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to generate a three-dimensional GVF field. Vectors and streamlines of the GVF field are shown in the (Z) zoomed region, (V) vertical plane,
and (H) horizontal plane.]]
The data and regularization terms in the integrand of the GVF functional can also be modified. A modification described
in <ref name =":XuxSP98"> {{Cite journal | first1 = C. | last1 = Xu | first2 = J.L. | last2 = Prince | title = Generalized gradient vector flow external forces for active contours | journal = Signal Processing | year = 1998 | volume = 71 | pages = 131-139 | issue = 2}}</ref>, called
''generalized gradient vector flow'' (GGVF) defines two scalar functions and reformulates the energy as
{{numBlk||
:<math display = "block">
\mathcal{E} = \int_{\mathbb{R}^n} g(|\nabla f|) |\nabla \mathbf{v}|^2
+ h(|\nabla f|) |\mathbf{v} - \nabla f|^2 d\mathbf{x} \,.
</math>| 4 | border=y}}
While the choices <math>\textstyle g(\nabla f|) = \mu</math> and <math>\textstyle h(|\nabla f|) = |\nabla f|^2</math> reduce GGVF to GVF,
the alternative choices <math>\textstyle g(|\nabla f|) = \exp\{-|\nabla f|/K\}</math> and <math>\textstyle h(\nabla f|) = 1 - g(|\nabla f|)</math>,
for <math>K</math> a user-selected constant, can improve the tradeoff between the data term and its regularization in some applications.
The GVF formulation has been further extended to vector-valued images in <ref name =":JaoxTIP14"> {{Cite journal | title= Variational segmentation of vector-valued images with gradient vector flow | journal= IEEE Transactions on Image Processing | volume=23 | issue = 11 | pages = 4773-4785 | year = 2014 | last1 = Jaouen | first1 = V. | last2 = Gonzalez | first2 = P. | last3 = Stute | first3 = S. | last4 = Guilloteau | first4 = D. | last5 = Chalon | first5 = S. | display-authors = etal}}</ref> where a weighted structure tensor of a vector-valued image is used. A learning based probabilistic weighted GVF extension was proposed in <ref name =":HafxCBM14"> {{Cite journal | title = Phase-based probabilistic active contour for nerve detection in ultrasound images for regional anesthesia | journal=Computers in Biology and Medicine | volume=52 | pages=88-95 | year=2014 | last1 = Hafiane | first1 = A. | last2 = Vieyres | first2 = P. | last3 = Delbos | first3 = A.}}</ref> to further improve the segmentation for images with severely cluttered textures or high levels of noise.
The variational formulation of GVF has also been modified in ''motion GVF'' (MGVF) to incorporate object motion in
an image sequence <ref name =":RayxTMI04"> {{Cite journal | title = Motion gradient vector flow: An external force for tracking rolling leukocytes with shape and size constrained active contours | journal = IEEE Transactions on Medical Imaging | year = 2004 | volume = 23 |
pages = 1466-1478 | issue = 12 | last1 = Ray | first1 = N. | last2 = Acton | first2 = S.T.}}</ref>.
Whereas the diffusion of GVF vectors from a conventional edge map acts in an isotropic manner, the formulation of
MGVF incorporates the expected object motion between image frames.
An alternative to GVF called vector field convolution (VFC) provides many of the advantages of GVF, has superior noise robustness, and
can be computed very fast <ref name =":LixTIP07"> {{Cite journal | title = Active contour external force using vector field convolution for image segmentation | journal = IEEE Transactions on Image Processing | year = 2007 | volume = 16 | pages = 2096-2106 | issue = 8 | last1 = Li | first1 = B. | last2 = Acton | first2 = S.T.}}</ref>. The VFC field <math>\textstyle\mathbf{v}_{\mathrm{VFC}}</math> is defined as the convolution of the edge map <math>f</math> with a vector field kernel <math>\mathbf{k}</math>
{{numBlk||
:<math display = "block">
\mathbf{v}_{\mathrm{VFC}}(x,y) = f(x,y) * \mathbf{k}(x,y) \,,
</math>| 5 | border=y}}
where
{{numBlk||
:<math display = "block">
\mathbf{k}(x,y) =
\left\{ \begin{array}{cl} m(x,y) \left( \frac{-x}{\sqrt{x^2
+ y^2}} , \frac{-y}{\sqrt{x^2 + y^2}} \right) & (x,y) \neq (0,0) \\
(0,0) & \mathrm{otherwise} \end{array} \right.
</math>| 6 | border=y}}
The vector field kernel <math>\textstyle\mathbf{k}</math> has vectors that always point toward the origin but their magnitudes, determined in detail by the
function <math>m</math>, decrease to zero with increasing distance from the origin.
The beauty of VFC is that it can be computed very rapidly using a fast Fourier transform (FFT), a multiplication, and an inverse FFT. The
capture range can be large and is explicitly given by the radius <math>R</math> of the vector field kernel. A possible drawback of VFC is that weak
edges might be overwhelmed by strong edges, but that problem can be alleviated by the use of a hybrid method that switches to conventional
forces when the snake gets close to the boundary.
'''Properties.''' GVF has characteristics that have made it useful in many diverse applications. It has already been noted that
its primary original purpose was to extend a local edge field throughout the image ___domain, far away from the actual edge in many
cases. This property has been described as an extension of the ''capture range'' of the external force of an active contour
model. It is also capable of moving active contours into concave regions of an object's boundary. These two properties are illustrated
in Figure 3.
[[File:GVF_Convergence.png|thumb|400px|right|Fig. 3. An active contour with traditional external forces (left) must be initialized very close to the boundary and it still will not converge to the true boundary in concave regions. An active contour using GVF external forces (right) can be initialized farther away and it will converge all the way to the true boundary, even in concave regions.]]
Previous forces that had been used as external forces (based on the edge map gradients and simply related variants) required pressure
forces in order to move boundaries from large distances and into concave regions. Pressure forces, also called balloon forces, provide
continuous force on the boundary in one direction (outward or inward), and tend to have the effect of pushing through
weak boundaries. GVF can often replace pressure forces and yield better performance in such situations.
Because the diffusion process is inherent in the GVF solution, vectors that point in opposite directions tend to compete as they meet at a
central ___location, thereby defining a type of geometric feature that is related to the boundary configuration, but not directly evident from
the edge map. For example, ''perceptual edges'' are gaps in the edge map which tend to be connected visually by human
perception~\cite{KasxIJCV88}. GVF helps to connect them by diffusing opposing edge gradient vectors across the gap; and even though there
is no actual edge map, active contour will converse to the perceptual edge because the GVF vectors drive them there (see \cite{GVF_Web}).
This property carries over when there are so-called ''weak edges'' identified by regions of edge maps having lower values.
GVF vectors also meet in opposition at central locations of objects thereby defining a type of medialness. This property has been
exploited as an alternative definition of the skeleton of objects~\cite{HasxPAMI09} and also as a way to initialize deformable
models within objects such that convergence to the boundary is more likely.
==Applications==
==Related Concepts==
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