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Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging.The study of images in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form {}, in which images can be dense scalar magnetic resonance or computed axial tomography images.
Two manifolds and are diffeomorphic (usually denoted ) if there is a diffeomorphism from to . Two C r {\displaystyle C^{r}} -differentiable manifolds are C r {\displaystyle C^{r}} -diffeomorphic if there is an r {\displaystyle r} times continuously differentiable bijective map between them whose inverse is also r {\displaystyle r} times ...
A diffeomorphic mapping system is a system designed to map, manipulate, and transfer information which is stored in many types of spatially distributed medical imagery. Diffeomorphic mapping is the underlying technology for mapping and analyzing information measured in human anatomical coordinate systems which have been measured via Medical ...
Recall that a topological manifold is a topological space which is locally homeomorphic to . Differentiable manifolds generalize the notion of smoothness on in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of to the manifold which are used to "pull back" the differential structure from to the ...
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
Problems of this sort are inevitable, since SO(3) is diffeomorphic to real projective space P 3 (R), which is a quotient of S 3 by identifying antipodal points, and charts try to model a manifold using R 3. This explains why, for example, the Euler angles appear to give a variable in the 3-torus, and the unit quaternions in a 3-sphere.
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The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.