Group actions in computational anatomy

Group actions are central to Riemannian geometry and defining orbits (control theory). 
The orbits of computational anatomy consist of anatomical shapes and medical images; the anatomical shapes are submanifolds of differential geometry consisting of points, curves, surfaces and subvolumes.
This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces. Medical images are scalar and tensor images from medical imaging. The group actions are used to define models of human shape which accommodate variation. These orbits are deformable templates as originally formulated more abstractly in pattern theory.
The orbit model of computational anatomy.
The central model of human anatomy in computational anatomy is a Groups and group action, a classic formulation from differential geometry. The orbit is called the space of shapes and forms. The space of shapes are denoted formula_1, with the group formula_2 with law of composition formula_3; the action of the group on shapes is denoted formula_4, where the action of the group formula_5 is defined to satisfy 
The orbit formula_7 of the template becomes the space of all shapes, formula_8.
Several group actions in computational anatomy.
The central group in CA defined on volumes in formula_9 are the diffeomorphism group formula_10 which are mappings with 3-components formula_11, law of composition of functions formula_12, with inverse formula_13.
Submanifolds: organs, subcortical structures, charts, and immersions.
For sub-manifolds formula_14, parametrized by a chart or immersion formula_15, the diffeomorphic action the flow of the position
Scalar images such as MRI, CT, PET.
Most popular are scalar images, formula_17, with action on the right via the inverse.
Oriented tangents on curves, eigenvectors of tensor matrices.
Many different imaging modalities are being used with various actions. For images such that formula_19 is a three-dimensional vector then
Tensor matrices.
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 
formula_22
of formula_23, termed frames, vector cross product denoted formula_24 then
The Frénet frame of three orthonormal vectors, formula_26 deforms as a tangent, formula_27 deforms like
a normal to the plane generated by formula_28, and formula_27. H is uniquely constrained by the
basis being positive and orthonormal.
For formula_30 non-negative symmetric matrices, an action would become formula_31.
For mapping MRI DTI images (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues. 
Given eigenelements
formula_32, then the action becomes
Orientation Distribution Function and High Angular Resolution HARDI.
Orientation distribution function (ODF) characterizes the angular profile of the diffusion probability density function of water molecules and can be reconstructed from High Angular Resolution Diffusion Imaging (HARDI). The ODF is a probability density function defined on a unit sphere, formula_35. In the field of information geometry, the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. For the purpose of LDDMM ODF mapping, the square-root representation is chosen because it is one of the most efficient representations found to date as the various Riemannian operations, such as geodesics, exponential maps, and logarithm maps, are available in closed form. In the following, denote square-root ODF (formula_36) as formula_37, where formula_37 is non-negative to ensure uniqueness and formula_39.
Denote diffeomorphic transformation as formula_40. Group action of diffeomorphism on formula_37, formula_42, needs to guarantee the non-negativity and formula_43. Based on the derivation in, this group action is defined as
where formula_45 is the Jacobian of formula_46.