Group actions in computational anatomy


Group actions are central to Riemannian geometry and defining orbits.
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, with the group with law of composition ; the action of the group on shapes is denoted, where the action of the group is defined to satisfy
The orbit of the template becomes the space of all shapes,.

Several group actions in computational anatomy

The central group in CA defined on volumes in are the diffeomorphism group which are mappings with 3-components, law of composition of functions, with inverse.

Submanifolds: organs, subcortical structures, charts, and immersions

For sub-manifolds, parametrized by a chart or immersion, the diffeomorphic action the flow of the position

Scalar images such as MRI, CT, PET

Most popular are scalar images,, 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 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
of, termed frames, vector cross product denoted then
The Fr\'enet frame of three orthonormal vectors, deforms as a tangent, deforms like
a normal to the plane generated by, and. H is uniquely constrained by the
basis being positive and orthonormal.
For non-negative symmetric matrices, an action would become.
For mapping MRI DTI images, then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues.
Given eigenelements
, then the action becomes

Orientation Distribution Function and High Angular Resolution HARDI

Orientation distribution function characterizes the angular profile of the diffusion probability density function of water molecules and can be reconstructed from High Angular Resolution Diffusion Imaging. The ODF is a probability density function defined on a unit sphere,. 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 as, where is non-negative to ensure uniqueness and.
Denote diffeomorphic transformation as. Group action of diffeomorphism on, , needs to guarantee the non-negativity and. Based on the derivation in, this group action is defined as
where is the Jacobian of.