Image Processing Reference
In-Depth Information
3
Image Thresholding using
Generalized Rough Sets
3.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
3.2
Generalized Rough Set based Entropy Measures with
respect to the Definability of a Set of Elements
3-3
. .
ρ
R
(X) and
{
)
•
ρ
R
(X
3.3
Measuring Grayness Ambiguity in Images
. . . . . . . . .
3-11
Debashis Sen
Center for Soft Computing Research, Indian
Statistical Institute
3.4
Image Thresholding based on Association Error
. .
3-15
3.5
Experimental Results
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3-19
Sankar K. Pal
Center for Soft Computing Research, Indian
Statistical Institute
3.6
Conclusion
3-26
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Bibliography
3-27
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3.1
Introduction
Real-life images are inherently embedded with various ambiguities. In order to perceive
the nature of ambiguities in images, let us consider a 1001×1001 grayscale image (see
Figure 3.1(a)) that has sinusoidal gray value gradations in horizontal direction. When
an attempt is made to mark the boundary of an arbitrary region in the image, an exact
boundary can not be defined as a consequence of the presence of steadily changing gray
values (gray value gradation). This is evident from Figure 3.1(b) that shows a portion of
the image, where it is known that the pixels in the 'white' shaded area uniquely belong
to a region. However, the boundary (on the left and right sides) of this region is vague as
it can lie anywhere in the gray value gradations present in the portion. Value gradation
is a common phenomenon in real-life images and hence it is widely accepted (Pal, 1982;
Pal, King, and Hashim, 1983; Udupa and Saha, 2003) that regions in an image have fuzzy
boundaries.
Moreover, the gray levels at various pixels in grayscale images are considered to be impre-
cise, which means that a gray level resembles other nearby gray levels to certain extents. It
is also true that pixels in a neighborhood with nearby gray levels have limited discernibility
due to the inadequacy of contrast. For example, Figure 3.1(c) shows a 6×6 portion cut
from the image in Figure 3.1(a). Although the portion contains gray values separated by
6 gray levels, it appears to be almost homogeneous.
The aforementioned ambiguities in
3-1
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