## 2 Results and Discussion  
If now we consider in Equation (5) an external force \(\bar{F}(\bar{v})\) significantly different from zero, the convergence of \(\bar{v}(x,y,t)\) to the steady state will result faster than in GGVF solution. To this end we have used the following anisotropic diffusion equation:  
\[\left\{ \begin{aligned} \bar{v}_{t}=div\left(g\left(\left|\
abla f \right|\right)\cdot\
abla\bar{v}\right)+\bar{F}\left(\bar{v}\right)\\ \bar{v}\left(x,y,0\right)=\
abla f\end{aligned} \right. \tag{8}\]  
with an external force for example as \(\bar{F}\left(\bar{v}\right)=g\left(\left|\
abla f\right|\right)\cdot\left( \bar{v}-\
abla f\right)\). In this way we will be able to speed up the convergence process, because \(\bar{F}\left(\bar{v}\right)\) is approximately null near edges but increases as moving away from them. An efficient numerical scheme to approximate the equilibrium solution of the diffusion Equation (8) may be the one proposed by [PERSON] [17] that recurs to the 4-neighbors discretization of the Laplacian operator:  
Figure 5: An arbitrary contour and divergence map. Figure 6: **(a)** Divergence Contour Map; **(b)** Initial contour automatically selected. \[\vec{v}_{i,j}^{t+1}=\vec{v}_{i,j}^{t}+\frac{1}{4}\Big{[}g_{N}\
abla_{N}\vec{v}_{i,j }^{t}+g_{S}\
abla_{S}\vec{v}_{i,j}^{t}+g_{E}\
abla_{E}\vec{v}_{i,j}^{t}+g_{W} \
abla_{W}\vec{v}_{i,j}^{t}\Big{]}+F\Big{(}\vec{v}_{i,j}^{t}\Big{)}\]  
\[\
abla_{N}\vec{v}_{i,j}=\vec{v}_{i-1,j}-\vec{v}_{i,j} \tag{9}\] \[\
abla_{S}\vec{v}_{i,j}=\vec{v}_{i+1,j}-\vec{v}_{i,j}\] \[\
abla_{E}\vec{v}_{i,j+1}=\vec{v}_{i,j+1}-\vec{v}_{i,j}\] \[\
abla_{W}\vec{v}_{i,j}=\vec{v}_{i,j-1}-\vec{v}_{i,j}\]  
with the conduction function \(g(. )\) numerically evaluated as follows:  
\[g_{N}=g\Big{(}f_{i-1,j}-f_{i,j}\Big{)} \tag{10}\] \[g_{S}=g\Big{(}f_{i+1,j}-f_{i,j}\Big{)}\] (11) \[g_{E}=g\Big{(}f_{i,j+1}-f_{i,j}\Big{)}\] (12) Figure 6:0  
through a gradient approximation of \(\big{|}\
abla f\big{|}\) along horizontal and vertical directions, so we have \(g\Big{(}\big{|}\frac{\partial f}{\partial x}\big{|}\Big{)}\) and \(g\Big{(}\big{|}\frac{\partial f}{\partial y}\big{|}\Big{)}\) instead of \(g\big{(}\big{|}\
abla f\big{|}\big{)}\). The vector field so determined, from now on called AVF field, will be used for image processing in the subsequent applications. In Figure 7a,b, we could compare the AVF field derived from Equation (8) with GGVF field obtained through Equation (7), for the same number of iterations. This model could provide us an extension of the capture range, specifically an enlargement of areas inside which any initial contour may be placed. These regions result as well defined in the divergence map because it is delimited by curves corresponding to opposite values of divergence. As a consequence, we have a complete characterization of the capture range for a vector field that will be of great help in snake initialization procedures. Figure 6:1 **(a)** Divergence map of AVF Field; **(b)** Divergence map of GGVF field after 100 iterations after 100 iterations after 100 iterations. These areas are varying according to the number of iterations after which the flow process has been interrupted in the numerical resolution of both Equations (7) and (8). Consequently, the evaluation of the AVF vector field will be less time consuming compared to the GGVF because we could choose a reduced number of iterations in order to have a comparable capture range size and to achieve similar results. Moreover, by increasing the number of iterations, the sides of areas that delimit parts of image inside which the field is null, collapse each other, especially in conjunction with deep and narrow concavities, as can be seen in Figure 8b. Therefore, the curves of high divergence allow us to clearly identify those parts of detected edges with high curvatures that result to be the most significant from the geometrical and morphological point of view.