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approach, while later papers introduced considerably faster implementations such as an octree-based method<ref name =":HerxCVIU2004> {{Cite journal | title = Silhouette and stereo fusion for 3D object modeling | journal = Computer Vision and Image Understanding | volume = 96 | issue = 3 | pages = 367-392 | year = 2004 | publisher = Elsevier | first1 = C. H. | last1 = Esteban | first2 = F. | last2 = Schmitt}}</ref>,
a multi-grid method<ref name=":HanxIETIP07"> {{Cite journal | title = Fast numerical scheme for gradient vector flow computation using a multigrid method | journal = IET Image Processing | year = 2007 | volume = 1 | pages = 48-55 | issue = 1 | first1 = X. | last1 = Han | first2 = C. | last2 = Xu | first3 = J.L. | last3 = Prince}}</ref>, and an augmented Lagrangian method<ref name=":RenxPRL12"> {{Cite journal | title = Fast gradient vector flow computation based on augmented Lagrangian method | last1 = Ren | first1 = D. | last2 = Zuo | first2 = W. | last3 = Zhao | first3 = X. | last4 = Lin | first4 = Z. | last5 = Zhang | first5 = D. | journal = Pattern Recognition Letters | volume = 34 | issue = 2 | pages = 219-225 | year = 2013 | publisher = Elsevier}}</ref>. In addition, very fast GPU implementations have been developed in<ref name=":SmixJRTIP15"> {{Cite journal | title = Real-time gradient vector flow on GPUs using OpenCL | last1 = Smistad | first1 = E. | last2 = Elster | first2 = A.C. | last3 = Lindseth | first3 = F. | journal = Journal of Real-Time Image Processing | volume = 10 | issue = 1 | pages = 67-74 | year = 2015 | publisher = Springer}}</ref><ref name=":SmixJRTIP16"> {{Cite journal | title = Multigrid gradient vector flow computation on the GPU | last1 = Smistad | first1 = E. | last2 = Lindseth | first2 = F. | journal = Journal of Real-Time Image Processing | volume = 12 | issue = 3 | pages = 593-601 | year = 2016 | publisher=Springer}}</ref>
'''Extensions and Advances.''' GVF is easily extended to higher dimensions. The energy function is readily written in a vector form as
{{numBlk||
:<math display = "block">
\mathcal{E} = \int_{\mathbb{R}^n} \mu|\nabla \mathbf{v}|^2 + |\nabla
f|^2 |\mathbf{v} - \nabla f|^2 d\mathbf{x} \,,
</math>| 3 | border=y}}
which can be solved by gradient descent or by finding and solving its
Euler equation. Figure 2 shows an illustration of a three-dimensional GVF field on the edge map of a simple object (see <ref name=":XuxHMIPA08"> {{Cite book | first1 = C. | last1 = Xu | first2 = X. | last2 = Han | first3 = J.L. | last3 = Prince | chapter = Gradient Vector Flow Deformable Models | title = Handbook of Medical Image Processing and Analysis | publisher = Academic Press | year = 2008 | editor = Isaac Bankman | pages = 181-194 | edition = 2nd }}</ref>).
[[File:Gradient Vector Flow 3D Metasphere Example Result.png|thumb|400px|right|Fig. 2. The object shown in the top left is used as an edge map
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.]]
==Related Concepts==
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