## 1 Introduction
### Generalized Gradient Vector Flow and Divergence Map
In the GGVF framework the external force field \(\vec{F}_{\text{Ext}}\left(\vec{x}\right)=\vec{v}\left(\vec{x}\right)\) can be found as a solution of the following diffusion equation [11]:  
\[\vec{v}_{t}=g\left(\left|\
abla f\right|\right)\cdot\
abla^{2}\vec{v}-h\left( \left|\
abla f\right|\right)\cdot\left(\vec{v}-\
abla f\right) \tag{4}\]  
where \(\
abla^{2}\) is the Laplacian operator, \(\
abla f\left(x,y\right)\) is the gradient of the edge map \(f\left(x,\,y\right)\), derived from the gradient of the brightness function \(I(x,\,y)\), for example as \(f\left(x,y\right)=\left|\
abla\left(I(x,y)\right)\right.\) or with any other edge detector. The field vectors point to the closest object boundaries with norms significantly different from zero in proximity of them. The functions \(g\left(\left|\
abla f\right|\right)\) and \(h\left(\left|\
abla f\right|\right)\) are non-negative and generally not uniform, \(g\left(\left|\
abla f\right|\right)\) is monotonically decreasing, since the vector field \(\vec{v}\left(\vec{x}\right)\) will be weakly variable far from the edges where image intensities are expected to be rather uniform. On the other hand, \(h\left(\left|\
abla f\right|\right)\) should be monotonically increasing thus, close to boundaries, the vector field \(\vec{v}\left(\vec{x}\right)\) should have a trend nearly equal to \(\
abla f\). In the generation of deformable contours, the main drawbacks are a weak convergence of models towards edges, specially in regions with highly variable concavities, the initialization problem, _i.e._, the excessive influence of shape and initial position of the active contour and the capture range, _i.e._, the size of area inside which an active contour can be initialized [13, 14, 15]. We would like to introduce a method that tries to reduce the initialization problem because, as we can see for the test image of Figure 2, the convergence of arbitrary initial contours for the GGVF leads only partially to the expected results, giving seemingly incomprehensible outcomes. In this work, we suggest a more general analysis for the diffusion process of the external force field \(\vec{v}\). To this end, we could see the GGVF Equation (4) as a special case of the following generalized parabolic equation [12, 16]:  
\[ \begin{cases}\vec{v}_{t}=div\big{(}g\big{(}\!\big{|}\!\
abla f\big{|}\!\big{)} \cdot\
abla\vec{v}\big{)}+\vec{F}\big{(}\vec{v}\big{)}\!=\!\
abla g\big{(}\! \big{|}\!\
abla f\big{|}\!\big{)}\cdot\
abla\vec{v}+g\big{(}\!\big{|}\!\
abla f \big{|}\!\big{)}\!\big{|}\
abla^{2}\vec{v}+\vec{F}\big{(}\vec{v}\big{)}\\ \vec{v}\big{(}x,y,0\big{)}\!=\!\vec{v}_{0}\big{(}x,y\big{)}\end{cases}  \tag{5}\]  
where div is the divergence operator, \(\vec{F}\big{(}\vec{v}\big{)}\) is a term generating an external force for the diffusion process of the field \(\vec{v}(x,y,t)\), \(g(. )\) is the conduction coefficient that must be null or tend to zero at boundaries. If the diffusivity function \(g(. )\) is monotonically decreasing to zero, diffusion will be stopped across the edges to be extracted, so the vector flow can take place inside or outside the region. We obtain the GGVF Equation (4) with initial condition \(\vec{v}_{0}\big{(}x,y\big{)}\!=\!\
abla f\), considering:  
\[\
abla g\big{(}\!\big{|}\!\
abla f\big{|}\!\big{)}\!\cdot\!\
abla\vec{v}=0\,, \,\,\,\vec{F}\big{(}\vec{v}\big{)}=-h\big{(}\!\big{|}\!\
abla f\big{|}\!\big{)} \cdot\big{(}\vec{v}-\
abla f\big{)}\,,\,\,\,g\big{(}\!\big{|}\!\
abla f\big{|} \!\big{)}\!=\!e^{\frac{\big{|}\!\
abla f\big{|}}{k}}\,,\,\,h\big{(}\!\big{|}\! \
abla f\big{|}\!\big{)}\!=\!1-g\big{(}\!\big{|}\!\
abla f\big{|}\!\big{)} \tag{6}\]  
where \(k\) is a constant positive value. If we have an anisotropic flow of the vector field \(\vec{v}\) with a null external force, _i.e._, \(\vec{F}\big{(}\vec{v}\big{)}\!=\!0\), the Equation (5) will become:  
\[ \begin{cases}\vec{v}_{t}=div\big{(}g\big{(}\!\big{|}\!\
abla f\big{|}\!\big{)} \cdot\
abla\vec{v}\big{)}\\ \vec{v}\big{(}x,y,0\big{)}\!=\!\vec{v}_{0}\big{(}x,y\big{)}\!=\!\
abla f\end{cases}