Computational anatomy (CA) is the study of shape and form in medical imaging.
The study of deformable shapes in CA rely on high-dimensional diffeomorphism groups
The diffeomorphisms in computational anatomy are generated to satisfy the Lagrangian and Eulerian specification of the flow fields,
, generated via the ordinary differential equation with the Eulerian vector fields
To ensure smooth flows of diffeomorphisms with inverse, the vector fields
[1][2] The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm: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 diffeomorphic transformations of some exemplar, termed the template
, resulting in the observed images to be elements of the random orbit model of CA.
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.
Take as the metric for Computational anatomy at each element of the tangent space
as a reproducing kernel Hilbert space (RKHS) defined by a 1-1, differential operator
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 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,[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.
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
, giving the following two Eqns: Equating the above two equations gives the perturbation of the vector field in terms of the Lie bracket adjustment.
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.
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
Equation (Euler-general) is the Euler-equation when diffeomorphic shape momentum is a generalized function.
[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.
then geodesic positioning with respect to the Riemannian metric of Computational anatomy solves for the flow of the Euler–Lagrange equation.
Matching information across coordinate systems is central to computational anatomy.
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.
The earliest large deformation diffeomorphic metric mapping (LDDMM) algorithms solved matching problems associated to images and registered landmarks.
The necessary conditions for the geodesic for image matching takes the form of the classic Equation (EL-Classic) of Euler–Lagrange with boundary condition: The registered landmark matching problem satisfies the dynamical equation for generalized functions with endpoint condition: Proof:[11] The variation