Gradient vector flow: Difference between revisions

Revision as of 14:23, 4 April 2020

Revision as of 14:38, 4 April 2020

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The most fundamental application of GVF is as an external force in a

deformable model. A typical application considers an image

$<math>\textstyle I(\mathbf{x})$</math> with an object delineated by intensity from its

background. Thus, a suitable edge map $<math>\textstyle f(\mathbf{x})$</math> could be defined by

{{numBlk||

\begin{equation}

:<math display = "block">

f(\mathbf{x}) = \frac{|\nabla (I(\mathbf{x}) *

G_{\sigma}(\mathbf{x}))|}{\max_{\mathbf{x}}|\nabla

(I(\mathbf{x}) * G_{\sigma}(\mathbf{x}))|} \,,

</math>| 7 | border=y}}

\end{equation}

where $<math>\textstyle G_{\sigma}$</math> is a Gaussian blurring kernel with standard

deviation $<math>\textstyle\sigma$</math> and $<math>*$</math> is convolution. This definition is

applicable in any dimension and yields an edge map that falls in the

range $<math>[0,1]$</math>. Gaussian blurring is used primarily so that a

meaningful gradient vector can always be computed, but $<math>\sigma$</math> is

generally kept fairly small so that true edge positions are not overly

distorted. Given this edge map, the GVF vector field

$<math>\textstyle\mathbf{v}(\mathbf{x})$</math> can be computed by solving (\ref{eq:gvf-euler}).

The deformable model itself can be implemented in a variety of ways

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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 $<math>\mathbf{v}$</math>

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 $<math>\mathbf{X}(s,t)$</math>, then a simple

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

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\begin{equation}

:<math display = "block">

\gamma \mathbf{X}_t = \alpha \mathbf{X}_{ss} - \mathbf{v}(\mathbf{X}) \,.

</math>| 8 | border=y}}

\end{equation}

Here, the subscripts indicate partial derivatives and $<math>\gamma$</math> and

$<math>\alpha$</math> are user-selected constants.

[[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).]]

In the case of geometric deformable models, then the GVF vector

field $<math>\mathbf{v}$</math> 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

$<math>\textstyle\phi_t(\mathbf{x})$</math> defining a simple geometric deformable contour

can be written as

{{numBlk||

\begin{equation}

:<math display = "block">

\gamma \phi_t = [\alpha \kappa - \mathbf{v} \cdot \frac{\nabla

\phi}{|\nabla \phi|} ] |\nabla \phi | \,,

</math>| 9 | border=y}}

\end{equation}

where $<math>\kappa$</math> is the curvature of the contour and $<math>\alpha$</math> is a

user-selected constant.

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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==