Riemannian geometry · Definition
Time derivative
Before you start you need to know
In LDDMM, the deformation \(\varphi_t\) evolves over time, and its derivative describes the speed of that change.
Definition
The time derivative \(\dot{\varphi}_t\) describes how a deformation changes as the time parameter \(t\) varies.
Properties: \(\dot{\varphi}_t\) is the derivative of \(\varphi_t\) with respect to \(t\) and satisfies \(\dot{\varphi}_t=v_t\circ\varphi_t\) with \(\varphi_0=\operatorname{id}\). If \(v_t\) is smooth, each \(\varphi_t\) is a diffeomorphism.
Guide example: if \(v_t=c\) is constant, then \(\varphi_t(x)=x+tc\) and \(\dot{\varphi}_t=c\), so the deformation moves at constant speed.
Typical error: confusing \(\dot{\varphi}_t\) with \(\varphi_t\) or with its inverse. The derivative measures how the deformation changes, not its value at a fixed instant.
Think about it
What relation does \(\dot{\varphi}_t\) satisfy with the field \(v_t\)?
\(\dot{\varphi}_t=v_t\circ\varphi_t\)
Exercises 11
- In the LDDMM equation, what does \(\dot{\varphi}_t\) represent?
- If \(\dot{\varphi}_t=0\) for all \(t\), what happens to the flow?
- The equality \(\dot{\varphi}_t=v_t\circ\varphi_t\) is a form of...
- What is the difference between \(\dot{\varphi}_t\) and an acceleration?
- What is the initial condition of the flow \(\varphi_t\) in an initial value problem?
- What is the time derivative of \(\varphi_t\)?
- How is the time derivative of \(\varphi_t\) written?
- What does the order of a time derivative indicate?
- What is the flow \(\varphi_t\) in LDDMM?
- If \(\dot{\varphi}_t=v_t\circ\varphi_t\) and \(v_t\) is zero, what happens to \(\varphi_t\)?
- If \(v_t(x)=x\) for all \(t\) and \(\varphi_0=\operatorname{id}\), what is \(\varphi_1(x)\)?
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.