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==Applications==
The most fundamental application of GVF is as an external force in a
deformable model. A typical application considers an image
$I(\mathbf{x})$ with an object delineated by intensity from its
background. Thus, a suitable edge map $f(\mathbf{x})$ could be defined by
\begin{equation}
f(\mathbf{x}) = \frac{|\nabla (I(\mathbf{x}) *
G_{\sigma}(\mathbf{x}))|}{\max_{\mathbf{x}}|\nabla
(I(\mathbf{x}) * G_{\sigma}(\mathbf{x}))|} \,,
\end{equation}
where $G_{\sigma}$ is a Gaussian blurring kernel with standard
deviation $\sigma$ and $*$ is convolution. This definition is
applicable in any dimension and yields an edge map that falls in the
range $[0,1]$. Gaussian blurring is used primarily so that a
meaningful gradient vector can always be computed, but $\sigma$ is
generally kept fairly small so that true edge positions are not overly
distorted. Given this edge map, the GVF vector field
$\mathbf{v}(\mathbf{x})$ can be computed by solving (\ref{eq:gvf-euler}).
The deformable model itself can be implemented in a variety of ways
including parametric models such as the original
snake~\cite{KasxIJCV88} or active surfaces and implicit models
including geometric deformable models~\cite{XuxCSSC00}. In the case
of parametric deformable models, the GVF vector field $\mathbf{v}$
can be used directly as the external forces in the model.
If the deformable model is defined by the evolution of the
(two-dimensional) active contour $\mathbf{X}(s,t)$, then a simple
parametric active contour evolution equation can be written as
\begin{equation}
\gamma \mathbf{X}_t = \alpha \mathbf{X}_{ss} - \mathbf{v}(\mathbf{X}) \,.
\end{equation}
Here, the subscripts indicate partial derivatives and $\gamma$ and
$\alpha$ are user-selected constants.
In the case of geometric deformable models, then the GVF vector
field $\mathbf{v}$ is first projected against the normal direction of
the implicit wavefront, which defines an additional speed function.
Accordingly, then the evolution of the signed distance function
$\phi_t(\mathbf{x})$ defining a simple geometric deformable contour
can be written as
\begin{equation}
\gamma \phi_t = [\alpha \kappa - \mathbf{v} \cdot \frac{\nabla
\phi}{|\nabla \phi|} ] |\nabla \phi | \,,
\end{equation}
where $\kappa$ is the curvature of the contour and $\alpha$ is a
user-selected constant.
A more sophisticated deformable model formulation that combines
the geodesic active contour flow with GVF forces was proposed
in~\cite{ParxTPAMI04}. This paper also shows how to apply the Additive
Operator Splitting schema~\cite{GolxTIP01} for rapid computation of
this segmentation method. The uniqueness and existence of this
combined model were proven in~\cite{GuixCPAA09}.
A further modification of this model by using an external force
term minimizing GVF divergence was proposed in~\cite{LixSP16}
to achieve even better segmentation for images with
complex geometric objects.
GVF has been used to find both inner,
central, and central cortical surfaces in the analysis of brain
images~\cite{HanxNI04}, as shown in Figure~\ref{fig:Cortex}. The
process first finds the inner surface using a three-dimensional
geometric deformable model with conventional forces. Then the central
surface is found by exploiting the central tendency property of GVF.
In particular, the cortical membership function of the human brain
cortex, derived using a fuzzy classifier, is used to compute GVF as if
itself were a thick edge map. The computed GVF vectors point towards
the center of the cortex and can then be used as external forces to
drive the inner surface to the central surface. Finally, another
geometric deformable model with conventional forces is used to drive
the central surface to a position on the outer surface of the cortex.
Several notable recent applications of GVF include constructing graphs for
optimal surface segmentation in spectral-___domain optical coherence
tomography volumes~\cite{MirxCMIG17}, a learning based probabilistic GVF active contour
formulation to give more weights to objects of interst in ultrasound image
segmentation~\cite{HafxCBM14}, and an adaptive multi-feature GVF active contour
for improved ultrasound image segmentation without hand tuned
paramaters~\cite{RodxJVCIR13}.
[[File:GVF_Cortex.png|thumb|400px|right|Fig. 4. The inner, central, and outer surfaces of the human brain cortex (top) are found sequentially using GVF forces in three geometric deformable models. The central surface uses the gray matter membership function (bottom left) as an edge map itself, which draws the central surface to the central layer of the cortical gray matter. The positions of the three surfaces are shown as nested surfaces in a coronal cutaway (bottom right).]]
==Related Concepts==
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