Large deformation diffeomorphic metric mapping: Difference between revisions

• Comment: Fix reference error before this article can be accepted. Robert McClenon (talk) 20:35, 13 February 2016 (UTC)

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Mapping 3-dimensional coordinate system in Medical imaging and in CA across high resolution dense image coordinates was championed in the small deformation setting by the University of Pennsylvania group lead by Ruzena Bajcy , and subsequently in the Grenander school with the HAND experiments and then brains with small deformation elasticity.

The earliest introduction of the use of flows of diffeomorphisms for transformation of coordinate systems in image analysis and medical imaging was by Christensen et. al.

In CA the geodesics and their coordinates are generated by solving inexact matching problems calculating the geodesic flows of diffeomorphisms from their start point onto targets. Inexact matching has been examined in many cases and have come to be called Large Deformation Diffeomorphic Metric Mapping (LDDMM) originally solved for dense image matching by Faisal Beg for his PhD at Johns Hopkins University. It was the first example in Computational Anatomy where a numerical code had been created whose fixed points satisfy the necessary conditions for geodesic shortest paths solving the Euler equation for Computational Anatomy and minimizing the dense image matching problem for which Dupuis, Grenander and Miller had derived the necessary Sobolev condition for existence of solutions of geodesic flows of diffeomorphisms in image matching. These methods have been extended to landmarks without registration via measure matching, curves, surfaces, dense vector and tensor imagery, and varifolds removing orientation.

The Diffeomorphism Model of Dense Image Matching

For the study of deformable shape in Computational Anatomy, the high-dimensional diffeomorphism group is used to study correspondences across desnse coordinate systems. In this setting, three dimensional medical images are modelled as a random unknown deformation of some exemplar, termed the template $I_{temp}$, with the set of observed images an element in the orbit $I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in Diff_{V}$. The diffeomorphisms are generated via smooth flows $\phi _{t},t\in [0,1]$ which satisfy the Lagrangian and Eulerian specification of the flow field satisfying the ordinary differential equation:

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

with $v_{t},t\in [0,1]$ the vector fields determining the flow.

The Dense Image Matching Problem

Construction of diffeomorphic correspondences between shapes calculates the initial vector field coordinates $v_{0}\in V$ and associated weights on the Greens kernels $p_{0}$. These initial coordinates are determined by matching of shapes, called Large Deformation Diffeomorphic Metric Mapping (LDDMM). LDDMM has been solved for landmarks with and without correspondence and for dense image matchings., curves, surfaces, dense vector and tensor imagery, and varifolds removing orientation. LDDMM calculates geodesic flows of the EL-General onto target coordinates, adding to the action integral ${\frac {1}{2}}\int _{0}^{1}(Av_{t}|v_{t})dt$ an endpoint matching condition $E:\phi _{1}\rightarrow R^{+}$ measuring the correspondence of elements in the orbit under coordinate system transformation. Existence of solutions were examined for image matching. The solution of the variational problem satisfies the EL-General for $t\in [0,1)$ with boundary condition.

The endpoint condition $E_{1}:\phi \rightarrow R^{+}$ which Beg solved for dense image matching with action $\phi \cdot I\doteq I\circ \phi ^{-1}\in {\mathcal {I}},\phi \in Diff_{V}$ and endpoint deviation measured via the $L^{2}$ squared-error metric$E(\phi _{1})\doteq \|I\circ \phi _{1}-J\|_{2}^{2}$. In the dense image setting, the Eulerian momentum has a density $Av_{t}=\mu _{t}dx$ so that the Euler equation for geodesic flows of diffeomorphisms in Computational Anatomy as a classical solution. Dense image matching illustrates one of the two extremes of $Av\in V^{*}$, the momentum having a vector density pointwise function, so that $(Av|w)=\int _{X}\mu \cdot wdx$ for $\mu$ a vector function.

Control Problem : The dense image matching problem satisfies the Euler equation with boundary condition at time t=1:

$\min _{v:{\dot {\phi }}=v\circ \phi ,\phi _{0}=id}C(\phi )\doteq {\frac {1}{2}}\int (Av_{t}|v_{t})dt+{\frac {1}{2}}\int _{X}|I\circ \phi ^{-1}-J|^{2}dx$

${\begin{cases}&{\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})\ ,\ \ t\in [0,1);\\&\ \ \ \ \ \ \ \ \ \ \ \mu _{t}=(D\phi _{t}^{-1})^{T}\mu _{0}\circ \phi _{t}^{-1}|D\phi _{t}^{-1}|,\\&Av_{1}=\mu _{1}dx,\mu _{1}={\frac {\partial E(\phi _{1})}{\partial \phi _{1}}}\\&\ \ \ \ \ \ \ \ \ \ \ \ \mu _{1}=(I\circ \phi _{1}^{-1}-J)\nabla (I\circ \phi _{1}^{-1})\ ,\ \ t=1\ .\end{cases}}$

The conservation equation implies that with the necessary fixed endpoint condition, the initial condition on the momentum is given by

$\mu _{0}=(I-J\circ \phi _{1})\nabla I|D\phi _{1}|$.

LDDMM for image matching via perturbation of the vector fields

The original large deformation diffeomorphic metric mapping (LDDMM) algorithms took variations with respect to the vector field parameterization of the group, since $v\doteq {\dot {\phi }}\circ \phi ^{-1}$ are in a vector spaces. Variations satisfy the necessary optimality conditions. Beg solved the Control Problem for dense image matching maximizing with respect to the vector fields$v_{t},t\in [0,1]$ via the following iterative algorithm:

• Beg's Iterative LDDMM Algorithm: Update until convergence, updating $\phi _{t}^{old}\leftarrow \phi _{t}^{new}$ each iteration:

${\begin{cases}&v_{t}^{new}(\cdot )=\int _{X}K(\cdot ,y)(I\circ \phi _{t}^{-1old}(y)-J\circ \phi _{t1}^{old}(y))\nabla (I\circ \phi _{t}^{-1old}(y))|D\phi _{t1}^{old}(y)|dy,t\in [0,1]\\&{\dot {\phi }}_{t}^{new}=v_{t}^{new}\circ \phi _{t}^{new},t\in [0,1]\end{cases}}$

• Fixed points satisfy the necessary maximizer condition:

$Av_{t}=(I\circ \phi _{t}^{-1}-J\circ \phi _{t1})\nabla (I\circ \phi _{t}^{-1})|D\phi _{t1}|dx$ where $\phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}$.

Proof of Gradient and Necesssary Maximizer Conditions

The perturbation in the vector field uses the Lie bracket of vector fields from the identity ${\frac {d}{dt}}\delta \phi _{|\phi }=(Dv)_{|\phi }\delta \phi _{|\phi }+\delta v_{|\phi }$. This has the solution

$(\delta \phi _{1})_{|{\phi _{1}}}=(D\phi _{1})\int _{0}^{1}(D\phi _{t})^{-1}\delta v_{t}\circ \phi _{t}dt$ implying $\delta \phi _{1}=(D\phi _{1})_{|\phi _{1}^{-1}}\int _{0}^{1}(D\phi _{t})_{|\phi _{1}^{-1}}^{-1}(\delta v_{t})_{\phi _{t}\circ \phi _{1}^{-1}}dt$

Taking the variation in the vector fields $v+\epsilon \delta v$ using the chain rule requires us to compute

${\frac {\partial E(\phi _{1})}{\partial v}}={\frac {\partial E(\phi _{1})}{\partial \phi ^{-1}}}{\frac {\partial \phi _{1}^{-1}}{\partial \phi }}{\frac {\partial \phi _{1}}{\partial v}}$ .

For dense images the action $\phi \cdot I\doteq I\circ \phi ^{-1}$ implies we will requires the variation of the inverse $\phi ^{-1}$ with respect to $\phi$ for the chain rule calculation ${\frac {\partial E(\phi )}{\partial \phi ^{-1}}}{\frac {\partial \phi ^{-1}}{\partial \phi }}$ . This requires the identity $\delta \phi ^{-1}=-(D\phi _{1})_{|_{\phi _{1}^{-1}}}^{-1}\delta \phi$ following from the identity $(\phi +\epsilon \delta \phi \circ \phi )\circ (\phi ^{-1}+\epsilon \delta \phi ^{-1})=id+o(\epsilon )$. This is for such function spaces the generalization of the classic matrix perturbation of the inverse which gives the first variation

${\frac {\partial E}{\partial v}}=(I\circ \phi _{1}^{-1}-J)\nabla I|_{\phi _{1}^{-1}}(-D\phi _{1})_{|\phi _{1}^{-1}}^{-1}(D\phi _{1})_{|\phi _{1}^{-1}})\int _{0}^{1}(D\phi _{t})_{|\phi _{1}^{-1}}^{-1}(\delta v_{t})_{|{\phi _{t}\circ \phi _{1}^{-1}}}dt$.

Taking the first variation of $C(v)\doteq {\frac {1}{2}}\int (Av_{t}|v_{t})dt+{\frac {1}{2}}\|I\circ \phi ^{-1}-J\|_{2}^{2}$ gives

$\partial C(v;\delta v)=\int _{0}^{1}(Av_{t}|\delta v_{t})dt-\int _{0}^{1}\int _{X}(I\circ \phi _{1}^{-1}-J)\nabla I|_{\phi _{1}^{-1}}(D\phi _{t})_{|\phi _{1}^{-1}}^{-1}(\delta v_{t})_{|{\phi _{t}\circ \phi _{1}^{-1}}}dxdt\ .$

Substituting $Av_{t}=\mu _{t}dx$ , $\phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}$

$\partial C(v;\delta v)=\int _{0}^{1}\int _{X}\left(\mu _{t}(x)-(I\circ \phi _{t}^{-1}-J\circ \phi _{t1})\nabla I|_{\phi _{t}^{-1}}(D\phi _{t})_{|\phi _{t}^{-1}}^{-1}|D\phi _{t1}|\right)\delta v_{t}dxdt\ .$

Category:Diffeomorphisms