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
Contents
- The diffeomorphisms group generated as Lagrangian and Eulerian flows
- The Riemannian orbit model
- The Riemannian metric
- The right invariant metric on diffeomorphisms
- The Lie bracket in the group of diffeomorphisms
- The generalized EulerLagrange equation for the metric on diffeomorphic flows
- Riemannian exponential for positioning
- The variation problem for matching or registering coordinate system information in computational anatomy
- EulerLagrange geodesic endpoint conditions for image matching
- EulerLagrange geodesic endpoint conditions for landmark matching
- References
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,
with the Eulerian vector fields
and the
To ensure smooth flows of diffeomorphisms with inverse, the vector fields
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
The Riemannian metric
The orbit of shapes and forms in Computational Anatomy are generated by the group action
with the vector fields modelled to be in a Hilbert space with the norm in the Hilbert space
where the integral is calculated by integration by parts for
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
This distance provides a right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all
The Lie bracket in the group of diffeomorphisms
The Lie bracket gives the adjustment of the velocity term resulting from a perturbation of the motion in the setting of curved spaces. Using Hamilton's principle of least-action derives the optimizing flows as a critical point for the action integral of the integral of the kinetic energy. The Lie bracket for vector fields in Computational Anatomy was first introduced in Miller, Trouve and Younes. The derivation calculates the perturbation
Proof: Proving Lie bracket of vector fields take a first order perturbation of the flow at point
The Lie bracket gives the first order variation of the vector field with respect to first order variation of the flow.
The generalized Euler–Lagrange equation for the metric on diffeomorphic flows
The Euler–Lagrange equation can be used to calculate geodesic flows through the group which form the basis for the metric. The action integral for the Lagrangian of the kinetic energy for Hamilton's principle becomes
The action integral in terms of the vector field corresponds to integrating the kinetic energy
The shortest paths geodesic connections in the orbit are defined via Hamilton's Principle of least action requires first order variations of the solutions in the orbits of Computational Anatomy which are based on computing critical points on the metric length or energy of the path. The original derivation of the Euler equation associated to the geodesic flow of diffeomorphisms exploits the was a generalized function equation when
Using the bracket
meaning for all smooth
Equation (Euler-general) is the Euler-equation when diffeomorphic shape momentum is a generalized function. This equation has been called EPDiff, Euler–Poincare equation for diffeomorphisms and has been studied in the context of fluid mechanics for incompressible fluids with
Riemannian exponential for positioning
In the random orbit model of Computational anatomy, the entire flow is reduced to the initial condition which forms the coordinates encoding the diffeomorphism, as well as providing the means of positioning information in the orbit. This was first terms a geodesic positioning system in Miller, Trouve, and Younes. From the initial condition
The Riemannian exponential satisfies
It is extended to the entire group,
The variation problem for matching or registering coordinate system information in computational anatomy
Matching information across coordinate systems is central to computational anatomy. Adding a matching term
The endpoint term adds a boundary condition for the Euler–Lagrange equation (EL-General) which gives the Euler equation with boundary term. Taking the variation gives
Proof: The Proof via variation calculus uses the perturbations from above and classic calculus of variation arguments.
Euler–Lagrange geodesic endpoint conditions for image matching
The earliest large deformation diffeomorphic metric mapping (LDDMM) algorithms solved matching problems associated to images and registered landmarks. are in a vector spaces. The image matching geodesic equation satisfies the classical dynamical equation with endpoint condition. The necessary conditions for the geodesic for image matching takes the form of the classic Equation (EL-Classic) of Euler–Lagrange with boundary condition:
Euler–Lagrange geodesic endpoint conditions for landmark matching
The registered landmark matching problem satisfies the dynamical equation for generalized functions with endpoint condition:
Proof:
The variation