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.
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 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.
[1] The space of shapes are denoted
; the action of the group on shapes is denoted
of the template becomes the space of all shapes,
The central group in CA defined on volumes in
, law of composition of functions
, the diffeomorphic action the flow of the position Most popular are scalar images,
Many different imaging modalities are being used with various actions.
is a three-dimensional vector then Cao et al. [2] 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
, termed frames, vector cross product denoted
then The Frénet frame of three orthonormal vectors,
deforms like a normal to the plane generated by
H is uniquely constrained by the basis being positive and orthonormal.
non-negative symmetric matrices, an action would become
For mapping MRI DTI images[3][4] (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues.
, then the action becomes 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,
In the field of information geometry,[5] 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 (
is non-negative to ensure uniqueness and
Group action of diffeomorphism on
Based on the derivation in,[6] this group action is defined as where