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