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measure obtained using the proposed classes of entropies. The strength of the proposed
methodology lies in the fact that it does not make any prior assumptions about the im-
age unlike many existing thresholding techniques. We present a novel bilevel thresholding
scheme that performs thresholding by assigning a bin in the graylevel histogram of an im-
age to one of the two classes based on the computation of certain association errors. In the
methodology, the graylevel histogram is first divided into three regions, say, bright (a region
of larger gray values), dark (a region of smaller gray values) and an undefined region. These
regions are obtained using two predefined gray values, which are called the seed values. It
is known (prior knowledge) that the bins of a graylevel histogram representing the smallest
and the largest gray value would belong to the dark and bright regions, respectively. Hence,
we consider that the graylevel bins of the histogram below the smaller seed value belong
to the dark region and those above the larger seed value belong to the bright region. Rest
of the graylevel bins form the undefined region. Then, each graylevel bin in the undefined
region is associated with the defined regions, dark and bright, followed by the use of gray-
ness ambiguity measure to obtain the errors due to the associations. The thresholding is
then achieved by comparing the association errors and assigning each graylevel bin of the
undefined region to one of the defined regions that corresponds to the lower association
error.
To carry out multilevel thresholding in a manner similar to the bilevel thresholding,
more than two seed values would be required. Unlike bilevel thresholding, in the case of
multilevel thresholding we do not posses the prior knowledge required to assign all the seed
values. Hence, we present a binary tree structured technique that uses the proposed bilevel
thresholding scheme in order to carry out multilevel thresholding. In this technique, each
region (node) obtained at a particular depth are further separated using the proposed bilevel
thresholding method to get the regions at the next higher depth. The required number of
regions are obtained by proceeding to a su
cient depth and then discarding some regions
at that depth using a certain criterion. As a region in the graylevel histogram of an image
corresponds to a region in the image, the aforementioned thresholding methodology would
divide the image into predefined (required) number of regions.
Image thresholding operations for segmentation and edge extraction are carried out in this
chapter employing grayness ambiguity measure obtained based on the proposed classes of
entropies. The aforesaid image thresholding operations are performed in two ways, namely,
by the ambiguity minimization method reported in (Pal et al., 1983) and by the proposed
image thresholding methodology. Qualitative and quantitative experimental results ob-
tained using aforementioned methods are compared to that obtained using a few popular
existing image thresholding techniques in order to demonstrate the utility of the proposed
entropy measures and the effectiveness of the proposed image thresholding methodology.
The organization of this chapter is as follows. In Section 3.2, the proposed entropy mea-
sures and their properties are presented after briefly mentioning the existing entropy mea-
sures based on rough set theory. The use of the proposed entropy measures for quantifying
grayness ambiguity in images is presented in Section 3.3. The explanation of the proposed
image thresholding methodology is given in Section 3.4. In Section 3.5, experimental results
are presented to demonstrate the utility and effectiveness of the proposed entropy measures
and image thresholding methodology. The chapter concludes with Section 3.6.
3.2
Generalized Rough Set based Entropy Measures with
respect to the Definability of a Set of Elements
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