Riemannian metric and Lie bracket in computational anatomy information

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Computational anatomy (CA) is the study of shape and form in medical imaging. The study of deformable shapes in computational anatomy rely on high-dimensional diffeomorphism groups ${\displaystyle \operatorname {Diff} _{V}}$ which generate orbits of the form ${\displaystyle {\mathcal {M}}\doteq \{\varphi \cdot m\mid \varphi \in \operatorname {Diff} _{V}\}}$. In CA, this orbit is in general considered a smooth Riemannian manifold since at every point of the manifold ${\displaystyle m\in {\mathcal {M}}}$ there is an inner product inducing the norm ${\displaystyle \|\cdot \|_{m}}$ on the tangent space that varies smoothly from point to point in the manifold of shapes ${\displaystyle m\in {\mathcal {M}}}$. This is generated by viewing the group of diffeomorphisms ${\displaystyle \varphi \in \operatorname {Diff} _{V}}$ as a Riemannian manifold with ${\displaystyle \|\cdot \|_{\varphi }}$, associated to the tangent space at ${\displaystyle \varphi \in \operatorname {Diff} _{V}}$ . This induces the norm and metric on the orbit ${\displaystyle m\in {\mathcal {M}}}$ under the action from the group of diffeomorphisms.