diffeomorphometry

The diffeomorphisms in computational anatomy are generated to satisfy the Lagrangian and Eulerian specification of the flow fields, $\backslash varphi\_t,\; t\; \backslash in\; [0,1]$, generated via the ordinary differential equation{{NumBlk|:|$$

\frac{d}{dt} \varphi_t = v_t \circ \varphi_t , \ \varphi_0 = \operatorname{id}; |{{EquationRef|Lagrangian flow}}}}

with the Eulerian vector fields $v\; \backslash doteq\; (v\_1,v\_2,v\_3)$ in $\{\backslash mathbb\; R\}^3$ for $v\_t\; =\; \backslash dot\; \backslash varphi\_t\; \backslash circ\; \backslash varphi\_t^\{-1\},\; t\; \backslash in\; [0,1]$. The inverse for the flow is given by

\frac{d}{dt} \varphi_t^{-1} = -(D \varphi_t^{-1}) v_t, \ \varphi_0^{-1} = \operatorname{id},

and the $3\; \backslash times\; 3$ Jacobian matrix for flows in $\backslash mathbb\{R\}^3$ given as $\backslash \; D\backslash varphi\; \backslash doteq\; \backslash left(\backslash frac\{\backslash partial\; \backslash varphi\_i\}\{\backslash partial\; x\_j\}\backslash right).$

To ensure smooth flows of diffeomorphisms with inverse, the vector fields $\{\backslash mathbb\; R\}^3$ must be at least 1-time continuously differentiable in spaceP. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.A. Trouvé. Action de groupe de dimension infinie et reconnaissance de formes. C R Acad Sci Paris Sér I Math, 321(8):1031– 1034, 1995. which are modelled as elements of the Hilbert space $(V,\; \backslash |\; \backslash cdot\; \backslash |\_V\; )$ using the Sobolev embedding theorems so that each element $v\_i\; \backslash in\; H\_0^3,\; i=1,2,3,$ has 3-square-integrable derivatives thusly implies $(V,\; \backslash |\; \backslash cdot\; \backslash |\_V\; )$ embeds smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:{{NumBlk|:|$$

\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 \} \ .

|{{EquationRef|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 $I\_\{temp\}$, resulting in the observed images to be elements of the random orbit model of CA. For images these are defined as $I\; \backslash in\; \backslash mathcal\; \{I\}$

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

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

:$\backslash |\; \backslash dot\; \backslash varphi\; \backslash |\_\backslash varphi\; \backslash doteq\; \backslash |\; \backslash dot\; \backslash varphi\; \backslash circ\; \backslash varphi^\{-1\}\; \backslash |\_V=\backslash |\; v\; \backslash |\_V,$

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

:$\backslash langle\; v\; ,\; w\; \backslash rangle\_V\; \backslash doteq\; \backslash int\_X\; A\; v\; \backslash cdot\; w\; \backslash ,\; dx,\; \backslash \; \backslash |\; v\backslash |\_V^2\; \backslash doteq\; \backslash int\_X\; A\; v\; \backslash cdot\; v\; \backslash ,\; dx,\; \backslash \; v,w\; \backslash in\; V\; \backslash \; .$

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

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 $A$ then $(V,\backslash |\; \backslash cdot\; \backslash |\_V)$ is an RKHS with the operator $K\; =\; A^\{-1\}:\; V^*\; \backslash rightarrow\; V$. The Green's kernels associated to the differential operator smooths since for controlling enough derivatives in the square-integral sense the kernel $k(\backslash cdot,\backslash cdot)$

is continuously differentiable in both variables implying

:$K\; Av\; (x)\_i\; \backslash doteq\; \backslash sum\_j\; \backslash int\_\{\{\backslash mathbb\; R\}^3\}\; k\_\{ij\}(x,y)\; Av\_j(y)\; \backslash ,dy\; \backslash in\; V\; \backslash \; .$

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

{{NumBlk|:|$$

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} \ .

|{{EquationRef|metric-diffeomorphisms}}}}This distance provides a right-invariant metric of diffeomorphometry,{{Cite journal|last1=Miller|first1=M. I.|last2=Younes|first2=L.|date=2001-01-01|title=Group Actions, Homeomorphisms, And Matching: A General Framework|journal=International Journal of Computer Vision|volume=41|pages=61–84|citeseerx=10.1.1.37.4816|doi=10.1023/A:1011161132514|s2cid=15423783}}{{cite journal|pmid=24904924|pmc=4041578|year=2014|last1=Miller|first1=M. I|title=Diffeomorphometry and geodesic positioning systems for human anatomy|journal=Technology|volume=2|issue=1|pages=36|last2=Younes|first2=L|last3=Trouvé|first3=A|doi=10.1142/S2339547814500010}}{{Cite journal|last1=Miller|first1=Michael I.|last2=Trouvé|first2=Alain|last3=Younes|first3=Laurent|date=2015-01-01|title=Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson|journal=Annual Review of Biomedical Engineering|volume=17|issue=1|pages=447–509|doi=10.1146/annurev-bioeng-071114-040601|pmid=26643025}} invariant to reparameterization of space since for all $\backslash phi\; \backslash in\; \backslash operatorname\{Diff\}\_V$,

:$d\_\{\backslash operatorname\{Diff\}\_V\}(\backslash psi,\; \backslash varphi)\; =\; d\_\{\backslash operatorname\{Diff\}\_V\}(\backslash psi\; \backslash circ\; \backslash phi,\; \backslash varphi\; \backslash circ\; \backslash phi).$

The distance on images,{{Cite journal|title = Group Actions, Homeomorphisms, And Matching: A General Framework|journal = International Journal of Computer Vision|date = 2001-01-01|pages = 61–84|volume = 41|first1 = M. I.|last1 = Miller|first2 = L.|last2 = Younes|citeseerx = 10.1.1.37.4816|doi = 10.1023/A:1011161132514|s2cid = 15423783}} $d\_\{\backslash mathcal\{I\}\}:\backslash mathcal\{I\}\; \backslash times\; \backslash mathcal\{I\}\backslash rightarrow\; \backslash R^+$,

{{NumBlk||$$

d_{\mathcal{I}}(I,J)=\inf_{\phi \in \operatorname{Diff}_V: \phi \cdot I = J } d_{\operatorname{Diff}_V}(id,\phi) \ ;

|{{EquationRef|metric-shapes-forms}}}}

The distance on shapes and forms,{{Cite journal|last1=Miller|first1=Michael I.|last2=Younes|first2=Laurent|last3=Trouvé|first3=Alain|date=March 2014|title=Diffeomorphometry and geodesic positioning systems for human anatomy|journal=Technology|volume=2|issue=1|pages=36|doi=10.1142/S2339547814500010|issn=2339-5478|pmc=4041578|pmid=24904924}} $d\_\{\backslash mathcal\{M\}\}:\backslash mathcal\{M\}\; \backslash times\; \backslash mathcal\{M\}\backslash rightarrow\; \backslash R^+$,

{{NumBlk||$$

d_{\mathcal{M}} (M,N) = \inf_{\phi \in \operatorname{Diff}_V: \phi \cdot M = N } d_{\mathrm{Diff}_V}(\operatorname{id},\phi) \ .

|{{EquationRef|metric-shapes-forms}}}}

For calculating the metric, the geodesics are a dynamical system, the flow of coordinates $t\; \backslash mapsto\; \backslash phi\_t\; \backslash in\; \backslash operatorname\{Diff\}\_V$ and the control the vector field $t\; \backslash mapsto\; v\_t\; \backslash in\; V$ related via $\backslash dot\; \backslash phi\_t\; =\; v\_t\; \backslash cdot\; \backslash phi\_t,\backslash phi\_0=\backslash operatorname\{id\}.$ The Hamiltonian view

{{Cite journal|title = Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'arcy Thompson |journal = Annual Review of Biomedical Engineering|date = 2015-01-01|volume = 17|issue = 1|doi = 10.1146/annurev-bioeng-071114-040601|first1 = Michael I.|last1 = Miller|first2 = Alain|last2 = Trouvé|first3 = Laurent|last3 = Younes|pages = 447–509|pmid=26643025}}

Glaunès J, Trouvé A, Younes L. 2006. [https://pdfs.semanticscholar.org/8e91/7605a2ebef438de633ca29898afb0ebea2c9.pdf Modeling planar shape variation via Hamiltonian flows of curves].

In Statistics and Analysis of Shapes, ed. H Krim, A Yezzi Jr, pp. 335–61. Model. Simul. Sci. Eng. Technol.

Boston: Birkhauser

Arguillère S, Trélat E, Trouvé A, Younes L. 2014. Shape deformation analysis from the optimal control

viewpoint. {{arXiv|1401.0661}} [math.OC]

{{cite journal | doi = 10.1142/S2339547814500010 | pmid=24904924 | pmc=4041578 | volume=2 | title=Diffeomorphometry and geodesic positioning systems for human anatomy | year=2014 | journal=Technology (Singap World Sci) | page=36 | last1 = Miller | first1 = MI | last2 = Younes | first2 = L | last3 = Trouvé | first3 = A| issue=1 }}

{{Cite journal|title = An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach |journal = Applied and Computational Harmonic Analysis|date = 2007-07-01|pages = 74–113|volume = 23|series = Special Issue on Mathematical Imaging|issue = 1|doi = 10.1016/j.acha.2006.07.004|first1 = Peter W.|last1 = Michor|first2 = David|last2 = Mumford|arxiv = math/0605009|s2cid = 732281}} reparameterizes the momentum distribution $Av\; \backslash in\; V^*$ in terms of the Hamiltonian momentum, a Lagrange multiplier $p:\; \backslash dot\; \backslash phi\; \backslash mapsto\; (p\backslash mid\backslash dot\; \backslash phi)$ constraining the Lagrangian velocity $\backslash dot\; \backslash phi\_t\; =\; v\_t\; \backslash circ\; \backslash phi\_t$.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 .

The Pontryagin maximum principle gives the Hamiltonian $H(\backslash phi\_t,p\_t)\; \backslash doteq\; \backslash max\_v\; H(\; \backslash phi\_t,\; p\_t,v)\; \backslash \; .$

The optimizing vector field $v\_t\; \backslash doteq\; \backslash operatorname\{argmax\}\_v\; H(\backslash phi\_t,p\_t,v)$ with dynamics $$

\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:{{Cite journal|title = Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson |journal = Annual Review of Biomedical Engineering|date = 2015-01-01|pmid = 26643025|pages = 447–509|volume = 17|issue = 1|doi = 10.1146/annurev-bioeng-071114-040601|first1 = Michael I.|last1 = Miller|first2 = Alain|last2 = Trouvé|first3 = Laurent|last3 = Younes}}

$H(\backslash phi\_t,p\_t)\; =\; H(\backslash operatorname\{id\},p\_0)=\backslash frac\{1\}\{2\}\; \backslash int\_X\; p\_0\; \backslash cdot\; v\_0\; \backslash ,\; dx$

. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

:$d\_\{\backslash mathrm\{Diff\}\_V\}(\backslash operatorname\{id\},\backslash varphi)\; =\backslash |\; v\_0\; \backslash |\_V\; =\; \backslash sqrt\{2H(\backslash operatorname\{id\},p\_0)\}$

For landmarks, $x\_i,\; i=1,\backslash dots,n$, the Hamiltonian momentum

: $p(i),\; i=1,\backslash dots,n$

with Hamiltonian dynamics taking the form

: $H(\backslash phi\_t,p\_t)\; =\backslash frac\{1\}\{2\}\backslash textstyle\; \backslash sum\_j\; \backslash sum\_i\; \backslash displaystyle\; p\_t(i)\backslash cdot\; K(\backslash phi\_t\; (x\_i),\backslash phi\_t\; (x\_j))\; p\_t(j)$

with

:$$

\begin{cases}

v_t = \textstyle \sum_i \displaystyle K(\cdot, \phi_t (x_i)) p_t(i) , \ \\

\dot p_t (i) = - (Dv_t)^T_{|_{\phi_t(x_i)}} p_t(i), i=1,2,\dots, n

\\

\end{cases}

The metric between landmarks $$

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

: $H(\backslash phi\_t,p\_t)\; =\backslash frac\{1\}\{2\}\; \backslash int\_U\; \backslash int\_U\; p\_t(u)\backslash cdot\; K(\backslash phi\_t\; (m(u)),\; \backslash phi\_t\; (m(v)))\; p\_t(v)\; \backslash ,\; du\; \backslash ,\; dv$

and 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)^T_{|_{\phi_t(m(u))} } p_t(u), u \in U

\end{cases}

:The metric between surface coordinates $d^2\; =\; (p\_0\; \backslash mid\; v\_0)\; =\backslash int\_U\; p\_0(u)\; \backslash cdot\; \backslash int\_U\; K(m(u),\; m(u^\backslash prime))\; p\_0(u^\backslash prime)\; \backslash ,\; du\; \backslash ,\; du^\backslash prime$

For volumes the Hamiltonian

: $H(\backslash phi\_t,p\_t)\; =\; \backslash frac\{1\}\{2\}\backslash int\_\{\{\backslash mathbb\; R\}^3\}\; \backslash int\_\{\{\backslash mathbb\; R\}^3\}\; p\_t(x)\backslash cdot\; K(\backslash phi\_t(x),\backslash phi\_t(y))\; p\_t(y)\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash displaystyle$

with dynamics

: $$

\begin{cases}

v_t=\textstyle \int_X \displaystyle K(\cdot, \phi_t(x))p_t(x)\,dx \ ,

\\

\dot p_t(x) = - (Dv_t)^T_{|_{\phi_t(x)} } p_t(x), x \in {\mathbb R}^3

\end{cases}

: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.

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

• Deformetrica{{cite report|title=Software - Stanley Durrleman|url=https://raweb.inria.fr/rapportsactivite/RA2015/aramis/uid32.html}}
• ANTS{{Cite journal|last1=Avants|first1=Brian B.|last2=Tustison|first2=Nicholas J.|last3=Song|first3=Gang|last4=Cook|first4=Philip A.|last5=Klein|first5=Arno|last6=Gee|first6=James C.|date=2011-02-01|title=A Reproducible Evaluation of ANTs Similarity Metric Performance in Brain Image Registration|journal=NeuroImage|volume=54|issue=3|pages=2033–2044|doi=10.1016/j.neuroimage.2010.09.025|issn=1053-8119|pmc=3065962|pmid=20851191}}
• DARTEL{{Cite journal|title = A fast diffeomorphic image registration algorithm |journal = NeuroImage|date = 2007-10-15|pmid = 17761438|pages = 95–113|volume = 38|issue = 1|doi = 10.1016/j.neuroimage.2007.07.007|first = John|last = Ashburner|s2cid = 545830}} Voxel-based morphometry(VBM)
• DEMONS{{cite web|title = Software - Tom Vercauteren|url = https://sites.google.com/site/tomvercauteren/software|website = sites.google.com|accessdate = 2015-12-11}}
• LDDMM{{Cite journal|last1=Beg|first1=M. Faisal|last2=Miller|first2=Michael I.|last3=Trouvé|first3=Alain|last4=Younes|first4=Laurent|date=2005-02-01|title=Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms|journal=International Journal of Computer Vision|language=en|volume=61|issue=2|pages=139–157|doi=10.1023/B:VISI.0000043755.93987.aa|s2cid=17772076|issn=0920-5691}}
• StationaryLDDMM{{Cite web|url=https://www.researchgate.net/publication/33419970|title=Comparing algorithms for diffeomorphic registration: Stationary LDDMM and Diffeomorphic Demons (PDF Download Available)|website=ResearchGate|language=en|access-date=2017-12-02}}

• MRICloud{{cite web|url=http://www.mricloud.org|title=MRICloud|website=|publisher=The Johns Hopkins University|access-date=1 January 2015}}

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