Diffeomorphometry

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

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

(Lagrangian flow)

with the Eulerian vector fields ${\displaystyle v\doteq (v_{1},v_{2},v_{3})}$  in ${\displaystyle {\mathbb {R} }^{3}}$  for ${\displaystyle v_{t}={\dot {\varphi }}_{t}\circ \varphi _{t}^{-1},t\in [0,1]}$ . The inverse for the flow is given by ${\displaystyle {\frac {d}{dt}}\varphi _{t}^{-1}=-(D\varphi _{t}^{-1})v_{t},\ \varphi _{0}^{-1}=\operatorname {id} ,}$  and the ${\displaystyle 3\times 3}$  Jacobian matrix for flows in ${\displaystyle \mathbb {R} ^{3}}$  given as ${\displaystyle \ D\varphi \doteq \left({\frac {\partial \varphi _{i}}{\partial x_{j}}}\right).}$ 

To ensure smooth flows of diffeomorphisms with inverse, the vector fields ${\displaystyle {\mathbb {R} }^{3}}$  must be at least 1-time continuously differentiable in space^[10][11] which are modelled as elements of the Hilbert space ${\displaystyle (V,\|\cdot \|_{V})}$  using the Sobolev embedding theorems so that each element ${\displaystyle v_{i}\in H_{0}^{3},i=1,2,3,}$  has 3-square-integrable derivatives thusly implies ${\displaystyle (V,\|\cdot \|_{V})}$  embeds smoothly in 1-time continuously differentiable functions.^[10][11] The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:

${\displaystyle \operatorname {Diff} _{V}\doteq \{\varphi =\varphi _{1}:{\dot {\varphi }}_{t}=v_{t}\circ \varphi _{t},\varphi _{0}=\operatorname {id} ,\int _{0}^{1}\|v_{t}\|_{V}\,dt<\infty \}\ .}$

(Diffeomorphism Group)

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 ${\displaystyle I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in \operatorname {Diff} _{V}\}}$ , with for charts representing sub-manifolds denoted as ${\displaystyle {\mathcal {M}}\doteq \{\varphi \cdot M_{temp}:\varphi \in \operatorname {Diff} _{V}\}}$ .

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 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 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)

For calculating the metric, the geodesics are a dynamical system, the flow of coordinates ${\displaystyle t\mapsto \phi _{t}\in \operatorname {Diff} _{V}}$  and the control the vector field ${\displaystyle t\mapsto v_{t}\in V}$  related via ${\displaystyle {\dot {\phi }}_{t}=v_{t}\cdot \phi _{t},\phi _{0}=\operatorname {id} .}$  The Hamiltonian view ^{[17] [18] [19] [20][21]} reparameterizes the momentum distribution ${\displaystyle Av\in V^{*}}$  in terms of the Hamiltonian momentum, a Lagrange multiplier ${\displaystyle p:{\dot {\phi }}\mapsto (p\mid {\dot {\phi }})}$  constraining the Lagrangian velocity ${\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t}}$ .accordingly:

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

The Pontryagin maximum principle^[17] gives the Hamiltonian ${\displaystyle H(\phi _{t},p_{t})\doteq \max _{v}H(\phi _{t},p_{t},v)\ .}$  The optimizing vector field ${\displaystyle v_{t}\doteq \operatorname {argmax} _{v}H(\phi _{t},p_{t},v)}$  with dynamics ${\displaystyle {\dot {\phi }}_{t}={\frac {\partial H(\phi _{t},p_{t})}{\partial p}},{\dot {p}}_{t}=-{\frac {\partial H(\phi _{t},p_{t})}{\partial \phi }}}$ . Along the geodesic the Hamiltonian is constant:^[22] ${\displaystyle H(\phi _{t},p_{t})=H(\operatorname {id} ,p_{0})={\frac {1}{2}}\int _{X}p_{0}\cdot v_{0}\,dx}$ . The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

${\displaystyle d_{\mathrm {Diff} _{V}}(\operatorname {id} ,\varphi )=\|v_{0}\|_{V}={\sqrt {2H(\operatorname {id} ,p_{0})}}}$ 

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.

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

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

Software suites containing a variety of diffeomorphic mapping algorithms include the following:

• Deformetrica^[23]
• ANTS^[24]
• DARTEL^[25] Voxel-based morphometry(VBM)
• DEMONS^[26]
• LDDMM^[27]
• StationaryLDDMM^[28]

• MRICloud^[29]