Group actions in computational anatomy

Main article: Computational anatomy

Group actions are central to Riemannian geometry and defining orbits (control theory). The orbits of computational anatomy consist of anatomical shapes and medical images; the anatomical shapes are submanifolds of differential geometry consisting of points, curves, surfaces and subvolumes,. This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces. Medical images are scalar and tensor images from medical imaging. The group actions are used to define models of human shape which accommodate variation. These orbits are deformable templates as originally formulated more abstractly in pattern theory.

The orbit model of computational anatomy

The central model of human anatomy in computational anatomy is a Groups and group action, a classic formulation from differential geometry. The orbit is called the space of shapes and forms. The space of shapes are denoted $m\in {\mathcal {M}}$, with the group $({\mathcal {G}},\circ )$ with law of composition $\circ$; the action of the group on shapes is denoted $g\cdot m$, where the action of the group $g\cdot m\in {\mathcal {M}},m\in {\mathcal {M}}$ is defined to satisfy

$(g\circ g^{\prime })\cdot m=g\cdot (g^{\prime }\cdot m)\in {\mathcal {M}}.$

The orbit ${\mathcal {M}}$ of the template becomes the space of all shapes, ${\mathcal {M}}\doteq \{m=g\cdot m_{\mathrm {temp} },g\in {\mathcal {G}}\}$.

Several group actions in computational anatomy

Main article: Computational anatomy

The central group in CA defined on volumes in ${\mathbb {R} }^{3}$ are the diffeomorphism group ${\mathcal {G}}\doteq \mathrm {Diff}$ which are mappings with 3-components $\phi (\cdot )=(\phi _{1}(\cdot ),\phi _{2}(\cdot ),\phi _{3}(\cdot ))$, law of composition of functions $\phi \circ \phi ^{\prime }(\cdot )\doteq \phi (\phi ^{\prime }(\cdot ))$, with inverse $\phi \circ \phi ^{-1}(\cdot )=\phi (\phi ^{-1}(\cdot ))=\operatorname {id}$.

Submanifolds: organs, subcortical structures, charts, and immersions

For sub-manifolds $X\subset {\mathbb {R} }^{3}\in {\mathcal {M}}$, parametrized by a chart or immersion $m(u),u\in U$, the diffeomorphic action the flow of the position

$\phi \cdot m(u)\doteq \phi \circ m(u),u\in U$.

Scalar images such as MRI, CT, PET

Most popular are scalar images, $I(x),x\in {\mathbb {R} }^{3}$, with action on the right via the inverse.

$\phi \cdot I(x)=I\circ \phi ^{-1}(x),x\in {\mathbb {R} }^{3}$.

Oriented tangents on curves, eigenvectors of tensor matrices

Many different imaging modalities are being used with various actions. For images such that $I(x)$ is a three-dimensional vector then

$\varphi \cdot I=((D\varphi )\,I)\circ \varphi ^{-1},$

$\varphi \star I=((D\varphi ^{T})^{-1}I)\circ \varphi ^{-1}$

Tensor matrices

Cao et al. examined actions for mapping MRI images measured via diffusion tensor imaging and represented via there principle eigenvector. For tensor fields a positively oriented orthonormal basis $I(x)=(I_{1}(x),I_{2}(x),I_{3}(x))$ of ${\mathbb {R} }^{3}$, termed frames, vector cross product denoted $I_{1}\times I_{2}$ then

$\varphi \cdot I=\left({\frac {D\varphi I_{1}}{\|D\varphi \,I_{1}\|}},{\frac {(D\varphi ^{T})^{-1}I_{3}\times D\varphi \,I_{1}}{\|(D\varphi ^{T})^{-1}I_{3}\times D\varphi \,I_{1}\|}},{\frac {(D\varphi ^{T})^{-1}I_{3}}{\|(D\varphi ^{T})^{-1}I_{3}\|}}\right)\circ \varphi ^{-1}\ ,$

The Fr\'enet frame of three orthonormal vectors, $I_{1}$ deforms as a tangent, $I_{3}$ deforms like a normal to the plane generated by $I_{1}\times I_{2}$, and $I_{3}$. H is uniquely constrained by the basis being positive and orthonormal.

For $3\times 3$ non-negative symmetric matrices, an action would become $\varphi \cdot I=(D\varphi \,ID\varphi ^{T})\circ \varphi ^{-1}$.

For mapping MRI DTI images (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues. Given eigenelements $\{\lambda _{i},e_{i},i=1,2,3\}$, then the action becomes

$\varphi \cdot I\doteq (\lambda _{1}{\hat {e}}_{1}{\hat {e}}_{1}^{T}+\lambda _{2}{\hat {e}}_{2}{\hat {e}}_{2}^{T}+\lambda _{3}{\hat {e}}_{3}{\hat {e}}_{3}^{T})\circ \varphi ^{-1}$

${\hat {e}}_{1}={\frac {D\varphi e_{1}}{\|D\varphi e_{1}\|}}\ ,{\hat {e}}_{2}={\frac {D\varphi e_{2}-\langle {\hat {e}}_{1},(D\varphi e_{2}\rangle {\hat {e}}_{1}}{\|D\varphi e_{2}-\langle {\hat {e}}_{1},(D\varphi e_{2}\rangle {\hat {e}}_{1}\|}}\ ,\ {\hat {e}}_{3}\doteq {\hat {e}}_{1}\times {\hat {e}}_{2}\ .$

Orientation Distribution Function and High Angular Resolution HARDI

Further information: Computational_anatomy § Diffusion_tensor_image_matching_in_computational_anatomy, and LDDMM ODF

Orientation distribution function (ODF) characterizes the angular profile of the diffusion probability density function of water molecules and can be reconstructed from High Angular Resolution Diffusion Imaging (HARDI). High angular resolution diffusion imaging (HARDI) addresses the well-known limitation of DTI, that is, DTI can only reveal one dominant fiber orientation at each location. HARDI measures diffusion along {\displaystyle n} uniformly distributed directions on the sphere and can characterize more complex fiber geometries. HARDI can be used to reconstruct an orientation distribution function (ODF) that characterizes the angular profile of the diffusion probability density function of water molecules.

The ODF is a function defined on a unit sphere, ${\mathbb {S} }^{2}$.

Dense LDDMM ODF matching takes the HARDI data as ODF at each voxel and solves the LDDMM variational problem in the space of ODF. In the field of information geometry, the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. This metric defines the distance between two ODF functions $\psi _{1},\psi _{2}\in \Psi$ as

${\begin{aligned}\rho (\psi _{1},\psi _{2})=\|\log _{\psi _{1}}(\psi _{2})\|_{\psi _{1}}=\cos ^{-1}\langle \psi _{1},\psi _{2}\rangle =\cos ^{-1}\left(\int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi _{1}({\bf {s}})\psi _{2}({\bf {s}})d{\bf {s}}\right),\end{aligned}}$

where $\langle \cdot ,\cdot \rangle$ is the normal dot product between points in the sphere under the $\mathrm {L} ^{2}$ metric.

Based on this metric of ODF, we define a variational problem assuming that two ODF volumes can be generated from one to another via flows of diffeomorphisms $\phi _{t}$, which are solutions of ordinary differential equations ${\dot {\phi }}_{t}=v_{t}(\phi _{t}),t\in [0,1],$ starting from the identity map $\phi _{0}={id}$. Denote the action of the diffeomorphism on template as $\phi _{1}\cdot \psi _{\mathrm {temp} }({\bf {s}},x)$, ${\bf {s}}\in {{\mathbb {S} }^{2}}$, $x\in X$ are respectively the coordinates of the unit sphere, ${{\mathbb {S} }^{2}}$ and the image domain, with the target indexed similarly, $\psi _{\mathrm {targ} }({\bf {s}},x)$,${\bf {s}}\in {{\mathbb {S} }^{2}}$,$x\in X$. The group action of the diffeomorphism on the template is given according to

$\phi _{1}\cdot \psi \doteq$. $(D\phi _{1})\psi \circ \phi _{1}^{-1}$,

where $(D\phi _{1})$ is the Jacobian of the affined transformed ODF and is defined as

${\begin{aligned}(D\phi _{1})\psi \circ \phi _{1}^{-1}(x)={\sqrt {\frac {\det {{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}}{\left\|{{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}{\bf {s}}\right\|^{3}}}}\quad \psi \left({\frac {(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}}{\|(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}\|}},\phi _{1}^{-1}(x)\right).\end{aligned}}$