Diffeomorphometry

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 diffeomorphic transformations of some exemplar, termed the template ${\displaystyle I_{temp}}$, resulting in the observed images to be elements of the random orbit model of CA. For images these are defined as ${\displaystyle I\in \mathcal{I}\doteq \{I=I_{temp}\circ \phi ,\phi \in {\mathrm{Diff}}_V\}}$, with for charts representing sub-manifolds denoted as ${\displaystyle \mathcal{M}\doteq \{\phi \cdot M_{temp}:\phi \in {\mathrm{Diff}}_V\}}$.

The Riemannian metric

The orbit of shapes and forms in Computational Anatomy are generated by the group action ${\displaystyle \mathcal{I}\doteq \{\phi \cdot I:\phi \in {\mathrm{Diff}}_V\}}$ , ${\displaystyle \mathcal{M}\doteq \{\phi \cdot M:\phi \in {\mathrm{Diff}}_V\}}$. 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 ${\displaystyle \phi \in {\mathrm{Diff}}_V}$ in the group of diffeomorphisms

${\displaystyle \Vert \dot{\phi }{\Vert }_{\phi }\doteq \Vert \dot{\phi }\circ {\phi }^{-1}{\Vert }_V=\Vert v{\Vert }_V,}$

with the vector fields modelled to be in a Hilbert space with the norm in the Hilbert space ${\displaystyle (V,\Vert \cdot {\Vert }_V)}$. We model ${\displaystyle V}$ as a reproducing kernel Hilbert space (RKHS) defined by a 1-1, differential operator ${\displaystyle A:V\to V^{\ast }}$, where ${\displaystyle V^{\ast }}$ is the dual-space. In general, ${\displaystyle \sigma \doteq Av\in V^{\ast }}$ 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 ${\displaystyle v,w\in V}$,

${\displaystyle ⟨v,w{⟩}_V\doteq {\int }_{X}Av\cdot wdx,\Vert v{\Vert }_V^2\doteq {\int }_{X}Av\cdot vdx,v,w\in V.}$

When ${\displaystyle Av\doteq \mu dx}$, a vector density, ${\displaystyle \int Av\cdot vdx\doteq \int \mu \cdot vdx=\sum _{i=1}^3{\mu }_i{v}_{i}dx.}$

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 ${\displaystyle A}$ then ${\displaystyle (V,\Vert \cdot {\Vert }_V)}$ is an RKHS with the operator ${\displaystyle K=A^{-1}:V^{\ast }\to V}$. The Green's kernels associated to the differential operator smooths since for controlling enough derivatives in the square-integral sense the kernel ${\displaystyle k(\cdot ,\cdot )}$ is continuously differentiable in both variables implying

${\displaystyle KAv(x{)}_i\doteq \sum _j{\int }_{{\mathbb{R}}^3}{k}_{ij}(x,y)A{v}_j(y)dy\in V.}$

The diffeomorphometry of the 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

This distance provides a right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all ${\displaystyle \varphi \in {\mathrm{Diff}}_V}$,

${\displaystyle d_{{\mathrm{Diff}}_V}(\psi ,\phi )=d_{{\mathrm{Diff}}_V}(\psi \circ \varphi ,\phi \circ \varphi ).}$

The metric on shapes and forms

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

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

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

For calculating the metric, the geodesics are a dynamical system, the flow of coordinates ${\displaystyle t\mapsto {\varphi }_t\in {\mathrm{Diff}}_V}$ and the control the vector field ${\displaystyle t\mapsto v_t\in V}$ related via ${\displaystyle {\dot{\varphi }}_t=v_t\cdot {\varphi }_t,{\varphi }_0=\mathrm{id}.}$ The Hamiltonian view reparameterizes the momentum distribution ${\displaystyle Av\in V^{\ast }}$ in terms of the Hamiltonian momentum, a Lagrange multiplier ${\displaystyle p:\dot{\varphi }\mapsto (p\mid \dot{\varphi })}$ constraining the Lagrangian velocity ${\displaystyle {\dot{\varphi }}_t=v_t\circ {\varphi }_t}$.accordingly:

${\displaystyle H({\varphi }_t,p_t,v_t)={\int }_X{p}_t\cdot (v_t\circ {\varphi }_t)dx-\frac{1}{2}{\int }_{X}A{v}_t\cdot v_{t}dx.}$

The Pontryagin maximum principle gives the Hamiltonian ${\displaystyle H({\varphi }_t,p_t)\doteq \underset{v}{max}H({\varphi }_t,p_t,v).}$ The optimizing vector field ${\displaystyle v_t\doteq {\mathrm{argmax}}_{v}H({\varphi }_t,p_t,v)}$ with dynamics ${\displaystyle {\dot{\varphi }}_t=\frac{\mathrm{\partial }H({\varphi }_t,p_t)}{\mathrm{\partial }p},{\dot{p}}_t=-\frac{\mathrm{\partial }H({\varphi }_t,p_t)}{\mathrm{\partial }\varphi }}$. Along the geodesic the Hamiltonian is constant: ${\displaystyle H({\varphi }_t,p_t)=H(\mathrm{id},p_0)=\frac{1}{2}{\int }_X{p}_0\cdot v_{0}dx}$. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

${\displaystyle d_{{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_V}(\mathrm{id},\phi )=\Vert v_0{\Vert }_V=\sqrt{2H(\mathrm{id},p_0)}}$

Landmark or pointset geodesics

For landmarks, ${\displaystyle x_i,i=1,\dots ,n}$, the Hamiltonian momentum

${\displaystyle p(i),i=1,\dots ,n}$

with Hamiltonian dynamics taking the form

${\displaystyle H({\varphi }_t,p_t)=\frac{1}{2}\sum _j\sum _i{\displaystyle p_t(i)\cdot K({\varphi }_t(x_i),{\varphi }_t(x_j))p_t(j)}}$

with

${\displaystyle \{\begin{array}{l}{v}_t=\sum _i{\displaystyle K(\cdot ,{\varphi }_t(x_i))p_t(i),}\\ {\dot{p}}_t(i)=-(D{v}_t{)}_{{|}_{{\varphi }_t(x_i)}}^T{p}_t(i),i=1,2,\dots ,n\end{array}}$

The metric between landmarks ${\displaystyle d^2=\sum _i{p}_0(i)\cdot \sum _j{\displaystyle K(x_i,x_j)p_0(j).}}$

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

Surface geodesics

For surfaces, the Hamiltonian momentum is defined across the surface has Hamiltonian

${\displaystyle H({\varphi }_t,p_t)=\frac{1}{2}{\int }_U{\int }_U{p}_t(u)\cdot K({\varphi }_t(m(u)),{\varphi }_t(m(v)))p_t(v)dudv}$

and dynamics

${\displaystyle \{\begin{array}{l}{v}_t={\int }_U{\displaystyle K(\cdot ,{\varphi }_t(m(u)))p_t(u)du,}\\ {\dot{p}}_t(u)=-(D{v}_t{)}_{{|}_{{\varphi }_t(m(u))}}^T{p}_t(u),u\in U\end{array}}$

The metric between surface coordinates ${\displaystyle d^2=(p_0\mid v_0)={\int }_U{p}_0(u)\cdot {\int }_{U}K(m(u),m(u^{\mathrm{\prime }}))p_0(u^{\mathrm{\prime }})dud{u}^{\mathrm{\prime }}}$

Volume geodesics

For volumes the Hamiltonian

${\displaystyle H({\varphi }_t,p_t)=\frac{1}{2}{\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}{p}_t(x)\cdot K({\varphi }_t(x),{\varphi }_t(y))p_t(y)dxdy{\displaystyle }}$

with dynamics

${\displaystyle \{\begin{array}{l}{v}_t={\int }_X{\displaystyle K(\cdot ,{\varphi }_t(x))p_t(x)dx,}\\ {\dot{p}}_t(x)=-(D{v}_t{)}_{{|}_{{\varphi }_t(x)}}^T{p}_t(x),x\in {\mathbb{R}}^3\end{array}}$

The metric between volumes ${\displaystyle {\displaystyle d^2=(p_0\mid v_0)={\int }_{{\mathbb{R}}^3}{p}_0(x)\cdot {\int }_{{\mathbb{R}}^3}K(x,y)p_0(y)dydx.}}$

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