Understanding the Level set Equation I have a question about the central equation of the Level Set method (e.g. https://math.berkeley.edu/~sethian/2006/Semiconductors/ieee_level_set_explain_technical.html, or Derivation of Basic Level Set Equations).
We are given a function $\Phi:\mathbb{R}^n \times [0,\infty) \rightarrow \mathbb{R}$.
Now we consider the trajectory of a particle on the level set at height 0 over time. So let $C_{t}:= \{ \mathbf{x} \in \mathbb{R}^n | \Phi(\mathbf{x},t) = 0 \}$ for $t \geq 0$. We consider a trajectory $\gamma: [0,\infty), t \mapsto p_{t}$ with $p_{t} \in C_{t}$.
Now let $\mathbf{s}(t) := (\gamma(t),t)^{\intercal}$ be the evolution of a point on the contour together with the time.
If we compute the derivative of $\Phi$ with respect to time $t$, we obtain by the chain rule:
$\frac{\partial \Phi \circ \mathbf{s}}{\partial t}(t)= \sum_{i=1}^{n}\frac{\partial \Phi}{\partial x_{i}}(\mathbf{s}(t))\frac{\partial \mathbf{s}_{i}}{\partial t}(t) + \frac{\partial \Phi}{\partial t}(\mathbf{s}(t)) \frac{\partial \mathbf{s}_{n+1}}{\partial t}(t)$, where $s_{n+1}(t) = t$ and $s_{i}(t) = \langle \gamma(t),e_{i} \rangle$ and $e_{i}$ is the $i$-th canonical basis vector for $i \in \{1,\ldots,n\}$.
So for n=2,
$\frac{\Phi \circ \mathbf{s}}{\partial t}(t)= \frac{\partial \Phi}{\partial x}(\mathbf{s}(t))\frac{\partial \mathbf{s}_{1}}{\partial t}(t)+\frac{\partial \Phi}{\partial y}(\mathbf{s}(t))\frac{\partial \mathbf{s}_{2}}{\partial t}(t) + \frac{\partial \Phi}{\partial t}(\mathbf{s}(t)) \frac{\partial \mathbf{s}_{3}}{\partial t}(t)$ and $s_{3}(t) = t$.
We can simplify this as:
$\frac{\partial \Phi \circ \mathbf{s}}{\partial t}(t)= \sum_{i=1}^{n}\frac{\partial \Phi}{\partial x_{i}}(\mathbf{s}(t))\frac{\partial \mathbf{s}_{i}}{\partial t}(t) + \frac{\partial \Phi}{\partial t}(\mathbf{s}(t)) $.
Now by construction $\Phi \circ \mathbf{s}$ is the 0 function, thus
$0 = \sum_{i=1}^{n}\frac{\partial \Phi}{\partial x_{i}}(\mathbf{s}(t))\frac{\partial \mathbf{s}_{i}}{\partial t}(t) + \frac{\partial \Phi}{\partial t}(\mathbf{s}(t)) $.
The equation is then used to obtain an update rule (see also Derivation of Basic Level Set Equations), which tells how to change a point over time. Essentially,
$\frac{\partial \Phi}{\partial t}(\mathbf{s}(t))$ is approximated by $\frac{1}{\Delta t} [-\Phi(\mathbf{s}(t+\Delta t))+\Phi(\mathbf{s}(t))]$, which can be used to see how to compute $\Phi(\mathbf{s}(t+\Delta t))$ given $\Phi(\mathbf{s}(t))$ and the $0$-equation just derived.
However, this equation or update rule is applied to all points.
Since the entire derivations assumes that we start with a point $p \in C_{0}$ on the contour, and use a trajectory of points that stay on the contour (so using a trajectory $\gamma$),  I dont understand why the update rule can be  applied to all points (e.g. of an image).
So in short,
instead of
$\Phi(\mathbf{s}(t+\Delta t)) := \Delta t [\sum_{i=1}^{n}\frac{\partial \Phi}{\partial x_{i}}(\mathbf{s}(t))\frac{\partial \mathbf{s}_{i}}{\partial t}(t)] + \frac{\partial \Phi}{\partial t}(\mathbf{s}(t)) $, I understand that the method is using
$\Phi(\mathbf{x}, t+\Delta t) := \Delta t [\sum_{i=1}^{n}\frac{\partial \Phi}{\partial x_{i}}(\mathbf{x}, t)\frac{\partial \mathbf{s}_{i}}{\partial t}(t)] + \frac{\partial \Phi}{\partial t}(\mathbf{x}, t) $, for all $\mathbf{x} \in \mathbb{R}^n$.
For example, have a look at https://profs.etsmtl.ca/hlombaert/levelset/ .In the paragraph "Level set equation" the equation I wrote down is shown. In the subsequent paragraph "Implementation" you see that the equation is applied to all pixels.
 A: Okay after reading the references, I'll offer my own interpretation.
Every pixel belongs to the domain of $\Phi(x,t)$; it is a globally defined function. Therefore, if we denote the image of all pixels by $\Phi_t(X) = U_t \subset \mathbb{R}$, the preimage $\Phi^{-1}_t(U_t)$ is a partition of $X$. Put more simply, every $(x,t) \in X \times [0,\infty)$ belongs to a contour $\mathbf{x}_c(t)$ at each time $t$ for some constant $c$.
However, we allow $\Phi(x,t)$ to take on any value outside of the zero-contour $\mathbf{x}_0(t)$. So long as it correctly predicts $\mathbf{x}_0(t)$, we don't care how it behaves elsewhere. Thus, you apply the update knowing that it will approximate the true solution on the zero-contour for all $x(t) \in \mathbf{x}_0(t)$, and will only generate a solution that is globally consistent with the $\mathbf{x}_0(t)$ for which you solved.
I believe this is what the author meant in the link that you provided when they said:

The question still remains: what is the function $\phi(\mathbf{x}(t),t)$? It can actually be anything we want as long as its zero
level set gives us the contour.

