Riemannian geometry · Definition
Velocity field
Before you start you need to know
A vector field assigns a velocity to each point; in LDDMM it is the object integrated over time to generate deformations.
Definition
A velocity field \(v_t\) is a vector field that changes with time: at each instant \(t\) it assigns to every point the velocity at which it moves. In LDDMM it guides the smooth motion of the deformation.
Properties: a smooth, bounded vector field \(v_t\) generates a flow \(\varphi_t\) of diffeomorphisms, and the equation \(\dot{\varphi}_t=v_t\circ\varphi_t\) defines the motion of each point. This regularity is what makes the field useful in LDDMM.
Guiding example: on \(\mathbb{R}\), with \(v_t(x)=x\) and \(\varphi_0=\mathrm{id}\), the equation \(\dot{\varphi}_t=\varphi_t\) gives \(\varphi_t(x)=e^{t}x\), which is invertible for each \(t\).
Common error: confusing \(v_t(\varphi_t(x))\) with \(v_t(x)\). If \(v_t(x)=x\) is used without evaluating at the current position, \(\dot{\varphi}_t=x\) gives \(\varphi_t(x)=(1+t)x\), a different flow.
Think about it
What changes if you use \(v_t(x)\) instead of \(v_t(\varphi_t(x))\) in the flow?
You get a different equation and flow, since the velocity is not evaluated at the current position.
Exercises 11
- In \(\dot{\varphi}_t=v_t\circ\varphi_t\), what object does \(v_t\) assign to each position?
- Why is it not equivalent to replace \(v_t(\varphi_t(x))\) by \(v_t(x)\) throughout the flow?
- If \(v_t\) depends on \(t\), what does it mean?
- A smooth vector field can generate a flow, but that does not mean the flow preserves all distances. What distinguishes the two ideas?
- Select the only correct statement about the field \(v_t\).
- What does a vector field \(v_t\) assign to each point \(x\) of the space?
- In the equation \(\dot{\varphi}_t=v_t\circ\varphi_t\), what does \(\varphi_t\) represent?
- What is the velocity at time \(t\) of a point following the flow \(\varphi_t\) from \(x\)?
- What does it mean for the field \(v_t\) to be smooth?
- If \(v_t\) is smooth and bounded, what can be guaranteed about \(\varphi_t\)?
- On \(\mathbb{R}\) take \(v(x)=x^2\), which is smooth but unbounded. With \(\varphi_0(x)=x\) for \(x>0\), on which interval of \(t\) is \(\varphi_t(x)\) defined?
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.