“Group actions in computational anatomy”的意思、由来-开放百科全书

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.^[1] 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

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

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

examined actions for mapping MRI images measured via diffusion tensor imaging and represented via there principle eigenvector.

The Fr\\'enet frame of three orthonormal vectors,

deforms as a tangent,

deforms like

For mapping MRI DTI images^[3]^[4] (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues.

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,

. 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 (

) 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,^[6] this group action is defined as

References

1. ^{{Cite journal|last=Miller|first=Michael I.|last2=Younes|first2=Laurent|last3=Trouvé|first3=Alain|date=2014-03-01|title=Diffeomorphometry and geodesic positioning systems for human anatomy|journal=Technology|volume=2|issue=1|pages=36|doi=10.1142/S2339547814500010|issn=2339-5478|pmc=4041578|pmid=24904924}}
2. ^Cao Y1, Miller MI, Winslow RL, Younes, Large deformation diffeomorphic metric mapping of vector fields. IEEE Trans Med Imaging. 2005 Sep;24(9):1216-30.
3. ^{{Cite journal|title = Spatial transformations of diffusion tensor magnetic resonance images|journal = IEEE Transactions on Medical Imaging|date = 2001-11-01|issn = 0278-0062|pmid = 11700739|pages = 1131–1139|volume = 20|issue = 11|doi = 10.1109/42.963816|first = D. C.|last = Alexander|first2 = C.|last2 = Pierpaoli|first3 = P. J.|last3 = Basser|first4 = J. C.|last4 = Gee}}
4. ^{{Cite book|title = Diffeomorphic Matching of Diffusion Tensor Images|journal = Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition|date = 2006-07-05|issn = 1063-6919|pmc = 2920614|pmid = 20711423|pages = 67|volume = 2006|doi = 10.1109/CVPRW.2006.65|first = Yan|last = Cao|first2 = Michael I.|last2 = Miller|first3 = Susumu|last3 = Mori|first4 = Raimond L.|last4 = Winslow|first5 = Laurent|last5 = Younes|isbn = 978-0-7695-2646-1}}
5. ^{{cite book|last1=Amari|first1=S|title=Differential-Geometrical Methods in Statistics|date=1985|publisher=Springer}}
6. ^{{cite journal|last1=Du|first1=J|last2=Goh|first2=A|last3=Qiu|first3=A|title=Diffeomorphic metric mapping of high angular resolution diffusion imaging based on Riemannian structure of orientation distribution functions|journal=IEEE Trans Med Imaging|date=2012|volume=31|issue=5|pages=1021–1033|doi=10.1109/TMI.2011.2178253|pmid=22156979}}