Riemannian geometry · Definition

Smooth diffeomorphic deformation

geometrydifferential geometryRiemannian geometry

Before you start you need to know

In LDDMM, image registration is modeled as a smooth diffeomorphic deformation generated by a velocity field. This section fixes the properties that deformation must satisfy.

Definition

A smooth diffeomorphic deformation \(\varphi\) is a bijective, smooth transformation whose inverse is also smooth.

\[\varphi^{-1}\text{ is smooth}\]

Smooth diffeomorphic deformations form a group: the composition of two of them is again a diffeomorphism, and the inverse of each one is also a diffeomorphism. This allows velocity fields to be integrated over time and yields deformations that do not break the topology of the image.

Guide example: in the registration of two brain images, a template is deformed toward a subject through \(\varphi\). If \(\det D\varphi\) stays positive over the whole domain, no folds appear, and if \(\varphi\) is also bijective, its inverse is smooth.

Typical error: believing that being bijective and continuous is enough for \(\varphi\) to be a diffeomorphism, or that regularity does not matter as long as the reference points match. Both ideas ignore the smoothness of \(\varphi^{-1}\) and can produce folds.

Think about it

Which condition on \(D\varphi\) guarantees that \(\varphi\) is locally a diffeomorphism?

That \(D\varphi\) is invertible at every point, i.e. \(\det D\varphi\neq 0\), by the inverse function theorem.

Exercises 11

  1. A map \(\varphi:M\to N\) is a candidate to be a diffeomorphism, but its inverse turns out to be only continuous and not smooth. Which conclusion is correct?
  2. During a registration, the Jacobian of \(\varphi\) vanishes in a region and folds appear. Which property is compromised?
  3. If \(f\) and \(g\) are composable diffeomorphisms, why does \(g\circ f\) keep that structure?
  4. Two deformations reach the same reference points, but one squashes a region until it forms a fold. Which one must be discarded?
  5. Which statement necessarily follows from \(\varphi\) being a smooth diffeomorphic deformation?
  6. What is called a smooth, bijective transformation with a smooth inverse?
  7. What does it mean for the Jacobian of \(\varphi\) to be nonzero at every point?
  8. What is the velocity field in LDDMM?
  9. What does the inverse function theorem guarantee for a smooth map with invertible Jacobian?
  10. If \(\varphi\) and \(\psi\) are diffeomorphisms, what is the inverse of \(\psi\circ\varphi\)?
  11. Let \(\varphi:\mathbb{R}\to\mathbb{R}\), \(\varphi(x)=x^3\). Which conclusion is correct?

References

  • John M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer, 2013.
  • M. Faisal Beg, Michael I. Miller, Alain Trouvé, Laurent Younes, Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms, 2005.