Gradient vector flow

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, segmentation, and edge detection.

GVF is defined as a diffusion process operating on the components of the input vector field.

Some newer purposes including defining a continuous medial axis representation,[3] regularizing image anisotropic diffusion algorithms,[4] finding the centers of ribbon-like objects,[5] constructing graphs for optimal surface segmentations,[6] creating a shape prior,[7] and much more.

For uniformity of results, it is important to restrict the edge map intensities to lie between 0 and 1, and by convention

that minimizes the energy functional In this equation, subscripts denote partial derivatives and the gradient of the edge map is given by the vector field

It encourages the spatial variations in the components of the solution to be small by penalizing the sum of all the partial derivatives of

is large, for example, then the resulting field will be very smooth and may not agree as well with the underlying edge gradients.

, which will produce a smooth scalar field entirely dependent on its boundary conditions.

Figure 2 shows an illustration of a three-dimensional GVF field on the edge map of a simple object (see [13]).

A modification described in ,[14] called generalized gradient vector flow (GGVF) defines two scalar functions and reformulates the energy as While the choices

a user-selected constant, can improve the tradeoff between the data term and its regularization in some applications.

A learning based probabilistic weighted GVF extension was proposed in [16] to further improve the segmentation for images with severely cluttered textures or high levels of noise.

[17] Whereas the diffusion of GVF vectors from a conventional edge map acts in an isotropic manner, the formulation of MGVF incorporates the expected object motion between image frames.

This property has been described as an extension of the capture range of the external force of an active contour model.

It is also capable of moving active contours into concave regions of an object's boundary.

Because the diffusion process is inherent in the GVF solution, vectors that point in opposite directions tend to compete as they meet at a central location, thereby defining a type of geometric feature that is related to the boundary configuration, but not directly evident from the edge map.

GVF vectors also meet in opposition at central locations of objects thereby defining a type of medialness.

This definition is applicable in any dimension and yields an edge map that falls in the range

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

[21] In the case of parametric deformable models, the GVF vector field

, then a simple parametric active contour evolution equation can be written as Here, the subscripts indicate partial derivatives and

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

is first projected against the normal direction of the implicit wavefront, which defines an additional speed function.

A more sophisticated deformable model formulation that combines the geodesic active contour flow with GVF forces was proposed in .

[22] This paper also shows how to apply the Additive Operator Splitting schema[23] for rapid computation of this segmentation method.

[24] A further modification of this model by using an external force term minimizing GVF divergence was proposed in [25] to achieve even better segmentation for images with complex geometric objects.

The process first finds the inner surface using a three-dimensional geometric deformable model with conventional forces.

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.

Several notable recent applications of GVF include constructing graphs for optimal surface segmentation in spectral-domain optical coherence tomography volumes,[6] a learning based probabilistic GVF active contour formulation to give more weights to objects of interest in ultrasound image segmentation,[16] and an adaptive multi-feature GVF active contour for improved ultrasound image segmentation without hand-tuned parameters.

Results from Gradient Vector Flow algorithm applied to 3-D Metasphere data
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.
Fig. 2. The object shown in the top left is used as an edge map to generate a three-dimensional GVF field. Vectors and streamlines of the GVF field are shown in the (Z) zoomed region, (V) vertical plane, and (H) horizontal plane.
Fig. 3. An active contour with traditional external forces (left) must be initialized very close to the boundary and it still will not converge to the true boundary in concave regions. An active contour using GVF external forces (right) can be initialized farther away and it will converge all the way to the true boundary, even in concave regions.
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).