13.3.3 Extending the Speed Function

As noted earlier, the values of the speed function at the zero level set must be extended to the whole area of active points such that the level set function retains the shape of the signed distanced function. This can be ensured by requiring

$$\nabla_\mathbf{x}u \cdot \nabla_\mathbf{x}F = 0.$$ (13.1)

Assuming that $F$ and $u$ are smooth, it is straightforward to show that the signed distance function is retained at all times under this requirement [122,154]. Initially we have $\Vert \nabla u (0,\mathbf{x}) \Vert=1$. Using the level set equation $u_t = - F \Vert\nabla u\Vert$, the calculation

$$\frac{\mathrm{d}\Vert\nabla u\Vert^2}{\mathrm{d}t} = 2 \nabla u \... ...abla F \Vert\nabla u\Vert - 2 \nabla u \cdot \nabla \Vert \nabla u \Vert F = 0$$

shows that $\Vert\nabla u\Vert = 1$ always holds. Indeed the first term is zero because of the assumption (13.1) and the second term is zero since $\Vert \nabla u (0,\mathbf{x}) \Vert=1$. Thus one solution is $\Vert\nabla u\Vert = 1$ and a uniqueness result for this differential equation implies that $\Vert\nabla u\Vert = 1$ always holds.

An algorithm extending the speed function so that (13.1) holds is described in Section 13.6. After the motivations and outline in the preceding sections, the details of the level set algorithm devised and used in the ELSA simulator (cf. Section 13.8) are presented in the following sections.