Table 1: Segmentation error assessed by function F
′
(I) of Borsotti et al.
Method Regions 13 34 101
Classic 57.3692 30.8142 23.0695
Area 57.4575 49.3614 23.1594
Disc 55.3578 49.4325 51.5475
Table 2: Segmentation error assessed by function Q(I) of Borsotti et al.
Method Regions 13 34 101
Classic 581.4834 229.1602 110.9979
Area 535.1354 319.6443 103.0404
Disc 524.6157 325.5783 270.1857
2010) was the first reported application of the uni-
form decomposition. This Section presents and dis-
cusses other applications of the uniform decomposi-
tion, such as hierarchical segmentation, stereo vision
pre-processing and interactive segmentation.
4.1 Hierarchical Segmentation
The first application presented in this paper is the
use of the uniform decomposition to do hierarchical
segmentation. Since it is a straightforward applica-
tion of the decomposition, it is preferable to present
the application along the comparison with the classi-
cal hierarchical segmentation by selection of regional
minima via an extinction value threshold. It will be
demonstrated here that the uniform decomposition is
comparable to the classical hierarchical segmentation
besides retaining the respect the decomposition crite-
rion.
The comparison will be done with three decom-
positions: the uniform decomposition under the area
criterion, the uniform decomposition under the disc
criterion and the classical one. The three decompo-
sitions require different kind of parameters, so we
decided to do the comparison according to the num-
ber of regions generated by the segmentation meth-
ods. The three methods will be applied to in or-
der to produce the same number of regions and then
the segmentation results will be assessed subjectively
and quantitatively by application of two functions de-
signed to evaluate the quality of a segmentation.
Classical decomposition is the easiest way to seg-
ment an image in n regions: after compute the extinc-
tion value for each regional minima, it is taken the n-
th greatest extinction value to be the threshold value to
select the n regional minima with the highest extinc-
tion values
1
. There is no straightforward way to seg-
ment an image in n regions by applying the uniform
1
Let us assume, for simplification, that each regional
minimum has a distinct extinction value.
decomposition and it is a drawback of the method. So,
in order to compare the uniform decomposition to the
classical one using the chosen criterion, we searched
for area and disc parameters that generated segmen-
tations with the same number of regions and, then,
applied this number to segment the image using the
classical method.
Three experiments were accomplished. In the first
one, two uniform decompositions were done by ap-
plying an area parameter equals to 5000 and a disc
parameter equals to 33. Both decompositions gener-
ated 13 regions. To complete the first experiment, it
was applied the classical method to achievea segmen-
tation with 13 regions. The second experiment ap-
plied area parameter equals to 1600 and disc parame-
ter equals to 20. The two segmentations provided 34
regions and were compared to the classical method.
Finally, the third experiment compares three decom-
positions that generated 101 regions. The applied area
and disc parameters to this experiment were, respec-
tively, 500 and 10. Figure 2 shows all images gener-
ated to the three experiments.
Following an subjective criteria, we can see that
the segmentation provided by the uniform decompo-
sition is comparable to the provided by the classi-
cal one. In some case, the uniform decomposition
presents a better representation of some objects. Be-
sides the uniform decomposition presents similar re-
sults to the classical one, the subsets of the uniform
results respect the criteria applied to generated them.
For instance, in Fig. 2 (second line-left), no region has
its area lower than 1600 pixels and, in Fig. 2 (second
line-center), all regions stands an erosion done using
a s.e. with radius equals to 10. Such structures preser-
vation does not occur in the classical decomposition.
In order to demonstrate the similarity among both
uniform decompositions and the classical one, it was
done and evaluation experiment. The segmentation
evaluations were done by applying two functions pro-
posed by Borsotti et al. (M. Borsotti, P. Campadelli
THE UNIFORM DECOMPOSITION - Properties and Applications
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