User:Mim.cis/sandbox/Group-Actions-Manifolds-CA

=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. 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 $$ \varphi \cdot I = (d\varphi\, I d\varphi^{T})\circ \varphi^{-1} $$.

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 \. $$