Riemannian metric and Lie bracket in computational anatomy

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 formula_1 which generate orbits of the form formula_2. In CA, this orbit is in general considered a smooth Riemannian manifold
since at every point of the manifold formula_3 there is an inner product inducing the norm formula_4 on the tangent space
that varies smoothly from point to point in the manifold of shapes formula_3. This is generated by viewing the
group of diffeomorphisms formula_1 as a Riemannian manifold with formula_7, associated to the tangent space at formula_8 . This induces the norm and metric on the orbit formula_3 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, formula_10, generated via the ordinary differential equation
with the Eulerian vector fields formula_11 in formula_12 for formula_13, with the inverse for the flow given by
and the formula_14 Jacobian matrix for flows in formula_15 given as formula_16
To ensure smooth flows of diffeomorphisms with inverse, the vector fields formula_12 must be at least 1-time continuously differentiable in space which are modelled as elements of the Hilbert space formula_18 using the Sobolev embedding theorems so that each element formula_19 has 3-square-integrable derivatives thusly implies formula_18 embeds smoothly in 1-time continuously differentiable functions. The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:
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 formula_21, resulting in the observed images to be elements of the random orbit model of CA. For images these are defined as formula_22, with for charts representing sub-manifolds denoted as formula_23.
The Riemannian metric.
The orbit of shapes and forms in Computational Anatomy are generated by the group actionformula_24. 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 formula_25 in the group of diffeomorphisms 
with the vector fields modelled to be in a Hilbert space with the norm in the Hilbert space formula_18. We model formula_28 as a reproducing kernel Hilbert space (RKHS) defined by a 1-1, differential operatorformula_29. For formula_30 a distribution or generalized function, the linear form formula_31 determines the norm:and inner product for formula_32 according to 
where the integral is calculated by integration by parts for formula_34 a generalized function formula_35 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
This distance provides a right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all formula_1,
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 formula_38 on the vector fields
formula_39 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:
Proof:
Proving Lie bracket of vector fields take a first order perturbation of the flow at point formula_1.
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 whenformula_43 is a distribution, or generalized function, take the first order variation of the action integral using the adjoint operator for the Lie bracket () gives for all smooth formula_44, 
Using the bracket formula_46 and formula_47 gives
meaning for all smooth formula_48
Equation () 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 formula_50 metric.
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 formula_51 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 formula_51 is termed the Riemannian-exponential, a mapping formula_53 at identity to the group.
The Riemannian exponential satisfies formula_54 for initial condition formula_55, vector field dynamics formula_56, 
It is
extended to the entire group,
formula_62.
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 formula_63 to the action integral of Equation ()
which represents the target endpoint 
The endpoint term adds a boundary condition for the Euler–Lagrange equation ()
which gives the Euler equation with boundary term. Taking the variation gives
Proof:<ref>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
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