## Computation procedure

The computation procedure for the 3D Fast Marching algorithm described in table 2 can be found in (Deschamps & Cohen, 2001). We give here additional calculation details to update the distance function estimate vk of an xtop's neighbour xk with a cost Tk.

Procedure FMComputeV(Neigh(xt0p))

01} Loop : for all configurations xk in Neigh(xtop)

{02} If xk is Unvisited, then remove it from Unvisited and insert it in Current with vk = <x>

{03} If xk is Current then apply case 1 or case 2 for the computation of vk.

{04} Sort Current list according to the priority assignment._

Table 2. Pseudo code of the FM procedure for updating Neigh(xtop)

One, two or three Accepted configurations are used to solve equation (8). We note {A1, A2}, {B1, B2} and {C1, C2} the three couples of opposite neighbours of xk (in 6-connexity) with the ordering u(Ai) < u(A2), u(Bi) < u(B2), u(Ci) < u(C2) and u(Ai) < u(Bi) < u(Ci). Two different cases are to be examined sequentially:

Case 1: considering that vk > u(C1) > u(B1) > u(A1), the upwind scheme (8) is equivalent to:

(vk - u(Aj ))2 + (vk - u(B ))2 + (vk - u(C ))2 = Tk (9)

Computing the discriminant of equation (9) there are two possibilities:

• if tk > u(Ai)2 + ufo)2 + 2u(C)(u(C)-ufo)-u^)) and ufo)<

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