Gradient vector flow: Difference between revisions

Revision as of 20:06, 31 March 2020

Revision as of 21:46, 31 March 2020

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</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 &nbsp;1 shows an edge map, the gradient

of the (slightly blurred) edge map, and the GVF field generated by

minimizing <math>\textstyle\mathcal{E}</math>.

[[File:UShape.png|thumb|400px|right|Fig. 1. An edge map (left) describes the boundary of an object. The gradient of the (slightly blurred) edge map (center) points towards the boundary, but is very local. The gradient vector flow (GVF) field (right) also points towards the boundary, but has a much larger capture range.]]

Equation&nbsp;1 is a variational formulation that has both a data term and a regularization term. The first term in the integrand is the data term. It encourages the solution <math>\textstyle\mathbf{v}</math> to closely agree with the gradients of the edge map since that will make

<math>\textstyle\mathbf{v} - \nabla f</math> small. However, this only needs to happen when the edge map gradients are large since

<math>\textstyle\mathbf{v} - \nabla f</math> is multiplied by the square of the length of these gradients. The second term in the integrand is a regularization term. It encourages the spatial variations in the components of the solution to be small by penalizing the sum of all the partial

derivatives of <math>\textstyle\mathbf{v}</math>. As is customary in these types of variational formulations, there is a regularization parameter

<math>\textstyle\mu > 0</math> that must be specified by the user in order to trade off the influence of each of the two terms.

If <math>\textstyle\mu</math> is large, for example, then the resulting field will be very smooth and may not agree as well

with the underlying edge gradients.

'''Theoretical Solution.''' Finding <math>\textstyle\mathbf{v}(x,y)</math> to minimize Equation&nbsp;1

requires the use of calculus of variations since <math>\textstyle\mathbf{v}(x,y)</math> is a function, not a

variable. Accordingly, the Euler equations, which provide the necessary conditions for <math>\textstyle\mathbf{v}</math>

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_x) (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.