We give here a geometrical intuition in two dimensions of how to convert equation (4) into equation (5). It is inspired by a level set formulation of the Eikonal equation in (Cohen & Kimmel, 1997) and a formal proof can be found in (Bruckstein, 1988).

Fig. 1. On a small surface dQ around a configuration x with a radius dx, one can approximate the distance function u as a plane wave, for which the level sets are parallel between them and perpendicular to the gradient Vu of u.

We start from the fact that the gradient Vu of u is normal to its level sets. Let n = Vu/|Vu||, where || is the Euclidean norm, be the outwards unit normal vector to level sets of u located in x (see figure 1). Express a variation du of u according to a variation dx of the position x:

where is the standard dot product in

Within the small region dQ of Q centered on x with a radius dx, we can assimilate t as a constant: Vp e dQ x(p) = t(x) = t .

Within dQ level sets of u are seen as straight lines:

From equations (6) and (7) we get ^Vu,dx^ = x^Vu,dx^/||Vu|| , which leads to the Eikonal equation (5).

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