Several Group Actions in CA

Many different imaging modalities are being used 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}

Cao et al. ^[1] 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={\Big (}{\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}\|}}{\Big )}\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^[2] ^[3] (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}

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

^ 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.
^ 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.
^ 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.

[1] 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.

[2] 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.

[3] 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.

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