User:Mim.cis/sandbox/Diffeomorphometry

Further information: LDDMM

Main article: Computational anatomy

Diffeomorphometry is the metric study of imagery, shape and form in the discipline of Computational anatomy (CA) in medical imaging. The study of images in computational anatomy rely on high-dimensional diffeomorphism groups $\varphi \in \operatorname {Diff} _{V}$ which generate orbits of the form ${\mathcal {I}}\doteq \{\varphi \cdot I\mid \varphi \in \operatorname {Diff} _{V}\}$, in which images $I\in {\mathcal {I}}$ can be dense scalar magnetic resonce or computed axial tomography images. For deformable shapes these are the collection of manifolds ${\mathcal {M}}\doteq \{\varphi \cdot M\mid \varphi \in \operatorname {Diff} _{V}\}$, points, curves and surfaces. In CA, these orbits are in general considered smooth Riemannian manifolds since at every point of the manifold $I\in {\mathcal {I}},M\in {\mathcal {M}}$ there is an inner product inducing the norm $\|\cdot \|_{I},\|\cdot \|_{M}$ on the tangent space that varies smoothly from point to point in the manifolds. This is generated by viewing the group of diffeomorphisms $\varphi \in \operatorname {Diff} _{V}$ as a Riemannian manifold with $\|\cdot \|_{\varphi }$, associated to the tangent space at $\varphi \in \operatorname {Diff} _{V}$ . This induces the norm and metric on the orbits of images and or shape manifolds under the action from the group of diffeomorphisms. The metric is constructed as the shortest length flow of diffeomorphisms connecting one image or shape to another.

The study of metrics on groups of diffeomorphisms and the study of metrics between manifolds and surfaces has been an area of significant investigation. In Computational anatomy, the diffeomorphometry metric measures how close and far two shapes or images are from each other. Informally, the metric length is the shortest length or geodesic flow which carries one coordinate system into the other.

Oftentimes, the familiar Euclidean metric is not directly applicable because the patterns of shapes and images don't form a vector space. In the Riemannian orbit model of Computational anatomy, diffeomorphisms acting on the forms $\phi \cdot I\in {\mathcal {I}},\phi \in Diff_{V},M\in {\mathcal {M}}$ don't act linearly. There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric is another. The method we use to induce the Riemannian metric is used to induce the metric on the orbit of shapes by defining it in terms of the metric length between diffeomorphic coordinate system transformations of the flows. Measuring the lengths of the geodesic flow between coordinates systems in the orbit of shapes is called diffeomorphometry.

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, $\varphi _{t},t\in [0,1]$, generated via the ordinary differential equation

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

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

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

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

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 thusly implies $(V,\|\cdot \|_{V})$ embeds smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:

$\operatorname {Diff} _{V}\doteq \{\varphi =\varphi _{1}:{\dot {\varphi }}_{t}=v_{t}\circ \varphi _{t},\varphi _{0}=\operatorname {id} ,\int _{0}^{1}\|v_{t}\|_{V}\,dt<\infty \}\ .$ 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 $I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in \operatorname {Diff} _{V}\}$, with for charts representing sub-manifolds denoted as ${\mathcal {M}}\doteq \{\varphi \cdot M_{temp}:\varphi \in \operatorname {Diff} _{V}\}$.

The Riemannian metric

The orbit of shapes and forms in Computational Anatomy are generated by the group action ${\mathcal {I}}\doteq \{\varphi \cdot I:\varphi \in \operatorname {Diff} _{V}\}$ , ${\mathcal {M}}\doteq \{\varphi \cdot M:\varphi \in \operatorname {Diff} _{V}\}$. These are made into a Riemannian orbits 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 $\varphi \in \operatorname {Diff} _{V}$ in the group of diffeomorphisms

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

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 $A:V\rightarrow V^{*}$, where $V^{*}$ is the dual-space. In general, $\sigma \doteq Av\in V^{*}$ is a generalized function or distribution, the linear form associated to the inner-product and norm for generalized functions are interpreted by integration by parts according to for $v,w\in V$,

$\langle v,w\rangle _{V}\doteq \int _{X}Av\cdot w\,dx,\ \|v\|_{V}^{2}\doteq \int _{X}Av\cdot v\,dx,\ v,w\in V\ .$

When $Av\doteq \mu dx$, a vector density, $\int Av\cdot vdx\doteq \int \mu \cdot vdx$.

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 Sobolev embedding theorem arguments were made in demonstrating that 1-continuous derivative is required for smooth flows. The Green's operator generated from the Green's function(scalar case) associated to the differential operator smooths. For proper choice of $A$ then $(V,\|\cdot \|_{V})$ is an RKHS with the operator $K=A^{-1}:V^{*}\rightarrow V$. The Green's kernels associated to the differential operator smooths since since for controlling enough derivatives in the square-integral sense the kernel $k(\cdot ,\cdot )$ is continuously differentiable in both variables implying

$KAv(x)_{i}\doteq \sum _{j}\int _{{\mathbb {R} }^{3}}k_{ij}(x,y)Av_{j}(y)dy\in V\ .$

Diffeomorphometry: The metric space of shapes and forms

The right-invariant metric on diffeomorphisms

The metric on the group of diffeomorphisms is defined by the distance as defined on pairs of elements in the group of diffeomorphisms according to

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

This distance provides a right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all $\phi \in \operatorname {Diff} _{V}$,

$d_{\operatorname {Diff} _{V}}(\psi ,\varphi )=d_{\operatorname {Diff} _{V}}(\psi \circ \phi ,\varphi \circ \phi ).$

The metric on shapes and forms

The distance on images , $d_{\mathcal {I}}:{\mathcal {I}}\times {\mathcal {I}}\rightarrow \mathbb {R} ^{+}$,

$d_{\mathcal {I}}(I,J)=\inf _{\phi \in \operatorname {Diff} _{V}:\phi \cdot I=J}d_{\operatorname {Diff} _{V}}(id,\phi )\ ;$ metric-shapes-forms

The distance on shapes and forms, $d_{\mathcal {M}}:{\mathcal {M}}\times {\mathcal {M}}\rightarrow \mathbb {R} ^{+}$,

$d_{\mathcal {M}}(M,N)=\inf _{\phi \in {Diff}_{V}:\phi \cdot M=N}d_{{Diff}_{V}}(id,\phi )\ .$ metric-shapes-forms

Hamiltonian formulation of computational anatomy

In Computational anatomy the diffeomorphisms are used to push the coordinate systems, and the vector fields are used as the control within the anatomical orbit or morphological space. The model is that of a dynamical system, the flow of coordinates $t\mapsto \phi _{t}\in \operatorname {Diff} _{V}$ and the control the vector field $t\mapsto v_{t}\in V$ related via ${\dot {\phi }}_{t}=v_{t}\cdot \phi _{t},\phi _{0}=id.$ The Hamiltonian view reparameterizes the momentum distribution $Av\in V^{*}$ in terms of the conjugate momentum or canonical momentum, introduced as a Lagrange multiplier $p:{\dot {\phi }}\mapsto (p\mid {\dot {\phi }})$ constraining the Lagrangian velocity ${\dot {\phi }}_{t}=v_{t}\circ \phi _{t}$.accordingly:

$H(\phi _{t},p_{t},v_{t})=\int _{X}p_{t}\cdot (v_{t}\circ \phi _{t})dx-{\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}dx.$

This function is the extended Hamiltonian. The Pontryagin maximum principle gives the optimizing vector field which determines the geodesic flow satisfying ${\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}=id,$ as well as the reduced Hamiltonian

$H(\phi _{t},p_{t})\doteq \max _{v}H(\phi _{t},p_{t},v)\ .$

The Lagrange multiplier in its action as a linear form has its own inner product of the canonical momentum acting on the velocity of the flow which is dependent on the shape, e.g. for landmarks a sum, for surfaces a surface integral, and. for volumes it is a volume integral with respect to $dx$ on ${\mathbb {R} }^{3}$. In all cases the Greens kernels carry weights which are the canonical momentum evolving according to an ordinary differential equation which corresponds to EL but is the geodesic reparameterization in canonical momentum. The optimizing vector field is given by

$v_{t}\doteq \arg max_{v}H(\phi _{t},p_{t},v)$

with dynamics of canonical momentum reparameterizing the vector field along the geodesic

${\begin{cases}{\dot {\phi }}_{t}={\frac {\partial H(\phi _{t},p_{t})}{\partial p}}\\{\dot {p}}_{t}=-{\frac {\partial H(\phi _{t},p_{t})}{\partial \phi }}\\\end{cases}}$ Hamiltonian-Dynamics

Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange

Whereas the vector fields are extended across the entire background space of ${\mathbb {R} }^{3}$, the geodesic flows associated to the submanifolds has Eulerian shape momentum which evolves as a generalized function $Av_{t}\in V^{*}$ concentrated to the submanifolds. For landmarks the geodesics have Eulerian shape momentum which are a superposition of delta distributions travelling with the finite numbers of particles; the diffeomorphic flow of coordinates have velocities in the range of weighted Green's Kernels. For surfaces, the momentum is a surface integral of delta distributions travelling with the surface.

The geodesics connecting coordinate systems satisfying EL-General have stationarity of the Lagrangian. The Hamiltonian is given by the extremum along the path $t\in [0,1]$, $H(\phi ,p)=\max _{v}H(\phi ,p,v)$, equalling the Lagrangian-Kinetic-Energy and is stationary along EL-General. Defining the geodesic velocity at the identity $v_{0}=\arg \max _{v}H(\phi _{0},p_{0},v)$, then along the geodesic

$H(\phi _{t},p_{t})=H(\phi _{0},p_{0})={\frac {1}{2}}\int _{X}p_{0}\cdot v_{0}dx={\frac {1}{2}}\int _{X}Av_{0}\cdot v_{0}dx={\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}dx$ Hamiltonian-Geodesics

The stationarity of the Hamiltonian demonstrates the interpretation of the Lagrange multiplier as momentum; integrated against velocity ${\dot {\phi }}$ gives energy density. The canonical momentum has many names. In optimal control, the flows $\phi$ is interpreted as the state, and $p$ is interpreted as conjugate state, or conjugate momentum. The geodesi of EL implies specification of the vector fields $v_{0}$ or Eulerian momentum $Av_{0}$ at $t=0$, or specification of canonical momentum $p_{0}$ determines the flow.

The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit

In Computational anatomy the submanifolds are pointsets, curves, surfaces and subvolumes which are the basic primitives. The geodesic flows between the submanifolds determine the distance, and form the basic measuring and transporting tools of diffeomorphometry. At $t=0$ the geodesic has vector field $v_{0}=Kp_{0}$ determined by the conjugate momentum and the Green's kernel of the inertial operator defining the Eulerian momentum $K=A^{-1}$. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

$d_{Diff_{V}}(id,\varphi )=\|Log_{id}(\varphi )\|_{V}=\|v_{0}\|_{V}={\sqrt {2H(id,p_{0})}}$

The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit

In Computational anatomy the submanifolds are pointsets, curves, surfaces and subvolumes which are the basic primitive forming the index sets or background space of medically imaged human anatomy. The geodesic flows connecting the submanifolds determine the metric distance. At $t=0$ the geodesic has vector field $v_{0}=Kp_{0}$ determined by the conjugate momentum and the Green's kernel of the inertial operator defining the Eulerian momentum $K=A^{-1}$. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

$d_{Diff_{V}}(id,\varphi )=\|Log_{id}(\varphi )\|_{V}=\|v_{0}\|_{V}={\sqrt {2H(id,p_{0})}}$

Landmark and surface submanifolds have Lagrange multiplier associated to a sum and surface integral, respectively; dense volumes an integral with respect to Lebesgue measure.

Landmark or pointset geodesics

What is so important about the RKHS norm defining the kinetic energy in the action principle is that the vector fields of the geodesic motions of the submanifolds are superpositions of Green's Kernel's. For landmarks the superposition is a sum of weight kernels weighted by the canonical momentum which determines the inner product, for surfaces it is a surface integral, and for dense volumes it is a volume integral. For Landmarks, the Hamiltonian momentum is defined on the indices, $p(i),i=1,\dots ,n$ with the inner product given by $(p_{t}|v_{t}\circ \phi _{t})\doteq \textstyle \sum _{i}\displaystyle p_{t}(i)\cdot v_{t}\circ \phi _{t}(x_{i})$ and Hamiltonian $H(\phi _{t},p_{t})={\frac {1}{2}}\textstyle \sum _{j}\sum _{i}\displaystyle p_{t}(i)\cdot K(\phi _{t}(x_{i}),\phi _{t}(x_{j}))p_{t}(j)$. The dynamics take the forms

${\begin{cases}v_{t}=\textstyle \sum _{i}\displaystyle K(\cdot ,\phi _{t}(x_{i}))p_{t}(i),\ \\{\dot {p}}_{t}(i)=-(Dv_{t})_{|_{\phi _{t}(x_{i})}}^{T}p_{t}(i),i=1,2,{\dot {,}}n\\\end{cases}}$

with the metric between landmarks $d^{2}=(p_{0}|v_{0})=\textstyle \sum _{i}p_{0}(i)\cdot \sum _{j}\displaystyle K(x_{i},x_{j})p_{0}(j).$

The dynamics associated to these geodesics is shown in the accompanying figure.

Illustration of geodesic flow for one landmark, demonstrating diffeomorphic motion of background space. Red arrow shows $p_{0}(1)$, blue curve shows $\varphi _{t}(x_{1})$, black grid shows $\varphi _{t}$

Surface geodesics

For surfaces, the Hamiltonian momentum is defined across the surface with the inner product $(p_{t}\mid v_{t}\circ \phi _{t})\doteq \textstyle \int _{U}\displaystyle p_{t}(u)\cdot v_{t}\circ \phi _{t}(m(u))du$, with $H(\phi _{t},p_{t})={\frac {1}{2}}\int _{U}\int _{U}p_{t}(u)\cdot K(\phi _{t}(m(u)),\phi _{t}(m(v)))p_{t}(v)\,du\,dv$. The dynamics

${\begin{cases}v_{t}=\textstyle \int _{U}\displaystyle K(\cdot ,\phi _{t}(m(u)))p_{t}(u)\,du\ ,\\{\dot {p}}_{t}(u)=-(Dv_{t})_{|_{\phi _{t}(m(u))}}^{T}p_{t}(u),u\in U\end{cases}}$

with the metric between surface coordinates $d^{2}=(p_{0}\mid v_{0})=\int _{U}p_{0}(u)\cdot \int _{U}K(m(u),m(u^{\prime }))p_{0}(u^{\prime })\,du\,du^{\prime }$

Volume geodesics

For volumes the Hamiltonian momentum is $(p_{t}\mid v_{t}\circ \phi _{t})\doteq \int _{{\mathbb {R} }^{3}}p_{t}(x)\cdot v_{t}\circ \phi _{t}(x)\,dx$ with $H(\phi _{t},p_{t})={\frac {1}{2}}\int _{{\mathbb {R} }^{3}}\int _{{\mathbb {R} }^{3}}p_{t}(x)\cdot K(\phi _{t}(x),\phi _{t}(y))p_{t}(y)\,dx\,dy\displaystyle$. The dynamics ${\begin{cases}v_{t}=\textstyle \int _{X}\displaystyle K(\cdot ,\phi _{t}(x))p_{t}(x)\,dx\ ,\\{\dot {p}}_{t}(x)=-(Dv_{t})_{|_{\phi _{t}(x)}}^{T}p_{t}(x),x\in {\mathbb {R} }^{3}\end{cases}}$

with the metric between volumes $\displaystyle d^{2}=(p_{0}\mid v_{0})=\int _{{\mathbb {R} }^{3}}p_{0}(x)\cdot \int _{{\mathbb {R} }^{3}}K(x,y)p_{0}(y)dy\ dx.$