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 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 ,
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.[6] The derivation calculates the perturbation on the vector fields
in terms of the derivative in time of the group perturbation adjusted by the correction of the Lie bracket of vector fields in this function setting involving the Jacobian matrix, unlike the matrix group case:
given by
| | (adjoint-Lie-bracket) |
Proof:
Proving Lie bracket of vector fields take a first order perturbation of the flow at point .
Lie bracket of vector fields
The Lie bracket gives the first order variation of the vector field with respect to first order variation of the flow.
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
| | (Hamilton's Action Integral) |
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[7] associated to the geodesic flow of diffeomorphisms exploits the was a generalized function equation when is a distribution, or generalized function, take the first order variation of the action integral using the adjoint operator for the Lie bracket (adjoint-Lie-bracket) gives for all smooth ,
Using the bracket and gives
| | (EL-General) |
meaning for all smooth
Equation (Euler-general) is the Euler-equation when diffeomorphic shape momentum is a generalized function.
[8]
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 metric.
[9]
[10]
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.[4] From the initial condition then geodesic positioning with respect to the Riemannian metric of Computational anatomy solves for the flow of the Euler–Lagrange equation. Solving the geodesic from the initial condition is termed the Riemannian-exponential, a mapping at identity to the group.
The Riemannian exponential satisfies for initial condition , vector field dynamics ,
- for classical equation on the diffeomorphic shape momentum as a smooth vector with the Euler equation exists in the classical sense as first derived for the density:[11]
- for generalized equation, , then
It is
extended to the entire group,
.
Matching information across coordinate systems is central to computational anatomy. Adding a matching term to the action integral of Equation (Hamilton's action integral)
which represents the target endpoint
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
- Necessary geodesic condition:
- ;\\[4pt]&Av_{1}+{\frac {\partial E(\varphi )}{\partial \varphi _{1}}}=0\end{cases}}}
Proof:[11] The Proof via variation calculus uses the perturbations from above and classic calculus of variation arguments.
Proof via calculus of variations with endpoint energy
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:
- :{\dot {\varphi }}=v_{t}\circ \varphi _{t}}C(\varphi )\doteq {\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}\,dx\,dt+{\frac {1}{2}}\int _{X}|I\circ \varphi _{1}^{-1}(x)-J(x)|^{2}\,dx}
- Necessary geodesic condition:
- ;\\[4pt]&Av_{1}=(I\circ \varphi _{1}^{-1}-J)\nabla (I\circ \varphi _{1}^{-1})\end{cases}}}
Euler–Lagrange geodesic endpoint conditions for landmark matching
The registered landmark matching problem satisfies the dynamical equation for generalized functions with endpoint condition:
- :{\dot {\varphi }}=v_{t}\circ \varphi _{t}}C(\varphi )\doteq {\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}\,dx\,dt+{\frac {1}{2}}\sum _{i}(\varphi _{1}(x_{i})-y_{i})\cdot (\varphi _{1}(x_{i})-y_{i}).}
- Necessary geodesic conditions:
Proof:[11]
The variation requires variation of the inverse generalizes the matrix perturbation of the inverse via giving
giving
P. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.
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, 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.
M.I. Miller, A. Trouve, L. Younes, Geodesic Shooting in Computational Anatomy, IJCV, 2006.
M.I. Miller, A. Trouve, L Younes,
On the Metrics and Euler–Lagrange equations of Computational Anatomy,
Annu. Rev. Biomed. Eng. 2002. 4:375–405
doi:10.1146/annurev.bioeng.4.092101.125733
Copyright °c 2002 by Annual Reviews.