}dx+\beta {\int }_{{\mathbb{R}}^3}(\Vert {\varphi }_1\cdot I\Vert -\Vert I^{\mathrm{\prime }}\Vert {)}^{2}dx.}$

Dense matching problem onto DTI MATRIX

The variational problem matching onto: ${\displaystyle M^{\mathrm{\prime }}(x),x\in {\mathbb{R}}^3}$ with endpoint

${\displaystyle E({\varphi }_1)\doteq {\int }_{{\mathbb{R}}^3}\Vert {\varphi }_1\cdot M(x)-M^{\mathrm{\prime }}(x){\Vert }_F^{2}dx}$

with ${\displaystyle \Vert \cdot {\Vert }_F}$ Frobenius norm, giving variational problem

LDDMM ODF

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, ${\displaystyle {\mathbb{S}}^2}$. Denote the square-root ODF (${\displaystyle \sqrt{\text{ODF}}}$) as ${\displaystyle \psi (\mathbf{s})}$, where ${\displaystyle \psi (\mathbf{s})}$ is non-negative to ensure uniqueness and ${\displaystyle {\int }_{\mathbf{s}\in {\mathbb{S}}^2}{\psi }^2(\mathbf{s})d\mathbf{s}=1}$. The metric defines the distance between two ${\displaystyle \sqrt{\text{ODF}}}$ functions ${\displaystyle {\psi }_1,{\psi }_2\in \mathrm{\Psi }}$ as

${\displaystyle \begin{array}{c}\rho ({\psi }_1,{\psi }_2)=\Vert {\log }_{{\psi }_1}({\psi }_2){\Vert }_{{\psi }_1}={\cos }^{-1}⟨{\psi }_1,{\psi }_2⟩={\cos }^{-1}\left({\int }_{\mathbf{s}\in {\mathbb{S}}^2}{\psi }_1(\mathbf{s}){\psi }_2(\mathbf{s})d\mathbf{s}\right),\end{array}}$

where ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is the normal dot product between points in the sphere under the ${\displaystyle {\mathrm{L}}^2}$ metric. The template and target are denoted ${\displaystyle {\psi }_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(\mathbf{s},x)}$, ${\displaystyle {\psi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}}(\mathbf{s},x)}$,${\displaystyle \mathbf{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 ${\displaystyle {\varphi }_t}$, which are solutions of ordinary differential equations ${\displaystyle {\dot{\varphi }}_t=v_t({\varphi }_t),t\in [0,1],{\varphi }_0=id}$. The group action of the diffeomorphism on the template is given according to ${\displaystyle {\varphi }_1\cdot \psi (x)\doteq (D{\varphi }_1)\psi \circ {\varphi }_1^{-1}(x),x\in X}$, where ${\displaystyle (D{\varphi }_1)}$ is the Jacobian of the affined transformed ODF and is defined as

${\displaystyle \begin{array}{c}(D{\varphi }_1)\psi \circ {\varphi }_1^{-1}(x)=\sqrt{\frac{det(D_{{\varphi }_1^{-1}}{\varphi }_1{)}^{-1}}{{\Vert (D_{{\varphi }_1^{-1}}{\varphi }_1{)}^{-1}\mathbf{s}\Vert }^3}}\psi \left(\frac{(D_{{\varphi }_1^{-1}}{\varphi }_1{)}^{-1}\mathbf{s}}{\Vert (D_{{\varphi }_1^{-1}}{\varphi }_1{)}^{-1}\mathbf{s}\Vert },{\varphi }_1^{-1}(x)\right).\end{array}}$

The LDDMM variational problem is defined as

${\displaystyle \begin{array}{c}\underset{v:{\dot{\varphi }}_t=v_t\circ {\varphi }_t,{\varphi }_0=id}{min}{\int }_0^1{\int }_{R^3}A{v}_t\cdot v_{t}dxdt+\lambda {\int }_{R^3}\Vert {\log }_{(D{\varphi }_1){\psi }_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}\circ {\varphi }_1^{-1}(x)}({\psi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}}(x)){\Vert }_{(D{\varphi }_1){\psi }_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}\circ {\varphi }_1^{-1}(x)}^{2}dx\end{array}}$.

Hamiltonian LDDMM for dense image matching

Beg solved the early LDDMM algorithms by solving the variational matching taking variations with respect to the vector fields. Another solution by Vialard, reparameterizes the optimization problem in terms of the state ${\displaystyle q_t\doteq I\circ {\varphi }_t^{-1},q_0=I}$, 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 ${\displaystyle {\dot{q}}_t=-\mathrm{\nabla }{q}_t\cdot v_t}$. The endpoint matching term ${\displaystyle E(q_1)\doteq \frac{1}{2}\Vert q_1-J{\Vert }^2}$ gives the variational problem:

Software for diffeomorphic mapping

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

• Deformetrica
• ANTS
• DARTEL Voxel-based morphometry(VBM)
• DEMONS
• LDDMM
• StationaryLDDMM

Cloud software

• MRICloud