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
Main article: 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 (tensors), 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
Further information: Computational_anatomy § Diffusion_tensor_image_matching_in_computational_anatomy, and LDDMM § LDDMM ODF
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, 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
.