I'm trying to get my head around level set based segmentation. I've considered many sources on that topic, but all of them use a formulation that confuses me. Let me describe briefly by quoting exemplarily from Cremers (2006), where we write $\Omega$ to denote the image domain:
Countours are represented as the zero level line of some emedding function $\varphi : \Omega \to \mathbb R$: $$\mathcal C = \left\{ x \in \Omega \middle| \varphi\left(x\right) = 0 \right\}$$ [...] A contour is propagated by evolving a time-dependent embedding function $\varphi\left(x, t\right)$ according to an appropriate partial differential equation. [...] Since $\varphi\left(\mathcal C\left(t\right), t\right) = 0$ at all times, the total time derivative of $\varphi$ at locations of the contour must vanish: $$\frac{\operatorname d}{\operatorname dt} \varphi\left(\mathcal C\left(t\right), t\right) \stackrel{(\ast)}{=} \nabla \varphi \frac{\partial \mathcal C}{\partial t} + \frac{\partial\varphi}{\partial t} = \text{[...]} = 0$$
Why does the equation $(\ast)$ hold? Obviously, $\varphi$ was extended to be a $\Omega \times \mathbb R \to \mathbb R$ function. So as far as I can tell from e.g. Wikipedia, the total time derivative of $\varphi$ should rather read $$\frac{\operatorname d}{\operatorname dt} \varphi\left(\mathcal C\left(t\right), t\right) = \frac{\partial \varphi}{\partial x} \frac{\partial \mathcal C}{\partial t} + \frac{\partial\varphi}{\partial t}\text,$$ but everybody seems to rely on the gradient $\nabla \varphi$ in place of $\partial \varphi / \partial x$. Why is that? Isn't it $$\nabla \varphi = \begin{pmatrix}\partial \varphi / \partial x \\ \partial \varphi / \partial t\end{pmatrix}\text?$$ Could it be, that $\nabla \varphi$ for some (confusing) reason is commonly used to denote $\partial \varphi / \partial x$?
In case somebody else will be wondering about this in the future: