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In the paper "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms" by Beg et al., the formula (4) states:

$\frac{d}{dt} \partial_h \phi = D_{\phi} v \partial_h \phi + h \circ \phi$

Where $\phi$ is a transformation field, $v$ the velocity field, $h$ the disturbance function. What does the first item in the right mean? I thought it means the partial derivative of $v$ followed by the composition of $\phi$.

I think the formula should be: (from definition that $\frac{d}{dt} \phi(x) = v \circ \phi(x)$)

$\frac{d}{dt} \partial_h \phi = D_{\phi} v \partial_h \phi + h \circ \phi \partial_h \phi $

Isn't it right?

Thank you!

silencej
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1 Answers1

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Firstly the notation $\partial_h$ is more or less misleading, explicitly it is $\frac{d}{d\epsilon}$ in such notation. And for making whole thing clear, we think $\phi$ being $\phi(\omega)$, with $\omega\in\Omega$ fixed. Thus $\phi$ is a vector variable, not a function.

$\frac{d}{dt}\partial_h \phi^{v+\epsilon h}=\frac{d}{dt}\frac{d}{d\epsilon}\phi^{v+\epsilon h}=\frac{d}{d\epsilon}(v+\epsilon h)\circ \phi^{v+\epsilon h}=\frac{d}{d\epsilon}v\circ \phi^{v+\epsilon h}+\frac{d}{d\epsilon}\epsilon h\circ \phi^{v+\epsilon h}$

The latter term naturally gives $h\circ\phi$. For the previous one is the gateaux-derivative part, it can be simply think of a derivative against a certain direction. More exactly in Einstein notation it is $\partial_i V \phi^i$

Peter Wu
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