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.
Contents
- The orbit model of computational anatomy
- Several group actions in computational anatomy
- Submanifolds organs subcortical structures charts and immersions
- Scalar images such as MRI CT PET
- Oriented tangents on curves eigenvectors of tensor matrices
- Tensor matrices
- Orientation Distribution Function and High Angular Resolution HARDI
- References
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
The orbit
Several group actions in computational anatomy
The central group in CA defined on volumes in
Submanifolds: organs, subcortical structures, charts, and immersions
For sub-manifolds
Scalar images such as MRI, CT, PET
Most popular are scalar images,
Oriented tangents on curves, eigenvectors of tensor matrices
Many different imaging modalities are being used with various actions. For images such that
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
The Fr\'enet frame of three orthonormal vectors,
For
For mapping MRI DTI images (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues. Given eigenelements
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,
Denote diffeomorphic transformation as
where