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This section concerns the infinite dimensional diffeomorphism group. It is not strictly relevant to this thesis, but the diffeomorphism group is commonly used for image reg- istration. The purpose of this section is to give an overview of diffeomorphic image registration for completeness.

The first attempt at computing a high-dimensional non-rigid registration was given by Broit, Bajcsy and co-workers in [7, 8, 14]. In this setting, the transformation ϕis generated by its linear approximation in a neighbourhood of the source,ϕ(x) =x+u(x) orϕ−1(x) =x−u(x), whereu: Ω→R2 is the displacement vector field. The distance functionE2 measures the square of theL2 difference between the images.

The optimal transformation is the one which minimizesE2among all possible solutions

and has the highest smoothness, where the smoothness ofuis measured byE1as follows,

E1(u) =Lu22,

where L is an operator on the space of vector fields. Commonly, L is chosen as L = (−α+γ)In_×n, where is the Laplacian operator and I is the identity matrix. In

the variational setting the optimal vector field is given by the minimisation of

arg min

u E1+

1

σ2E2,

where σ is a parameter. This approach is known as small deformation matching [2], because for sufficiently smallu,ϕis a diffeomorphism. One limitation of this approach is that the transformation does not necessarily lie in the diffeomorphism group for larger u. To overcome this limitation, the large deformation model was developed [19]. In this model, the transformation ϕ is the endpoint of a path φt in the space

of transformations, where φt is the flow of time-dependent vector field vt : Ω → Rn,

t∈[0,1] and is specified by the ODE, ˙φt=vt(φt), withφ0 =id(idis the identity map)

and the endpoint ϕ=φ1 =φ0+₀1vt(φt)dt. The large deformation algorithm is given

as follows (see [75] for more details):

• Start with ϕ0 =id.

• Solve the evolution equation :

∂tϕ(t, y) =−2

Ω(I◦ϕ

−1_{(t, x)−J)∇I◦ϕ}−1_{(t, x)K(ϕ(t, y), x)dx.}

HereK is areproducing kernel of Hilbert space and is commonly chosen to be a Gaus- sian, i.e. K(x, y) = exp(−αx−y2). This algorithm is also known as greedy image matching.

One of the principal aspects of diffeomorphic image registration is to measure the distance between images in the diffeomorphism group. The diffeomorphism group is also an infinite dimensional manifold and can be equipped with a Riemannian metric. The large diffeomorphic method introduced by [19] connects the source to the target, but the orbit is not the shortest path. A method called Large Deformation Diffeomorphic Metric Mapping (LDDMM) is introduced in [9] such that its solution is similar to the flow of the large deformation model, but in contrast to the large deformation method, the path connecting the source and target is the shortest path. The transformation

is determined via the basic variational problem that in the space of smooth velocity vector fieldsV on domain Ω takes the form:

ˆ v= arg min v: ˙ϕt=vt(ϕt) 1 0 vt 2 Vdt+I0◦ϕ1−1−I122 , (1.8)

whereI0◦ϕ−11 =I1, ˙ϕt=vt(ϕt) andvtV =Lv2, whereL is a differential operator

to enforce the vector field to be smooth. The second term in Equation (1.8) enforces matching of the images with .2₂, which is the squared-error norm. The length of the shortest path is inf₀1vtVdt, which defines a metric on the image orbit.

Another method for diffeomorphic image registration is the Stationary velocity field

(SVF) method. In this setting, the diffeomorphism is parameterized by the one- parameter subgroups generated by stationary velocity fields through the Lie group exponential. In contrast to LDDMM in which the diffeomorphism lies on a geodesic, in the SVF method the diffeomorphism may not lie on a geodesic, because one-parameter subgroups may not be geodesic. (On geodesics, the acceleration is zero.) To measure the acceleration on geodesics an affine connection (∇) between the tangent spaces is defined. If the velocityX = ˙γ(0) is transported along a curveγ by an affine connection parallel, so that ∇γ˙X = 0, then γ is a geodesic. An affine connection is called the Levi-Civita connection if the parallel transport is geodesic using the Riemannian met- ric. In [39] they investigated when one-parameter subgroups coincide with Riemannian geodesics. They found that with the Cartan connection one-parameter subgroups are Riemannian geodesics, and based on this, they proposed a diffeomorphic registration method.

Image registration with the infinite dimensional diffeomorphism group has been well studied during the last decade. However, relatively little attention has been devoted to image registration in finite dimensional groups other than the similarity group. This is the focus of this thesis. We present the following motivations for the study of image registration by finite dimensional groups.

Thompson’s idea of simplest transformations: The remarkable idea of Thomp- son that simple groups are to be preferred is a strong justification to study finite dimensional groups for image registration.

Few groups: As far as we are aware, only a few finite dimensional groups (rigid, similarity, affine, projective) have been employed in image registration [15, 49, 76]. Are there other groups, either subgroups of the full diffeomorphism group or not,

that can be usefully employed in registration?

More information: In the standard approach to diffeomorphic image registration, affine registration is performed before full diffeomorphism, in order to match gross features of the images and get the coordinate frames to line up. However, there may well be information in the affine part (e.g., growth). Are there other groups, either subgroups of the full diffeomorphism group, or not, that can provide useful information?

Occam’s razor: Occam’s razor states that among competing hypotheses, the one with the fewest assumptions should be selected. By Occam’s razor, simpler theories are preferable to more complex ones because they are better testable and falsifiable. Finite dimensional groups are less complicated than infinite dimensional groups, so by Occam’s razor, they are preferable.

Faster implementation: Diffeomorphic registration may not be quick enough for some uses, while finite dimensional registration is faster due to having fewer unknowns. Also for this reason, among the finite dimensional groups, groups with lower dimension are preferred for the registration.