In each of these examples, we either create a geometrical object on a grid or load a binary image / volume onto the grid. The geometrical objects are initially defined by the following indicator function

\[ \phi_{\text{indicator}} = \begin{cases} +1 & \text{point lies inside the object} \\ -1 & \text{point lies outside the object} \\ \end{cases} \]

This indicator function is already some form of a level-set-function. In order to convert Phi_indicator into a signed-distance function (SDF), where

\[|\nabla\phi| = 1\]

we will perform Sussman redistancing (see: M. Sussman and E. Fatemi, “Efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow” (1999), Appendix A).

In the Sussman-redistancing, a level-set-function, which can be as simple as a step-function but requires a switch of sign accross the surface of the object, is reinitialized to a SDF by finding the steady-state solution of the following PDE:

\[ \phi_{t} + sgn(\phi_{0})(|\nabla\phi| - 1) = 0 \]

For improved numerical stability and faster convergence, the sign function is replaced by:

\[ sgn_{\epsilon}(\phi) = \frac{\phi}{ \sqrt{\phi^2 + |\nabla\phi|^2 \Delta x^2} }\]

where the φ is the approximation of the SDF from the last iteration (see: Peng et al. "A PDE-Based Fast Local Level Set Method", equation (36)).