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==Applications==
The most fundamental application of GVF is as an external force in a deformable model. A typical application considers an image
<math>\textstyle I(\mathbf{x})</math> with an object delineated by intensity from its background. Thus, a suitable edge map
<math>\textstyle
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:<math display = "block">
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(I(\mathbf{x}) * G_{\sigma}(\mathbf{x}))|} \,,
</math>| 7 | border=y}}
where <math>\textstyle G_{\sigma}</math> is a Gaussian blurring kernel with standard deviation <math>\textstyle\sigma</math> and <math>*</math> is convolution. This definition is applicable in any dimension and yields an edge map that falls in the range <math>[0,1]</math>. Gaussian blurring is used primarily so that a
meaningful gradient vector can always be computed, but <math>\sigma</math> is generally kept fairly small so that true edge positions are not overly▼
distorted. Given this edge map, the GVF vector field <math>\textstyle\mathbf{v}(\mathbf{x})</math> can be computed by solving (2). ▼
▲meaningful gradient vector can always be computed, but <math>\sigma</math> is
▲<math>\textstyle\mathbf{v}(\mathbf{x})</math> can be computed by solving (2).
The deformable model itself can be implemented in a variety of ways including parametric models such as the original
snake <ref name=":KasxIJCV88/> or active surfaces and implicit models including geometric deformable models <ref name=":XuxCSSC00">{{Cite conference | first1 = C. | last1 = Xu | first2 = A. | last2 = Yezzi | first3 = J.L. | last3 = Prince | title = On the relationship between parametric and geometric active contours and its applications | book-title = 34th Asilomar Conference on Signals, Systems and Computers | volume = 1 | pages = 483-489 | date = October 2000}}</ref>. In the case▼
of parametric deformable models, the GVF vector field <math>\mathbf{v}</math> can be used directly as the external forces in the model. ▼
If the deformable model is defined by the evolution of the (two-dimensional) active contour <math>\mathbf{X}(s,t)</math>, then a simple▼
▲including geometric deformable models <ref name=":XuxCSSC00">{{Cite conference | first1 = C. | last1 = Xu | first2 = A. | last2 = Yezzi | first3 = J.L. | last3 = Prince | title = On the relationship between parametric and geometric active contours and its applications | book-title = 34th Asilomar Conference on Signals, Systems and Computers | volume = 1 | pages = 483-489 | date = October 2000}}</ref>. In the case
▲of parametric deformable models, the GVF vector field <math>\mathbf{v}</math>
▲(two-dimensional) active contour <math>\mathbf{X}(s,t)</math>, then a simple
parametric active contour evolution equation can be written as
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[[File:GVF_Cortex.png|thumb|400px|right|Fig. 4. The inner, central, and outer surfaces of the human brain cortex (top) are found sequentially using GVF forces in three geometric deformable models. The central surface uses the gray matter membership function (bottom left) as an edge map itself, which draws the central surface to the central layer of the cortical gray matter. The positions of the three surfaces are shown as nested surfaces in a coronal cutaway (bottom right).]]
In the case of geometric deformable models, then the GVF vector field <math>\mathbf{v}</math> is first projected against the normal direction of
the implicit wavefront, which defines an additional speed function. Accordingly, then the evolution of the signed distance function▼
<math>\textstyle\phi_t(\mathbf{x})</math> defining a simple geometric deformable contour can be written as ▼
▲the implicit wavefront, which defines an additional speed function.
▲<math>\textstyle\phi_t(\mathbf{x})</math> defining a simple geometric deformable contour
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\phi}{|\nabla \phi|} ] |\nabla \phi | \,,
</math>| 9 | border=y}}
where <math>\kappa</math> is the curvature of the contour and <math>\alpha</math> is a user-selected constant.
A more sophisticated deformable model formulation that combines
the geodesic active contour flow with GVF forces was proposed
in <ref name=":ParxTPAMI04">{{Cite journal | first1 = N. | last1 = Paragios | first2 = O. | last2 = Mellina-Gottardo | first3 = V. | last3 = Ramesh | title = Gradient vector flow fast geometric active contours | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence | year = 2004 | volume = 26 | pages = 402-407 | issue = 3}}</ref>. This paper also shows how to apply the Additive
Operator Splitting schema
| first4 = M. | last4 = Rudzsky | title = Fast geodesic active contours | journal = IEEE Transactions on Image Processing | year = 2001
| volume = {10 | pages = 1467-1475 | issue = 10}}</ref> for rapid computation of this segmentation method. The uniqueness and existence of this
combined model were proven in :<ref name=":GuixCPAA09">{{Cite journal | first1 = L. | last1 = Guilot | first2 = M. | last2 = Bergounioux
| title = Existence and uniqueness results for the gradient vector flow and geodesic active contours mixed model |
journal = Communications on Pure and Applied Analysis | year = 2009 | volume = 8 | issue = 4 | pages = 1333-1349}}</ref>.
to achieve even better segmentation for images with ▼
A further modification of this model by using an external force term minimizing GVF divergence was proposed in <ref name=":LixSP16">{{Cite journal | title=Active contours driven by divergence of gradient vector flow | journal=Signal Processing |
volume=120 | pages=185-199 | year=2016 | publisher=Elsevier}}</ref>
▲to achieve even better segmentation for images with complex geometric objects.
GVF has been used to find both inner,
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