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Although GVF was designed originally for the purpose of segmenting objects using active contours attracted to edges, it has been since
adapted and used for many alternative purposes. Some newer purposes including defining a continuous medial axis representation<ref name=":3">{{Cite journal | title = Variational curve skeletons using gradient vector flow | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence | volume = 31| issue = 12| pages = 2257–2274| year = 2009| last1 = Hassouna | first1 = M.S.| last2 = Farag | first2 = A.Y. }}</ref>, regularizing image anisotropic diffusion algorithms<ref name=":YuxTIP06">{{Cite journal | title = GVF-based anisotropic diffusion models | journal = IEEE Transactions on Image Processing | volume = 15 | issue = 6 | pages = 1517--1524 | year = 2006 | last1 = Yu | first1 = H. | last2 = Chua | first2 = C.S. }}</ref>, finding the centers of ribbon-like objects<ref name=":HanxNI04">{{Cite journal | title = CRUISE: cortical reconstruction using implicit surface evolution | journal = NeuroImage | volume = 23 | number = 3 | pages = 997--1012 | year = 2004 | last1 = Han | first1 = X. | last2 = Pham | first2 = D.L. | last3 = Tosun | first3 = D. | last4 = Rettmann | first4 = M.E. | last5 = Xu | first5 = C. | last6 = Prince | first6 = J.L. | display-authors = etal }}</ref>, constructing graphs for optimal surface segmentations<ref name=":MirxCMIG17"> {{Cite journal | title = Incorporation of gradient vector flow field in a multimodal graph-theoretic approach for segmenting the internal limiting membrane from glaucomatous optic nerve head-centered SD-OCT volumes | journal = Computerized Medical Imaging and Graphics | volume = 55 | pages = 87-94 | year = 2017 | last1 = Miri | first1 = M.S., | last2 = Robles | first2 = V.A. | last3 = Abràmoff | first3 = M.D. | last4 = Kwon | first4 = Y.H. | last5 = Garvin | first5 = M.K.}}</ref>, creating a shape prior<ref name=":BaixCMIG18"> {{Cite journal | title = Optimal multi-object segmentation with novel gradient vector flow based shape priors | journal = Computerized Medical Imaging and Graphics | volume=69 | pages= 96-111 | year= 2018 | publisher=Elsevier | last1 = Bai | first1 = J. | last2 = Shah | first2 = A. | last3 = Wu | first3 = X.}}</ref>, and much more.
==Theory==
The theory of GVF was originally described in<ref name=":2"></ref>. Let <math>\textstyle f(x,y)</math> be an edge map defined on the image ___domain. For uniformity of results, it is important to restrict the edge map intensities to lie between 0 and 1, and by convention <math>\textstyle f(x,y)</math> takes on larger values (close to 1) on the object edges. The gradient vector flow (GVF) field is given by the vector field <math>\textstyle \mathbf{v}(x,y) = [u(x,y),v(x,y)]</math> that minimizes the energy functional
{{numBlk||
:<math display = "block">
\mathcal{E} = \iint_{\mathbb{R}^2} |\nabla f|^2 |\mathbf{v} - \nabla
f|^2 + \mu (u_x^2 + u_y^2 + v_x^2 + v_y^2) \, dx\,dy \,.
</math>| 1 | border=y}}
In this equation, subscripts denote partial derivatives and the gradient of the edge map is given by the vector field
<math>\textstyle \nabla f =(f_x, f_y)</math>. Figure~\ref{fig:UShape} shows an edge map, the gradient
of the (slightly blurred) edge map, and the GVF field generated by
minimizing <math>\textstyle\mathcal{E}</math>.
[[Image:UShape.png]]
==Related Concepts==
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