Group actions in computational anatomy

Group actions are central to Riemannian geometry and defining orbits (control theory). Unlike the orbits of Linear algebra which are linear vector spaces, 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,. The 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 an Groups and group action, a classic formulation from differential geometry. The orbit is called the space of shapes and forms.^[1] 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

The central group in CA defined on volumes in ${\mathbb R}^3$ are the the diffeomorphism group $\mathcal{G} \doteq Diff$ 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 $\phi \circ \phi^{-1}(\cdot) =\phi ( \phi^{-1}(\cdot))= id$.

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, Eigen Vectors 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. ^[2] 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 $\varphi \cdot I = (D\varphi\, I D\varphi^{T})\circ \varphi^{-1}$.

For mapping MRI DTI images^[3][4] (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

$\varphi \cdot I \doteq( \lambda_1 \hat e_1 \hat e_1^T + \lambda_2 \hat e_2 \hat e_2^T + \lambda_3 \hat e_3 \hat e_3^T ) \circ \varphi^{-1}$

$\hat e_1 = \frac{ D \varphi e_1 }{\| D \varphi e_1 \|} \ , \hat e_2 = \frac{ D \varphi e_2 - \langle \hat e_1 ,(D \varphi e_2 \rangle \hat e_1 }{\|D \varphi e_2 - \langle \hat e_1 , (D \varphi e_2 \rangle \hat e_1\| } \ , \ \hat e_3 \doteq \hat e_1 \times \hat e_2 \ .$

References

1. Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2014-03-01). "Diffeomorphometry and geodesic positioning systems for human anatomy". Technology 2 (1): 36. doi:10.1142/S2339547814500010. ISSN 2339-5478. PMC 4041578. PMID 24904924.
2. Cao Y1, Miller MI, Winslow RL, Younes, Large deformation diffeomorphic metric mapping of vector fields. IEEE Trans Med Imaging. 2005 Sep;24(9):1216-30.
3. Alexander, D. C.; Pierpaoli, C.; Basser, P. J.; Gee, J. C. (2001-11-01). "Spatial transformations of diffusion tensor magnetic resonance images". IEEE transactions on medical imaging 20 (11): 1131–1139. doi:10.1109/42.963816. ISSN 0278-0062. PMID 11700739.
4. Cao, Yan; Miller, Michael I.; Mori, Susumu; Winslow, Raimond L.; Younes, Laurent (2006-07-05). "Diffeomorphic Matching of Diffusion Tensor Images". Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition 2006: 67. doi:10.1109/CVPRW.2006.65. ISSN 1063-6919. PMC 2920614. PMID 20711423.