Riemannian geometry · Definition

LDDMM

geometryRiemannian geometrydeformable registrationcomputational anatomy

Before you start you need to know

LDDMM organizes image registration as a geodesic problem on the group of diffeomorphisms.

Definition

LDDMM (Large Deformation Diffeomorphic Metric Mapping) is a registration framework that looks for a smooth diffeomorphic deformation \(\varphi_t\), generated by a vector field \(v_t\), with minimum energy \(E_{\min}\).

\[\begin{gathered}\dot{\varphi}_t=v_t\circ\varphi_t\\\varphi_0=\operatorname{id}\end{gathered}\]

LDDMM is not a manifold by itself: it is a framework for registering images and shapes through smooth deformations.

The geometry appears in the group of diffeomorphisms, which can be treated as an infinite-dimensional Riemannian space. The optimal deformation is sought as a minimum-energy geodesic.

In our atlas, a template image is transported toward another through a flow \(\varphi_t\). In the equation, \(\dot{\varphi}_t\) is the time derivative, \(v_t\) the vector field that generates the motion, \(\circ\) the composition of functions and \(\operatorname{id}\) the initial identity map. The images reachable by those transformations form an orbit; that is why LDDMM connects differential geometry, Lie groups and image registration.

Think about it

Is LDDMM a manifold?

No. It is a registration framework; the central geometric object is the group of diffeomorphisms equipped with a Riemannian metric.

What does a geodesic represent in LDDMM?

A smooth deformation trajectory connecting the template with the target at minimum cost or energy.

Exercises 10

  1. If \(\varphi_0=\operatorname{id}\) and \(\varphi_1\) is fixed, what distinguishes the solution that LDDMM prefers?
  2. In \(\dot{\varphi}_t=v_t\circ\varphi_t\), what does evaluating the right-hand side at a point \(x\) mean?
  3. A discretization produces a velocity field that creates folds and stops being invertible. What was lost?
  4. Two flows reach the same target and have energies \(E_A=9\) and \(E_B=4\). If both are admissible, which criterion does LDDMM favor?
  5. In geodesic shooting, why can an initial velocity determine the whole trajectory?
  6. A method chains many pairwise registrations and reaches the target. Which conclusion is NOT justified by that fact alone?
  7. If \(\varphi_1\) takes the template to the target, what does \(\varphi_1^{-1}\) naturally represent?
  8. Which property by itself characterizes an admissible diffeomorphic deformation?
  9. What change would directly weaken the interpretation of LDDMM as a geometric problem with a metric?
  10. Which of these elements appears explicitly in the equation \(\dot{\varphi}_t=v_t\circ\varphi_t\), \(\varphi_0=\operatorname{id}\)?

References

  • M. Faisal Beg, Michael I. Miller, Alain Trouvé, Laurent Younes, Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms, 2005.
  • Center for Imaging Science, Johns Hopkins University, Large Deformation Diffeomorphic Metric Mapping, Johns Hopkins University, 2005.