Biomedical Engineering Reference
In-Depth Information
Table 1.
Computing times and speed-up of parallel program
running on 2 to 32 processors.
# processors
2
4
8
16
32
time(secs)
373.4
198.15
112.5
62.85
38.5
speed-up
2
3.77
6.64
11.88
19.4
rel. speed-up
1
0.94
0.83
0.74
0.61
In the next example, we solve the same problem but with an image resolution
given by 128
3
voxels. This 3Dexamplewe solved on anMPP cluster at CINECA in
Bologna. Since due to the huge amount of unknowns we cannot solve the problem
on a single processor, we started the report on the results with computation on two
processors. Table 1 shows the computing times in seconds for 34 time steps when
the segmentation was achieved using the same parameters as above. As we can
see, the computation times are well scaled using a larger number of processors. As
expected, due to the increasing complexity of communication using a large number
of processors, the relative speed-up (i.e., speed-up over a number of processors)
is decreasing.
Next, we present an example of subjective surface segmentation of a 3D
echocardiographic image of size 81
×
87
×
166 voxels. We use
τ
=0
.
001,
K
=1,
TOL
=0
.
001, and
δ
=10
−
5
. As one can see from the volume rendering
visualization in Figure 22, the 3D image is very noisy; however, the surface of
the left ventricle is observable. How noisy is the image intensity can be seen also
from Figure 23, where we plot intensity and its graph in one 2D slice. Due to the
high complexity of this image, we start the segmentation process with an initial
function with maxima in several “points of view” inside the desired object. We
again evolve the segmentation function until the L
2
norm of the difference of the
two subsequent time steps is less than the prescribed threshold
δ
. We then check
a 2D slice with relatively good ventricular boundary edges (Figure 24), where we
can see an accumulation of level sets along the inner boundary of the ventricle
(Figure 24, left). The largest gap in the histogram (Figure 24, right) indicates
the shock in the segmentation function, which can be used for segmentation. We
choose one level inside the gap, and plot it inside the slice (Figure 25, left). We
can check what this level set looks like in other noisy slices (Figure 25, right,
Figure 27), and then we visualize the corresponding 3D isosurface (Figure 28),
which gives a realistic representation of the left ventricle.
