Diffeomorphometry

The Riemannian metric

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

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

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

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

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

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 ${\displaystyle (V,\|\cdot \|_{V})}$ is an RKHS with the operator ${\displaystyle K=A^{-1}:V^{*}\rightarrow V}$. 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

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

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

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

(metric-diffeomorphisms)

This distance provides a right-invariant metric of diffeomorphometry,^[12][13][14] invariant to reparameterization of space since for all ${\displaystyle \phi \in \operatorname {Diff} _{V}}$,

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

The metric on shapes and forms

The distance on images,^[15] ${\displaystyle d_{\mathcal {I}}:{\mathcal {I}}\times {\mathcal {I}}\rightarrow \mathbb {R} ^{+}}$,

${\displaystyle 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,^[16] ${\displaystyle d_{\mathcal {M}}:{\mathcal {M}}\times {\mathcal {M}}\rightarrow \mathbb {R} ^{+}}$,

${\displaystyle d_{\mathcal {M}}(M,N)=\inf _{\phi \in \operatorname {Diff} _{V}:\phi \cdot M=N}d_{\mathrm {Diff} _{V}}(\operatorname {id} ,\phi )\ .}$

(metric-shapes-forms)

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

${\displaystyle H(\phi _{t},p_{t})={\frac {1}{2}}\textstyle \sum _{j}\sum _{i}\displaystyle p_{t}(i)\cdot K(\phi _{t}(x_{i}),\phi _{t}(x_{j}))p_{t}(j)}$

with

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

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

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

Surface geodesics

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

${\displaystyle H(\phi _{t},p_{t})={\frac {1}{2}}\int _{U}\int _{U}p_{t}(u)\cdot K(\phi _{t}(m(u)),\phi _{t}(m(v)))p_{t}(v)\,du\,dv}$

and dynamics

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

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

Volume geodesics

For volumes the Hamiltonian

${\displaystyle H(\phi _{t},p_{t})={\frac {1}{2}}\int _{{\mathbb {R} }^{3}}\int _{{\mathbb {R} }^{3}}p_{t}(x)\cdot K(\phi _{t}(x),\phi _{t}(y))p_{t}(y)\,dx\,dy\displaystyle }$

with dynamics

${\displaystyle {\begin{cases}v_{t}=\textstyle \int _{X}\displaystyle K(\cdot ,\phi _{t}(x))p_{t}(x)\,dx\ ,\\{\dot {p}}_{t}(x)=-(Dv_{t})_{|_{\phi _{t}(x)}}^{T}p_{t}(x),x\in {\mathbb {R} }^{3}\end{cases}}}$

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

Cloud software

• MRICloud^[29]