We define the gradient vector flow
(GVF) field to be the vector field
that minimizes the
We note that the smoothing term -- the first term within the integrand of (10) -- is the same term used by Horn and Schunk in their classical formulation of optical flow . On one hand, it is known that this term leads to the Laplacian operator in the corresponding Euler equations. On the other hand, it has recently been shown that this term corresponds to an equal penalty on the divergence and curl of the vector field . Therefore, the external field resulting from this minimization can be expected to be neither entirely irrotational (as are the usual snake potential fields) nor entirely solenoidal.
Using the calculus of variations , it can be shown that the GVF can be found by solving the following Euler equations where is the Laplacian operator. These equations give us another intuition behind the GVF formulation. We note that in homogeneous regions, the right hand side of both equations (11a) and (11b) is zero (because the gradient of f(x,y) is zero). Therefore, within these regions, u and v are each determined by Laplace's equation. This results in a type of ``filling-in'' of information taken from the boundaries of the region.