Hi

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).

Was this article helpful?

0 0
Learn Photoshop Now

Learn Photoshop Now

This first volume will guide you through the basics of Photoshop. Well start at the beginning and slowly be working our way through to the more advanced stuff but dont worry its all aimed at the total newbie.

Get My Free Ebook


Post a comment