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could split the pipes unless the junction box is vented. The ancient Romans
knew of this problem and their solution was to relieve the pressures in their
underground aquaducts with a series of vertical vents and fountains. Vents
allow the excess water to rise up the vent pipe, providing a restoring force to
balance the flow. Thus the water level in the vent pipe is equivalent to the
potential function in the GMS algorithm.
Now consider the following system of differential equations.
∂P
∂t
div
F
,
=
−
(10.6)
∂
F
∂t
=
−∇
P,
(10.7)
F
2
≤
g.
(10.8)
These first two equations, taken together, form a simple system of wave equa-
tions. They may be interpreted as a linear model of the dynamics of an ide-
alized fluid with pressure
P
and velocity
F
. Without loss of generality and to
maintain symmetry between source and sink, we fix the scalar potential field
P
at the source
s
and sink
t
such that
P
s
=1and
P
t
=
1.
It can be shown [7] that at convergence the potential field is an isosurface of
value +1 in the region connected to the source and -1 in the region connected
to the sink. Thus the potential field becomes an indicator function that tells
us whether we are inside or outside the minimal surface. Without loss of
generality, we choose level set 0 as the minimal surface.
−
10.4.3 Applications of the GMS Algorithm
The evolution of the potential function to determine the minimal surface cor-
responding to a human lung is shown in Figure 10.18. Note how the potential
function evolves to an indicator function separating the interior region of the
lung from the exterior. Figure 10.19 shows the segmentation of volumetric
MRI data to segment the hippocampus.
A less obvious application is the use of GMS to find the optimal 3D recon-
struction from multiview images. Now the use of a stereo pair of images to de-
termine ground elevation from image disparity is a well known technique from
aerial photogrammetry. Unfortunately, stereo image pair photogrammetry can
only provide so-called 2-1/2D rather than true 3D reconstruction—with just
two frontal images, it is impossible to reconstruct the back of an object. So
true 3D model reconstruction requires multiple images—hence the term mul-
tiview reconstruction.
Leung [105] developed a technique called Embedded Voxel Coloring (EVC),
which employed space carving and photoconsistency contraints to the 3D re-
construction problem. He determines the 3D surface that optimally satis-
fied all the reconstruction constraints using the GMS algorithm. Figure 10.21
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