GVF can be easily generalized to higher dimensions. Let
be an edge map defined in
.
The GVF field in
is defined as the vector
field
which
minimizes the energy functional
(14) |
A solution to these Euler equations can be found by introducing a time
variable t and finding the steady-state solution of the following linear
parabolic partial differential equation
A preliminary experiment using GVF in three dimensions was carried out using the object shown in Fig. 7a. This shape was created on a 64^{3} grid, and rendered using an isosurface algorithm. The roughness of its surface reflects the relative coarseness of the grid, although the isosurface algorithm makes it somewhat smoother because of its ability for sub-pixel resolution. We computed the 3-D GVF field for this data set using a numerical approximation to (20) and . This result on the two planes shown in Fig. 7b, is shown projected onto these planes in Figs. 7c and d. The same characteristic properties that we observed about GVF in 2-D are apparent here as well.
Next, a deformable surface [3] was initialized as the sphere shown in Fig. 7e; it is neither entirely inside nor entirely outside the object. The surface was then allowed to move in response to the GVF external forces and its own internal forces. Intermediate results after 10 and 40 iterations are shown in Figs. 7f and g. The final result after 100 iterations is shown in Fig. 7h. It is interesting to note that the final active surface is much smoother than the isosurface rendering. This is because the internal forces of the active surface favor smoother surfaces.