Image Thresholding using Generalized Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 51

lating roughness values (see (3.3)). The weights considered are the number of occurrences
of gray values given by the graylevel histogram O I of the image I. Therefore, the weighted
cardinality of the underlying set (in G) gives the number of pixels in the image I that take
the gray values belonging to that set. From (3.30) and (3.31), we see that the grayness
ambiguity measure lies in the range [0, 1], where a larger value means higher ambiguity.

3.4 Image Thresholding based on Association Error

In this section, we propose a new methodology to perform image thresholding using the
grayness ambiguity measure presented in the previous section. The proposed methodology
does not make any prior assumptions about the image unlike many existing thresholding
techniques. As boundaries of regions in an image are in general not well-defined and nearby
gray values are indiscernible, we consider here that the various areas in an image are am-
biguous in nature. We then use grayness ambiguity measures of regions in an image to
perform thresholding in that image.

3.4.1 Bilevel Thresholding
Here we propose a methodology to carry out bilevel image thresholding based on the analy-
sis of the graylevel histogram of the image under consideration. Let us consider two regions
in the graylevel histogram of an image I containing a few graylevel bins corresponding to
the dark and bright areas of the image, respectively. These regions are obtained using two
predefined gray values, say g d and g b , with the graylevel bins in the range [g b , g max ] repre-
senting the initial bright region and the graylevel bins in the range [g min , g d ] representing
the initial dark region. The symbols g min and g max represent the lowest and highest gray
value of the image, respectively. A third region given by the graylevel bins in the range (g d ,
g b ) is referred to as the undefined region.
Now, let the association of a graylevel bin from the undefined region to the initial bright
region causes an error of Err b units and the association of a graylevel bin from the undefined
region to the initial dark region results in an error of Err d units. Then, if Err d > Err b
(Err b > Err d ), it would be appropriate to assign the graylevel bin from the undefined
region to the bright (dark) region.

The Proposed Methodology
Here we present the methodology to calculate the error caused due to the association of
a graylevel bin from the undefined region to a defined region. Using this method we shall
obtain the association errors corresponding to the dark and bright regions, that is, Err d and
Err b . Each of these association errors comprise of two constituent error measure referred
to as the proximity error and the change error.
Let H i represent the value of the i th bin of the graylevel histogram of a grayscale image
I. We may define S b , the array of all the graylevel bins in the initial bright region as

S b = [H i : i∈G b ], where G b = [g b , g b + 1, . . . , g max ] (3.32)

and S d , the array of all the graylevel bins in the initial dark region as

S d = [H i : i∈G d ], where G d = [g min , . . . , g d −1, g d ] (3.33)

Now, consider that a graylevel bin from the undefined region corresponding to a gray value
g a has been associated to the initial bright region. The bright region after the association