Riemannian metric and Lie bracket in computational anatomy: Difference between revisions

Main articles: Computational anatomy and Computational anatomy § The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms

Computational anatomy (CA) is the study of shape and form in Medical imaging. The study of deformable shapes in Computational Anatomy rely on high-dimensional diffeomorphism groups $\varphi \in Diff_{V}$ which generate orbits of the form ${\mathcal {M}}\doteq \{\varphi \cdot m\ |\ \varphi \in Diff_{V}\}$. In CA, this orbit is in general considered a smooth Riemannian manifold since at every point of the manifold $m\in {\mathcal {M}}$ there is an inner product inducing the norm $\|\cdot \|_{m}$ on the tangent space that varies smoothly from point to point in the manifold of shapes $m\in {\mathcal {M}}$. This is generated by viewing the group of diffeomorphisms $\varphi \in Diff_{V}$ as a Riemannian manifold with $\|\cdot \|_{\phi }$, associated to the tangent space at $\phi \in Diff_{V}$ . This induces the norm and metric on the orbit $m\in {\mathcal {M}}$ under the action from the group of diffeomorphisms.

The diffeomorphisms group generated as Lagrangian and Eulerian flows

The diffeomorphisms in Computational anatomy are generated to satisfy the Lagrangian and Eulerian specification of the flow fields, $\phi _{t},t\in [0,1]$, generated via the ordinary differential equation

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

with the Eulerian vector fields $v\doteq (v_{1},v_{2},v_{3})$ in ${\mathbb {R} }^{3}$ for $v_{t}={\dot {\phi }}_{t}\circ \phi _{t}^{-1},t\in [0,1]$, with the inverse for the flow given by

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

and the $3\times 3$ Jacobian matrix for flows in $\mathbb {R} ^{3}$ given as $\ D\phi \doteq ({\frac {\partial \phi _{i}}{\partial x_{j}}})$.

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

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

The Riemannian orbit model

Shapes in Computational Anatomy (CA) are studied via the use of diffeomorphic mapping for establishing correspondences between anatomical coordinate systems. In this setting, 3-dimensional medical images are modelled as diffemorphic transformations of some exemplar, termed the template $I_{temp}$, resulting in the observed images to be elements of the random orbit model of CA. For images these are defined as $I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in Diff_{V}\}$, with for charts representing sub-manifolds denoted as ${\mathcal {M}}\doteq \{\varphi \cdot m:\varphi \in Diff_{V}\}$.

The Riemannian Metric

The orbit of shapes and forms in Computational Anatomy are generated by the group action${\mathcal {M}}\doteq \{\varphi \cdot m:\varphi \in Diff_{V}\}$. This is made into a Riemannian orbit by introducing a metric associated to each point and associated tangent space. For this a metric is defined on the group which induces the metric on the orbit. Take as the metric for Computational anatomy at each element of the tangent space $\phi \in Diff_{V}$ in the group of diffeomorphisms

$\|{\dot {\phi }}\|_{\phi }\doteq \|{\dot {\phi }}\circ \phi ^{-1}\|_{V}=\|v\|_{V}$,

with the vector fields modelled to be in a Hilbert space with the norm in the Hilbert space $(V,\|\cdot \|_{V})$. We model $V$ as a reproducing kernel Hilbert space (RKHS) defined by a 1-1, differential operator$A:V\rightarrow V^{*}$. For $\sigma (v)\doteq Av\in V^{*}$ a distribution or generalized function, the linear form $(\sigma |w)\doteq \int _{{\mathbb {R} }^{3}}\sum _{i}w_{i}(x)\sigma _{i}(dx)$ determines the norm:and inner product for $v\in V$ according to

$\langle v,w\rangle _{V}\doteq (Av|w),\ \|v\|_{V}^{2}\doteq (Av|v),\ v,w\in V\ .$

The differential operator is selected so that the Green's kernel associated to the inverse is sufficiently smooth so that the vector fields support 1-continuous derivative.

The Right Invariant Metric on Diffeomorphisms

Define the distance on the group of diffeomorphisms

:$d_{Diff_{V}}(\psi ,\varphi )=\inf _{v_{t}}\left(\int _{0}^{1}(Av_{t}|v_{t})dt:\phi _{0}=\psi ,\phi _{1}=\varphi ,{\dot {\phi }}_{t}=v_{t}\circ \phi _{t}\right)^{1/2}\ ;$ metric-diffeomorphisms

this is is the right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all $\phi \in Diff_{V}\,\,\,$,

$d_{Diff_{V}}(\psi ,\varphi )=d_{Diff_{V}}(\psi \circ \phi ,\varphi \circ \phi )$.

The Lie bracket in the group of diffeomorphisms

The Lie bracket gives the adjustment of the velocity term resulting from a perturbation of the motion in the setting of curved spaces. Using Hamilton's principle of least-action derives the optimizing flows as a critical point for the action integral of the integral of the kinetic energy. The derivation calculates the perturbation $\delta v$ on the vector fields $v^{\epsilon }=v+\epsilon \delta v\$ in terms of the derivative in time of the group perturbation adjusted by the correction of the Lie bracket of vector fields in this function setting involving the Jacobian matrix, unlike the matrix group case:

$ad_{v}:V\mapsto V$ given by $ad_{v}(w)\doteq (Dv)w-(Dw)v,v,w\in V.$ adjoint-Lie-bracket

ProvingLie bracket of vector fields take a first order perturbation of the flow at point $\phi \in Diff_{V}$ according to $\phi _{t}^{\epsilon }\doteq (id+\epsilon w)\circ \phi =\phi +\epsilon w\circ \phi$, with fixed boundary $w_{0}=w_{1}=0$, with ${\frac {d}{dt}}\phi _{t}^{\epsilon }=v_{t}^{\epsilon }\circ \phi _{t}^{\epsilon },\phi _{0}^{\epsilon }=id,\phi _{1}^{\epsilon }=\phi _{1}$, giving the following two Eqns:

• ${\frac {d}{dt}}\phi _{t}^{\epsilon }={\frac {d}{dt}}\phi _{t}+\epsilon {\frac {d}{dt}}(w_{t}\circ \phi _{t})=v_{t}\circ \phi _{t}+(Dw_{t})\circ \phi _{t}v_{t}\circ \phi _{t}+o(\epsilon )\ .$
• ${\dot {\phi }}_{t}^{\epsilon }=(v_{t}+\epsilon \delta v_{t})\circ (\phi _{t}+\epsilon w_{t}\circ \phi _{t})\simeq v_{t}\circ \phi _{t}+\epsilon (Dv_{t})\circ \phi _{t}w_{t}\circ \phi _{t}+\delta v_{t}\circ \phi _{t}+o(\epsilon )\ .$

Equating the above two equations gives the perturbation of the vector field in terms of the Lie bracket adjustment

$\delta v_{t}={\frac {d}{dt}}w_{t}-ad_{v_{t}}(w_{t})={\frac {d}{dt}}w_{t}-((Dv_{t})w_{t}-(Dw_{t})v_{t})\ .$

The generalized Euler–Lagrange equation for the Metric on diffeomorphic flows

Main articles: Computational anatomy and Computational anatomy § The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms

The action integral for the Lagrangian of the kinetic energy for Hamilton's principle becomes

$J(\phi )\doteq {\frac {1}{2}}\int _{0}^{1}\|{\dot {\phi }}_{t}\|_{\phi _{t}}^{2}dt={\frac {1}{2}}\int _{0}^{1}\|{\dot {\phi }}_{t}\circ \phi _{t}^{-1}\|_{V}^{2}dt={\frac {1}{2}}\int _{0}^{1}\|v_{t}\|_{V}^{2}dt\ .$

The shortest paths geodesic connections in the orbit are defined via Hamilton's Principle of least action requires first order variations of the solutions in the orbits of Computational Anatomy which are based on computing critical points on the metric length or energy of the path. The original derivation of the Euler equation associated to the geodesic flow of diffeomorphisms exploits the was a generalized function equation when$Av\in V^{*}$ is a distribution, or generalized function, take the first order variation of the action integral using the adjoint operator for the Lie bracket (adjoint-Lie-bracket) gives for all smooth $w\in V$,

${\frac {d}{d\varepsilon }}J(\phi ^{\epsilon })|_{\varepsilon =0}=\int _{0}^{1}(Av_{t}\mid \delta v_{t})\,dt=\int _{0}^{1}(Av_{t}\mid {\frac {d}{dt}}w_{t}-((Dv_{t})w-(Dw)v_{t}))dt$.

Using the bracket $ad_{v}:w\in V\mapsto V$ and $ad_{v}^{*}:V^{*}\rightarrow V^{*}$ gives

${\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})=0\ ,\ t\in [0,1]\ .$ EL-General

Riemannian Exponential for Positioning

In the random orbit model of Computational anatomy, the entire flow is reduced to the initial condition which forms the coordinates encoding the diffeomorphism. From the initial condition $v_{0}$ then geodesic positioning with respect to the Riemannian metric of Computational anatomy solves for the flow of the Euler-Lagrange equation. Solving the geodesic from the initial condition $v_{0}$ is termed the Riemannian-exponential, a mapping $Exp_{\rm {id}}(\cdot ):V\to Diff_{V}$ at identity to the group.

• For classical equation $(Av_{t}\mid w)=\int _{X}Av_{t}\cdot w\,dx$, $Av\in V$with the diffeomorphic shape momentum a density, then $Exp_{id}(v_{0})=\phi _{1},{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},t\in [0,1]$ satisfying

${\frac {d}{dt}}Av_{t}+(Dv_{t})^{T}Av_{t}+(DAv_{t})v_{t}+(\nabla \cdot v)Av_{t}=0\ ,\ {\text{for initial condition}}\ {\dot {\phi }}_{0}=v_{0}\ .$

EL-Classic

• For generalized equation, the bracket $ad_{v}:V\mapsto V$, $ad_{v}(w)\doteq (Dv)w-(Dw)v,v,w\in V$ and $ad_{v}^{*}:V^{*}\rightarrow V^{*}$ then $Exp_{id}(v_{0})=\phi _{1}$, satisfying

$({\frac {d}{dt}}Av_{t}\mid w)+(Av_{t}\mid (Dv_{t})w-(Dw)v_{t})=0.$

It is extended to the entire group, $\phi =Exp_{\varphi }(v_{0}\circ \varphi )\doteq Exp_{id}(v_{0})\circ \varphi$.For the smooth vector density case, $Av_{t}=\mu _{t}\,dx$ with $(Av_{t}\mid w)=\int _{X}\mu _{t}\cdot w\,dx\ ,w\in V$ the Euler equation exists in the classical sense as fderived for the density in Equation (Euler-general) is the Euler-equation when diffeomorphic shape momentum is a generalized function. This equation has been called EPDiff, Euler–Poincare equation for diffeomorphisms and has been studied in the context of fluid mechanics for incompressible fluids with $L^{2}$ metric.

The variation problem for matching or registering coordinate system information in computational anatomy

Matching information across coordinate systems is central to Computational Anatomy. Adding a matching term $E:\phi \in Diff_{V}\rightarrow R^{+}$ which represents the target endpoint add a secondary boundary condition to Hamilton's principle for variation using the Lie brack for vector fields: for matching gives the cost

$C(\phi )\doteq \int _{0}^{1}(Av_{t}|v_{t})dt+E(\phi _{1}),$

which gives the Euler equation with boundary term. Taking the variation gives

$\int _{0}^{1}(Av_{t}\mid {\frac {\partial }{\partial t}}\delta \phi _{t}-(Dv\delta \phi -D\delta \phi v))\,dt+({\frac {\partial E(\phi )}{\partial \phi _{1}}}\mid \delta \phi _{1})=-\int _{0}^{1}({\frac {\partial Av_{t}}{\partial t}}+ad_{v_{t}}^{*}(Av_{t})\mid \delta \phi _{t})\,dt+(Av_{1}+{\frac {\partial E(\phi )}{\partial \phi _{1}}}\mid \delta \phi _{1}).$

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