Diffeomorphometry

Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging. The study of images in computational anatomy rely on high-dimensional diffeomorphism groups formula_1 which generate orbits of the form formula_2, in which images formula_3 can be dense scalar magnetic resonance or computed axial tomography images. For deformable shapes these are the collection of manifolds formula_4, points, curves and surfaces. The diffeomorphisms move the images and shapes through the orbit according to formula_5 which are defined as the group actions of computational anatomy.
The orbit of shapes and forms is made into a metric space by inducing a metric on the group of diffeomorphisms. The study of metrics on groups of diffeomorphisms and the study of metrics between manifolds and surfaces has been an area of significant investigation. In Computational anatomy, the diffeomorphometry metric measures how close and far two shapes or images are from each other. Informally, the metric is constructed by defining a flow of diffemorphisms formula_6 which connect the group elements from one to another, so for formula_7 then formula_8. The metric between two coordinate systems or diffeomorphisms is then the shortest length or geodesic flow connecting them. The metric on the space associated to the geodesics is given byformula_9. The metrics on the orbits formula_10 are inherited from the metric induced on the diffeomorphism group.
The group formula_1 is thusly made into a smooth Riemannian manifold with Riemannian metric formula_12 associated to the tangent spaces at all formula_13. The Riemannian metric satisfies at every point of the manifold formula_14 there is an inner product inducing the norm on the tangent space formula_15 that varies smoothly across formula_16.
Oftentimes, the familiar Euclidean metric is not directly applicable because the patterns of shapes and images don't form a vector space. In the Riemannian orbit model of Computational anatomy, diffeomorphisms acting on the forms formula_17 don't act linearly. There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric is another. The method used to induce the Riemannian metric is to induce the metric on the orbit of shapes by defining it in terms of the metric length between diffeomorphic coordinate system transformations of the flows. Measuring the lengths of the geodesic flow between coordinates systems in the orbit of shapes is called diffeomorphometry.
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_18, generated via the ordinary differential equation
with the Eulerian vector fields formula_19 in formula_20 for formula_21. The inverse for the flow is given by
formula_22
and the formula_23 Jacobian matrix for flows in formula_24 given as formula_25
To ensure smooth flows of diffeomorphisms with inverse, the vector fields formula_20 must be at least 1-time continuously differentiable in space which are modelled as elements of the Hilbert space formula_27 using the Sobolev embedding theorems so that each element formula_28 has 3-square-integrable derivatives thusly implies formula_27 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_30, resulting in the observed images to be elements of the random orbit model of CA. For images these are defined as formula_31, with for charts representing sub-manifolds denoted as formula_32.
The Riemannian metric.
The orbit of shapes and forms in Computational Anatomy are generated by the group action formula_33 , formula_34. These are made into a Riemannian orbits 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_35 in the group of diffeomorphisms
with the vector fields modelled to be in a Hilbert space with the norm in the Hilbert space formula_27. We model formula_38 as a reproducing kernel Hilbert space (RKHS) defined by a 1-1, differential operator formula_39, where formula_40 is the dual-space. In general, formula_41 is a generalized function or distribution, the linear form associated to the inner-product and norm for generalized functions are interpreted by integration by parts according to for formula_42,
When formula_44, a vector density, formula_45
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 Sobolev embedding theorem arguments were made in demonstrating that 1-continuous derivative is required for smooth flows. The Green's operator generated from the Green's function(scalar case) associated to the differential operator smooths.
For proper choice of formula_46 then formula_47 is an RKHS with the operator formula_48. The Green's kernels associated to the differential operator smooths since for controlling enough derivatives in the square-integral sense the kernel formula_49 is continuously differentiable in both variables implying
(I,J)=\inf_{\phi \in \operatorname{Diff}_V: \phi \cdot I = J } d_{\operatorname{Diff}_V}(id,\phi) \ ;
The distance on shapes and forms, formula_51,
 (M,N) = \inf_{\phi \in \operatorname{Diff}_V: \phi \cdot M = N } d_{\mathrm{Diff}_V}(\operatorname{id},\phi) \ .
The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit.
For calculating the metric, the geodesics are a dynamical system, the flow of coordinates formula_52 and the control the vector field formula_53 related via formula_54 The Hamiltonian view
 reparameterizes the momentum distribution formula_55 in terms of the "Hamiltonian momentum," a Lagrange multiplier formula_56 constraining the Lagrangian velocity formula_57.accordingly:
The Pontryagin maximum principle gives the Hamiltonian formula_59
The optimizing vector field formula_60 with dynamics formula_61. Along the geodesic the Hamiltonian is constant:
formula_62. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:
Landmark or pointset geodesics.
For landmarks, formula_64, the Hamiltonian momentum
with Hamiltonian dynamics taking the form
with
The metric between landmarks formula_68
The dynamics associated to these geodesics is shown in the accompanying figure.
Surface geodesics.
For surfaces, the Hamiltonian momentum is defined across the surface has Hamiltonian
and dynamics
Volume geodesics.
For volumes the Hamiltonian