Exercise · ZW-VECTOR-FIELD-002 · ★★☆ Intermediate
Why is it not equivalent to replace \(v_t(\varphi_t(x))\) by \(v_t(x)\) throughout the flow?
Choose one option.
A Because composition is always commutative B Because the field stops being a vector field C Because the point has already moved and the velocity must be evaluated at its current position D Because \(v_t\) only accepts integers Show answer C. Because the point has already moved and the velocity must be evaluated at its current position← Back to the question
Explanation Since in general \(\varphi_t(x)\neq x\), the velocity must be evaluated at the current position. Replacing it by \(v_t(x)\) changes the flow.
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.
\[v_t(x)\]
More practice problems: velocity field