Gradient vector flow: Difference between revisions

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{{Short description|Computer vision framework}}

'''Gradient vector flow''' ('''GVF'''), a [[computer vision]] framework introduced by Chenyang Xu and [[Jerry L. Prince]],<ref name=":1">{{ Cite conference | last1 = Xu | first1 = C. | last2 = Prince | first2 = J.L. | title = Gradient Vector Flow: A New External Force for Snakes | book-title = Proc. IEEE Conf. on Comp. Vis. Patt. Recog. (CVPR) | place = Los Alamitos | publisher = Comp. Soc. Press | pages = 66–71 | date = June 1997 | url = http://iacl.ece.jhu.edu/pubs/p087c.pdf}}</ref><ref name=":2">{{Cite journal | title = Snakes, Shapes, and Gradient Vector Flow| journal = IEEE Transactions on Image Processing | volume = 7| issue = 3| pages = 359–369| year = 1998| last1 = Xu | first1 = C.| last2 = Prince | first2 = J.L. | url = http://iacl.ece.jhu.edu/pubs/p084j.pdf}}</ref> is the vector field that is produced by a process that smooths and diffuses an input vector field. It is usually used to create a vector field from images that points to object edges from a distance. It is widely used in image analysis and computer vision applications for object tracking, shape recognition, [[Image segmentation|segmentation]], and [[edge detection]]. In particular, it is commonly used in conjunction with [[active contour model]].

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==Theory==

The theory of GVF was originally described inby Xu and Prince.<ref name=":2" /> 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

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:<math display = "block">

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to be a solution can be found by calculus of variations, yielding

{{NumBlk|:|<math display = "block">\mu \nabla^2 u - (u - f_x) (f_x^2 + f_y^2) = 0 \,,</math> | 2a}}

{{NumBlk|:|<math display = "block">\mu \nabla^2 v - (v - f_xf_y) (f_x^2 + f_y^2) = 0 \,,</math> | 2b}}

where <math>\textstyle\nabla^2</math> is the Laplacian operator. It is instructive to examine the form of the equations in&nbsp;(2). Each is a partial differential equation that the components <math>u</math> and <math>v</math> of <math>\mathbf{v}</math> must satisfy. If the magnitude of the edge gradient is small, then the solution of each equation is guided entirely by Laplace's equation, for example <math>\textstyle\nabla^2 u = 0</math>, which will produce a smooth scalar field entirely dependent on its boundary conditions. The boundary conditions are effectively provided by the locations in the image where the magnitude of the edge gradient is large, where the solution is driven to agree more with the edge gradients.