User:Mim.cis/sandbox/Diffeomorphometry
Computational anatomy (CA) is the study of shape and form in medical imaging. The study of deformable shapes in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form . In CA, this orbit is in general considered a smooth Riemannian manifold since at every point of the manifold there is an inner product inducing the norm on the tangent space that varies smoothly from point to point in the manifold of shapes . This is generated by viewing the group of diffeomorphisms as a Riemannian manifold with , associated to the tangent space at . This induces the norm and metric on the orbit under the action from the group of diffeomorphisms.
The diffeomorphisms group generated as Lagrangian and Eulerian flows
The diffeomorphisms in computational anatomy are generated to satisfy the Lagrangian and Eulerian specification of the flow fields, , generated via the ordinary differential equation
Lagrangian flow |
with the Eulerian vector fields in for , with the inverse for the flow given by
Eulerianflow |
and the Jacobian matrix for flows in given as
To ensure smooth flows of diffeomorphisms with inverse, the vector fields must be at least 1-time continuously differentiable in space[1][2] which are modelled as elements of the Hilbert space using the Sobolev embedding theorems so that each element has 3-square-integrable derivatives thusly implies embeds smoothly in 1-time continuously differentiable functions.[1][2] The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:
Diffeomorphism Group |
The Riemannian orbit model
Shapes in Computational Anatomy (CA) are studied via the use of diffeomorphic mapping for establishing correspondences between anatomical coordinate systems. In this setting, 3-dimensional medical images are modelled as diffemorphic transformations of some exemplar, termed the template , resulting in the observed images to be elements of the random orbit model of CA. For images these are defined as , with for charts representing sub-manifolds denoted as .
The Riemannian metric
The orbit of shapes and forms in Computational Anatomy are generated by the group action. This is made into a Riemannian orbit by introducing a metric associated to each point and associated tangent space. For this a metric is defined on the group which induces the metric on the orbit. Take as the metric for Computational anatomy at each element of the tangent space in the group of diffeomorphisms
- ,
with the vector fields modelled to be in a Hilbert space with the norm in the Hilbert space . We model as a reproducing kernel Hilbert space (RKHS) defined by a 1-1, differential operator. For a distribution or generalized function, the linear form determines the norm:and inner product for according to
where the integral is calculated by integration by parts for a generalized function the dual-space. The differential operator is selected so that the Green's kernel associated to the inverse is sufficiently smooth so that the vector fields support 1-continuous derivative.
The right-invariant metric on diffeomorphisms
The metric on the group of diffeomorphisms is defined by the distance as defined on pairs of elements in the group of diffeomorphisms according to
metric-diffeomorphisms |
This distance provides a right-invariant metric of diffeomorphometry,[3][4][5] invariant to reparameterization of space since for all ,
- ^ a b P. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.
- ^ a b A. Trouvé. Action de groupe de dimension infinie et reconnaissance de formes. C R Acad Sci Paris Sér I Math, 321(8):1031– 1034, 1995.
- ^ Miller, M. I.; Younes, L. (2001-01-01). "Group Actions, Homeomorphisms, And Matching: A General Framework". International Journal of Computer Vision. 41: 61–84.
- ^ Cite error: The named reference
pmid24904924
was invoked but never defined (see the help page). - ^ Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2015-01-01). "Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson". Annual Review of Biomedical Engineering. 17 (1): 447–509. doi:10.1146/annurev-bioeng-071114-040601. PMID 26643025.