User:Mim.cis/Beg's LDDMM for Dense Image Matching

Beg's LDDMM for Dense Image Matching

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 equatoin 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 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. In the dense image seting, the Eulerian momentum has a density $Av_{t}=\mu _{t}dx$ so that the Euler equation for geodesic flows of diffeomorphisms in Computational Anatomy has a classical solution.

Dense image matching

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

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

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

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

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

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. 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.

Landmark matching with correspondence

Landmark matching demonstrates the singularity of $Av\in V^{*}$ as a generalized function or delta-distribution, since for landmarks,$q_{0}\doteq \{x_{1},x_{2},\dots \}$ all of boundary mass is conscentrated on singular subsets of the volume space. The optimizer of Control Problem 1 satisfies the weak Euler equation for delta-distributions:

 $\min _{\phi :{\dot {\phi }}=v\circ \phi ,\phi _{0}=id}C(\phi )\doteq {\frac {1}{2}}\int \|{\dot {\phi }}_{t}\circ \phi _{t}^{-1}\|_{V}^{2}dt+{\frac {1}{2}}\sum _{i}\|\phi _{1}(x_{i})-y_{i}\|^{2}$

${\begin{cases}&{\text{Necessary condition:}}\ \ \ \ \ \ \ \ \ \ \ \ \ Av_{1}=\sum _{i=1}^{n}p_{1}(i)\delta _{\phi _{1}(x_{i})},p_{1}(i)=(y_{i}-\phi _{1}(x_{i})),\\&{\text{Conservation equation:}}\ \ \ \ \ \ \ Av_{t}=\sum _{i=1}^{n}p_{t}(i)\delta _{\phi _{t}(x_{i})},p_{t}(i)=(D\phi _{t1})_{|\phi _{t}(x_{i})}^{T}p_{1}(i)\ ,\phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}\\\end{cases}}$

Notice this gives in particular the initial condition $Av_{0}=\sum _{i=1}^{n}p_{0}(i)\delta _{x_{i}},p_{0}(i)=(D\phi _{1})_{|x_{i}}^{T}(y_{i}-\phi (x_{i}))$.

Taking $Av_{t}=\sum _{i=1}^{n}p_{t}(i)\delta _{\phi _{t}(x_{i})}$; conservation $(Av_{t}|(D\phi _{t}w)_{|\phi _{t}^{-1}})=const.$ , $\sum _{i=1}^{n}(D\phi _{t})_{x_{i}}^{T}p_{t}(i)\cdot w(x_{i})=\sum _{i=1}^{n}(D\phi _{1})_{x_{i}}^{T}p_{1}(i)\cdot w(x_{i})$

. $p_{t}(i)=((D\phi _{t})_{|x_{i}}^{-1})^{T}(D\phi _{1})_{x_{i}}^{T}p_{1}(i)$.

LDDMM for image and landmark matching: perturbation in 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 velocity field; as did Joshi for Landmark matching.

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

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

• Joshi solved the landmark matching problem taking variations with respect ot the vector field:

$\min _{v:\phi _{1}=\int _{0}^{1}v_{t}\circ \phi _{t}dt}C(v)\doteq {\frac {1}{2}}\int \|v_{t}\|_{V}^{2}dt+{\frac {1}{2}}\sum _{i}\|\phi _{1}(x_{i})-y_{i}\|^{2}$;

necessary conditions become $Av_{t}=\sum _{i}(D\phi _{t1})^{T}|_{\phi _{t}(x_{i})}(y_{i}-\phi _{1}(x_{i}))\delta _{\phi _{t}(x_{i})}$.

The perturbation in the vector field requires the identity ${\frac {d}{dt}}\delta \phi _{|\phi }=(Dv)_{|\phi }\delta \phi _{|\phi }+\delta v_{|\phi }$ which implies $(\delta \phi _{1})_{|{\phi _{1}}}=(D\phi _{1})\int _{0}^{1}(D\phi _{t})^{-1}\delta v_{t}\circ \phi _{t}dt$ Take the variation in the vector fields $v+\epsilon \delta v$ using the chain rule ${\frac {\partial E(\phi _{1})}{\partial v}}={\frac {\partial E(\phi _{1})}{\partial \phi ^{-1}}}{\frac {\partial \phi _{1}^{-1}}{\partial \phi }}{\frac {\partial \phi _{1}}{\partial v}}$ which gives the first variation

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

Joshi solved the landmark matching problem optimizing with respect to the velocity field:

${\frac {\partial E}{\partial v}}=\sum _{i}(\phi _{1}(x_{i})-y_{i})\cdot D\phi _{1}|_{\phi _{1}^{-1}(\phi _{1}(x_{i}))}\int _{0}^{1}(D\phi _{t})_{|\phi _{1}^{-1}(\phi _{1}(x_{i}))}^{-1}\delta v_{t}|_{\phi _{t}\circ \phi _{1}^{-1}(\phi _{1}(x_{i}))}dt$

${\frac {\partial E}{\partial v}}=\int _{0}^{1}\int _{X}\sum _{i}\delta _{\phi _{t}(x_{i})}(x)(\phi _{1}(x_{i})-y_{i})\cdot (D\phi _{1})_{\phi _{t}^{-1}(x)}(D\phi _{t})_{\phi _{t}^{-1}(x)}^{-1}\delta v_{t}(x)dxdt=\int _{0}^{1}\int _{X}\sum _{i}\delta _{\phi _{t}(x_{i})}(y)(D\phi _{t1})_{\phi _{t}(x_{i})}^{T}(\phi _{1}(x_{i})-y_{i})\cdot \delta v_{t}(x)dxdt$

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