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Geometry of LDDMM Registration

Hidden behind the complex formulae (2.164), (2.165), (2.166), and (2.167) is an intuitive geometric picture, which helps to understand the key characteristics of the LDDMM framework.

Riemannian structure The Riemannian structure of LDDMM can be understood

from the abstract formulation of the diffeomorphic deformation in (2.164).

  • • Images We can think of the vector space V as containing different types of images encountered in CA by selecting Q, M. The inner product < •, • >V can be used to measure the difference between two images. For example, brain MRI images can be represented as V = {I : Q ! Rg,Q e R3, and the difference between I0 and I1 is given by ||I0I11|2 =< I0 —11, I0 —11 >V= fa lI0(^) — h(x)2dx.
  • • Transformation Diffeomorphic transformations on V is a Lie group G e Diff (Q) essentially determined by V through the image domain Q. A transformation g e G deforms I e V by the action l given in (1). For intensity MRI images, e.g., T1-weighted images, the action is gI = I о g_1. For more information about how transformations act on different images, the reader is directed to [95, 96].

• Riemannian manifold Lie group G e Diff (Q) with the inner product < •, • >0 on its Lie algebra g define a Riemannian structure on G with a right invariant metric [81, 83]. Obviously by selecting different G and <, >0, we are in fact working on different Riemannian manifolds, which may lead to different registration results as will be explained later.

Geodesics The objective of LDDMM registration (2.165) and (2.166) is to find the optimal path gt e G continuously parametrized by time t that smoothly deforms I0 through It = gtI0. The optimal path is defined as the path that costs the least in time-integrated kinetic energy i < ut, u, >0 dt for a given error tolerance < gik — h, gik — h >v- The optimal solution given by (2.166) and (2.167) is simply a geodesic connecting I0 and a point in the near neighborhood of I1 determined by (2.166) and (2.167), equivalent to the Euler-Poincare equation [89]. It can be solved by gradient flow [87] or geodesic shooting methods [82, 97].

Riemannian exponential and shape space linearization Besides a distance metric on shape space, by modeling optimal transformations as geodesics, LDDMM also linearizes the shape space to validate statistical analysis on it. Any geodesic fulfilling differential equations (2.167) and (2.168) is uniquely determined by I0, I and the initial momentum ub0 e g* or equivalently the initial vector field u0 e g. Taking I0 as a common reference, any image I e V can be reached within an error tolerance by a geodesic determined by a vector field u0(I) in the vector space g. So the Riemannian exponential map exp and its inverse log linearize the shape space by representing any image I = gI0 e G with u0(I) = log(g) e g [98].

Further considerations To consolidate the foundations of the LDDMM framework, the following questions also need to be answered:

  • When does the geodesic provide a mathematically and physically valid distance metric between shapes, for example, satisfying the triangle relationship?
  • Is the distance metric between shapes a smooth function? Does a small change in images lead to a small change of the distance between them?
  • Does a geodesic with a limited length always exist between two shapes? Is it unique?
  • How reliable is the linearized representation of the shape space? For example, starting from a reference shape I0, does a small deviation of the initial vector field u0 result in a small change of the correspondent destination shape giI0?

Answers to these questions lie in the Riemannian structure on G defined by G, Q, and < •, • >fl, especially the curvature of the Riemannian manifold. A few interesting facts about these questions include: [99, 100]

• In the standard LDDMM framework, the group G is set as the flow of all time-dependent vector fields ut defined on an admissible Banach space V. The distance metric defined there is a valid distance on GV .

  • • Right-invariant L2 metric on the full diffeomorphic group Diff (Q) leads to vanishing geodesic distance so that any two shapes can be deformed into each other by a deformation that is arbitrarily small with this metric.
  • • An H1 metric on Diff (Q) introduces nondegenerate geodesic distance.
  • • For a unit n-dimensional cube Mn in Rn, the diameter (maximal geodesic distance on the manifold) of the smooth volume-preserving diffeomorphism group SDiff (Mn) is finite with an L2 right-invariant metric for n > 3. And the diameter is infinite for n = 2.

These are only a few examples to remind the readers how complex the situation

can be in the LDDMM framework with different Riemannian structures.

For more information about related topics on the LDDMM framework, such as the metamorphosis, currents, inner and outer metrics, and curvature of the shape space, the reader is referred to [85, 86, 88, 99].

 
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