Riemannian geometry · Definition

Diffeomorphic flow

differential geometryRiemannian geometryLDDMM

Before you start you need to know

A diffeomorphic flow describes a continuous deformation of space through a family of smooth, invertible transformations indexed by time. It is the basis of LDDMM, where the deformation is generated by integrating a velocity field.

Definition

A diffeomorphic flow \(\varphi_t\) is a family of deformations parametrized by time \(t\), where at each instant the transformation remains invertible and smooth.

\[\{\varphi_t\}_{t\in[0,1]},\quad \varphi_0=\operatorname{id}\]

The basic properties are that \(\varphi_0=\operatorname{id}\), that each \(\varphi_t\) is bijective and smooth, and that its inverse is also smooth. These conditions let each point be followed forward and backward in time without the space tearing or folding.

Guide example: on \(\mathbb{R}\) with constant field \(v_t(x)=1\), the flow is \(\varphi_t(x)=x+t\). Every point moves at the same speed, so \(\varphi_t\) is a translation, a diffeomorphism with inverse \(x-t\).

Typical error: confusing the field \(v_t\) with the total displacement \(\varphi_t(x)-x\). The field is an instantaneous velocity; the displacement comes from integrating \(v_t\) over the whole time interval.

Think about it

Why must a diffeomorphic flow have an inverse at each instant?

Because each point can be traced back, so the deformation neither tears nor folds the space.

Exercises 11

  1. For a point \(x\), the flow equation is read as \(\dot{\varphi}_t(x)=v_t(\varphi_t(x))\). What does it describe?
  2. What role does \(\varphi_0=\operatorname{id}\) play in a flow?
  3. If two flows have the same final map but travel different paths, what can vary?
  4. Which hypothesis on \(v_t\) helps the flow equation have a smooth, well-determined trajectory?
  5. Which description is compatible with a diffeomorphic flow \(\{\varphi_t\}\)?
  6. What does it mean for \(\varphi_t\) to be a diffeomorphism?
  7. What is the field \(v_t\) in LDDMM?
  8. What does it mean for the flow to be parametrized by \(t\in[0,1]\)?
  9. What does the final map \(\varphi_1\) do in LDDMM?
  10. If the field \(v_t\) is identically zero for all \(t\), what happens to the flow \(\varphi_t\)?
  11. Let \(v_t(x)=2x\) on \(\mathbb{R}\) with \(\varphi_0=\operatorname{id}\). What is \(\varphi_1\circ\varphi_{1/2}^{-1}(3)\)?

What comes next

References

  • M. Faisal Beg, Michael I. Miller, Alain Trouvé, Laurent Younes, Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms, 2005.