FUZZY SET THEORETIC APPROACH TO IMAGE THRESHOLDING

International Journal of Computer Science, Engineering and Applications (IJCSEA) Vol.1, No.6, December 2011
DOI : 10.5121/ijcsea.2011.1605 63
FUZZY SET THEORETIC APPROACH TO IMAGE
THRESHOLDING
Ambar Dutta1
, Avijit Kar2
and B. N. Chatterji3
1
Department of Computer Engineering,
Birla Institute of Technology, Mesra, Kolkata Campus, India
adutta@bitmesra.ac.in
2
Department of Computer Science and Engineering,
Jadavpur University, Kolkata, India
avijit_kar@cse.jdvu.ac.in
3
Department of Electronics and Communication Engineering,
B. P. Poddar Institute of Management and Technology, Kolkata, India
bnchatterji@gmail.com
ABSTRACT
Thresholding is a fast, popular and computationally inexpensive segmentation technique that is always
critical and decisive in some image processing applications. The result of image thresholding is not always
satisfactory because of the presence of noise and vagueness and ambiguity among the classes. Since the
theory of fuzzy sets is a generalization of the classical set theory, it has greater flexibility to capture
faithfully the various aspects of incompleteness or imperfectness in information of situation. To overcome
this problem, in this paper we proposed a two-stage fuzzy set theoretic approach to image thresholding
utilizing the measure of fuzziness to evaluate the fuzziness of an image and to determine an adequate
threshold value. At first, images are preprocessed to reduce noise without any loss of image details by fuzzy
rule-based filtering and then in the final stage a suitable threshold is determined with the help of a
fuzziness measure as a criterion function. Experimental results on test images have demonstrated the
effectiveness of this method.
KEYWORDS
Fuzzy Sets, Thresholding, Measures of fuzziness, Noise Removal
1. INTRODUCTION
Image segmentation is that phase of image processing which partitions an image into a set of non-
overlapping meaningful homogeneous regions whose union is the entire image. Its objective is to
extract the semantic objects lying in images either by dividing any given image into meaningful
contiguous regions, or by extracting one or more important objects in images. The level upto
which the partitioning is carried out depends on the problem being solved. There are various
reasons for which segmentation is considered to be an extremely difficult task in image
processing for non-trivial images: (1) lack of a priori knowledge about the number of segments
present in the image; (2) variation of the gray scale values and their distributions from an image
to another; and (3) inherent limitations of imaging sensors. Thus, vagueness in image processing
arises from grayness ambiguity and uncertain knowledge about image. Uncertainty may arise at
any stage in an image analysis system. Any decision made at a particular phase will have an

64
impact on the subsequent phases. Hence, it is essential for an image analysis system to have
sufficient provision for representing the uncertainties involved at every phase, so that the ultimate
output of the system can be associated with the least uncertainties. One of the most popular tools
for image segmentation is thresholding or gray level segmentation which consists of classification
of the pixels as belonging to either the set of background pixels or the set of object pixels as a
function of the intensity level. The threshold is determined by the histogram of the gray levels of
pixels in the image. Thresholding is a transformation of an input image g(x, y) into a segmented
binary output image b(x, y) as follow:
where the pixels having gray level intensity greater than T will belong to the object and are
represented with a color b1, while the pixels belonging to the background are represented with
another color b0.
The optimal choice of the threshold T
*
of T based on a designed criterion function is a difficult
task because of the presence of noise and vagueness and ambiguity among the classes. Recent
advances in the fuzzy set theory provide the possibilities for developing new image segmentation
techniques. Fuzzy models are capable to handle ambiguity and noise in the images and hence can
improve the selection of optimal threshold for better image segmentation. Bezdek et al [1]
classified fuzzy approaches for image segmentation into four categories: segmentation via
thresholding, segmentation via clustering, supervised segmentation, supervised segmentation and
rule-based segmentation. On the other hand, Tizhoosh [2] categorized fuzzy segmentation
techniques into fuzzy thresholding, fuzzy rule, fuzzy clustering, fuzzy geometry and fuzzy
integral based segmentation. Chaira and Ray [3] used fuzzy set theory to threshold an image and
they introduced four types of fuzzy thresholding methods. Qiao et al [4] formulated a new
criterion for segmenting small objects by exploring the knowledge about intensity contrast. Saha
and Ray [5] introduced an adaptive thresholding technique via minimax optimization.
The remainder of this paper is structured as follows. Section 2 deals with basic related theories
such as fuzzy set, membership function and several fuzziness measures those are useful for better
understanding. We demonstrate our new fuzzy set-theoretic approach to threshold determination
in section 3. In section 4, the superiority of our approach is illustrated with the help of
experimental results. Finally, we conclude in section 5.
2. RELATED THEORIES
In this section, we explain a few relevant theories that will be helpful for further understanding
and discussion.
2.1. Fuzzy Set and Membership Function
Fuzzy sets were introduced by Zadeh [6] in 1965 as a new way of representing vagueness in
everyday life. Conventional sets contain objects that satisfy precise properties required for
membership, whereas fuzzy sets contain objects that satisfy imprecisely defined properties to
varying degrees. A fuzzy set A of the universe of disclosure X is defined as a collection of
ordered pairs

65
where )1)(0)(( ≤≤ xx AA µµ gives the degree of belonging of the element x to the set A. The
flexibility of fuzzy set theory is associated with the elasticity property of the concept of its
membership function. The grade of membership is a measure of the compatibility of an object
with the concept represented by a fuzzy set. Uncertainty in an image pattern may be explained in
terms of grayness ambiguity or spatial ambiguity or both. Grayness ambiguity means
‘indefiniteness’ in deciding whether a pixel is white or black. Spatial ambiguity refers to
‘indefiniteness’ in the shape and geometry of a region within the image.
In order to threshold an image, it is essential to determine a membership function )( fFµ
associated with each gray level of the histogram. The membership function defines the degree of
belonging of any pixel f either to the background or the object set depending on the relationship
between its gray level and the threshold T. There are various ways in which a membership
function can be defined. Irrespective of the measure of fuzziness used, an estimate of the mean
gray level of the background Mb and that of the objects Mo is needed, where both values depend
on the threshold T:
The smaller the difference between the gray level of any pixel and the mean for its class, the
greater will be the value of the membership function.
2.2. Measures of Fuzziness
There are two types of uncertainty: (1) vagueness that corresponds to the uncertainty associated
with the problem of finding well defined borders between the objects to be segmented; and (2)
ambiguity that corresponds to the uncertainty related to the difficulty of making the correct choice
among two or more alternatives. The measure of fuzziness provides a way to measure the degree
of fuzziness of a fuzzy set. A measure of fuzziness is a function
where F(X) denotes the set of all fuzzy subsets of X. There are few properties of the measures of
fuzziness I:
Below a few measures of fuzziness are discussed.
2.2.1. De Luca and Termini's Entropy Measure of Fuzziness
Motivated by Shannon's entropy measure [7], which is considered to be the fundamental base of
the information theory, De Luca and Termini [8] proposed a non-probabilistic entropy function
f(A) as a measure of fuzziness as follows.

66
The normalized measure )(
^
Af of )(Af is defined as:
where |X| is the cardinality of the universal set X.
2.2.2. Kauffmann's Measure of Fuzziness
Kauffmann [9] proposed a measure of fuzziness as
where )(xCµ is defined as
2.2.3. Yager's Measure of Fuzziness
Yager [10] proposed a measure of fuzziness that depends on the relationship between the fuzzy
set and its complement. The distance between a fuzzy set A and its complement is defined as:
The measure of fuzziness is defined as
3. OUR APPROACH TO THRESHOLD DETERMINATION
The thresholding scheme, proposed in this paper, works in two stages. In the first stage, images
are preprocessed with the noise removal and edge preservation by fuzzy rule-based filtering. In
the second and final stage, a suitable threshold is determined with the help of a fuzziness measure
as a criterion function. Below we give a detailed description of the two stages.
3.1. Preprocessing with Noise Removal
The presence of noise affects the accuracy of image processing. In order to reduce the noise, non-
linear filtering techniques provide better result than linear filters as they do not degrade the edges
and the details of the image. Recent advances in fuzzy logic gives new dimensions for developing
new noise removal techniques. In this paper, in order to reduce noise we used a Gaussian fuzzy
filter with mean center (GFFMC). The filters used in the image processing that employs fuzzy

67
logic model are known as fuzzy filters. Let x(i, j) be the input of a two-dimensional fuzzy filter,
the output y(i, j) of the fuzzy filter is defined as:
where F[x(i, j)] is the general window function and A is the area of the window. For a square
window of dimension NN × , the range of m and n are: MsrM ≤≤− , , where N = 2M + 1.
Moreover, the Gaussian fuzzy filter with mean center (GFFMC) is defined as follows:
where
_
x (i, j) and σ (i, j) denote the mean and standard deviation of all input values x (i + m, j +
n) respectively for m, n in A at discrete indices (i, j). The reason for application of Gaussian fuzzy
filter is to reduce both impulse noise and random noise from the image at the same time without
degrading any image details.
3.2. Determination of Threshold
Huang and Wang [11] selected an optimal threshold by using the entropy measure as a criterion
function. In this paper, we used an entropy-based fuzzy thresholding technique that uses the
entropy of the background and foreground (object) regions, the entropy between the original and
binarized image, relative entropy etc. The entropy of a random variable measures the uncertainty
of the random variable and hence is a measure of the amount of information required to describe
the random variable. Entropy serves as a measure of separation in an image that finds out the
information of the two regions - above and below the threshold separately and calculates the
entropy of the two classes. The entropies for all the gray levels below and above the threshold are
added separately and then the entropy of the two classes is added to obtain the final entropy
expression. The optimal threshold value is that gray level which corresponds to the maximum of
all the calculated entropy expressions. Optimal threshold can also be obtained by minimizing the
cross entropy of the two regions [12].
An L-level image G ( NM × ) can be considered as an array of fuzzy singletons, where each
singleton corresponds to an image pixel, each having a membership value denoting its degree of
possessing some property (brightness, darkness etc). In the fuzzy set theoretic notation, it may be
written as
Then Shannon's entropy for an image of size NM × is given as

68
which is used to obtain an optimum threshold. The entropy is calculated for each threshold gray
level and the gray level that corresponds to the minimum entropy value represents the optimum
threshold.
The normalized fuzziness function of an image can be calculated as:
where h(f) represents the histogram of the image which shows the frequency distribution of the
gray levels and
Here in this paper, we have used Shannon's measure of fuzziness, but other fuzziness measures
can also be used instead of it. The fuzziness function g(f, t) is calculated for all the values of t.
The optimal threshold t* is the gray level that minimizes g(f, t) and hence it is the gray level that
maximizes the separation of between the means of two classes as much as possible.
Mathematically, it can be expressed as:
The detailed algorithm is given below:
1. Gaussian fuzzy filter is applied to reduce both impulse and random noise from the image
at the same time.
2. The mean of the foreground and background regions are calculated for each threshold
gray level.
3. The membership degree of the pixels of the image is calculated for each threshold gray
level.
4. The fuzzy entropy is calculated for each threshold gray level.
5. The minimum value of the fuzzy threshold is selected.
6. The image is thresholded with the gray level corresponding to the minimum value of the
fuzzy entropy.

69
Figure 1: Test Images and their histograms
4. EXPERIMENTAL RESULTS
In order to evaluate the effectiveness of the proposed method, we have tested our algorithm with
the help of a number of images, but due to limitation of space, in this paper we have shown the
experimental results for only three test images given in Figure 1. Here, we have compared our
algorithm with two well-known global thresholding techniques, proposed by Otsu [13] and
Kittler-Illingworth [14] and one relatively new technique, proposed by Huang and Wang [11].
Otsu's method selected the threshold t in such a way that the between-class variance is maximized
and the intra-class variance is minimized. The algorithm positioned t midway between the means
of the two classes. On the other hand, Kittler and Illingworth assumed a Gaussian mixture model,
that is, the pixels in the two categories came from a normal distribution and the gray level that
minimized the number of misclassifications between the two normal distributions with the given
means, variances, and proportions was considered as the threshold. Whereas, Huang and Wang
proposed an algorithm for image thresholding by minimizing different measures of fuzziness.
Figure 2: Thresholding results for Block image

70
Figure 3: Thresholding results for House image
It is observed from the figures Figure 2, Figure 3 and Figure 4 that our algorithm performs better
than the two established thresholding techniques - Otsu's and Kittler-Illingworth's algorithms, and
performs almost similarly (in fact, better for few test images) if we compare our algorithm with
Huang-Wang's algorithm. In the presence of noise, our algorithm outperforms the three remaining
algorithms considered in this paper.
Figure 4: Thresholding results for Tank image
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5. CONCLUSIONS
Image thresholding based on gray level histogram is an efficient and important tool for image
segmentation. In this paper, we suggested a two-stage fuzzy set theoretic approach to image
thresholding that uses the measure of fuzziness to evaluate the fuzziness of an image and to find
an optimal threshold value. Fuzzy based methods consider the uncertainty in the image due to the
imprecise pixel gray levels and vagueness in the image regions and boundaries which is
incorporated in the form of membership function. Initially, images are preprocessed to reduce
noise components without affecting much image details by fuzzy rule-based filtering and then in
the second phase, a suitable threshold is determined with the help of a fuzziness measure as a

71
criterion function. We demonstrated the effectiveness of our algorithm with the help of a number
of test images. We shall try to extend this work to color images.
REFERENCES
[1] J. C. Bezdek, J. Keller, K. Raghu and N. R. Pal, Fuzzy models and algorithms for pattern recognition
and image processing, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers,
Boston/London/Dordrecht, 1999.
[2] H. R. Tizhoosh, Image thresholding using type II fuzzy sets, Pattern Recognition, Vol. 38, No. 12,
2363 - 2372, 2005.
[3] T. Chaira and A. K. Ray, Threshold selection using fuzzy set theory, Pattern Recognition Letters, Vol.
25, No. 8, 865 – 874, 2004.
[4] Y. Qiao, Q. Hu, G. Qian, S. Luo and W. L. Nowinski, Thresholding based on variance and intensity
contrast, Pattern Recognition, Vol. 40, Iss. 2, 596 - 608, 2007
[5] B. N. Saha and N. Ray, Image thresholding by variational minimax optimization, Pattern
Recognition, Vol. 42, Iss. 5, 843 - 856, 2009.
[6] L. A. Zadeh, Fuzzy Sets, Information Control, Vol. 8, 338 - 353, 1965
[7] C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27,
379 - 423, 1948
[8] A. De Luca and S. Termini, A definition of a non-probabilistic entropy in the setting of fuzzy set
theory, Information and Control, Vol. 20, 301 - 312, 1972.
[9] Kaufmann, Introduction to the Theory of Fuzzy Subsets - Fundamental Theoretical Elements, Vol. 1,
Academic Press, New York, 1975.
[10] R. R. Yager, On the measure of fuzziness and negation, Part - 1: Membership in the unit interval,
International Journal of General Systems, Vol. 5, 221 - 229, 1979.
[11] L.K. Huang and M.J. Wang, Image thresholding by minimizing the measure of fuzziness, Pattern
Recognition, Vol. 28, No. 1, 41 - 51, 1995.
[12] C.H. Li and C.K. Lee, Minimum cross entropy thresholding, Pattern Recognition, Vol. 26, No. 4, 617
- 625, 1993.
[13] N. Otsu, A threshold selection method from gray-level histogram, IEEE Transactions on Systems,
Man and Cybernetics, Vol. 9, No. 1, 62 - 66, 1979.
[14] J. Kittler and J. Illingworth, Minimum error thresholding, Pattern Recognition, Vol. 19, No. 1, 41 -
47, 1986.
Authors
Ambar Dutta did his B.Sc. (Honors) in Mathematics from Presidency College,
Kolkata in 1999 and MCA from Jadavpur University, Kolkata in 2002. He is working
as a Lecturer in the department of Computer Science and Engineering, Birla Institute of
Technology, Mesra, Kolkata Extension Centre. He s ubmitted his PhD thesis in
December, 2010 in Jadavpur University, Kolkata in the area of image processing
(corner detection and matching). His research interest includes Image Processing,
Pattern Recognition, Data Mining and Soft Computing.

72
Avijit Kar did his M.Sc. and Ph.D in 1980 and 1984 respectively from IIT Kharagpur.
He is currently a professor in the department of Computer Science and Engineering in
Jadavpur University, Kolkata. He has supervised several PhD theses and is actively
involved in many R & D activities and IT related consultancy for Government of India
and the private sector. His research interest includes biomedical imaging as well as SAR
imaging. He is also into computer systems reliability. He is involved in a large number
of industry sponsored development projects.
B. N. Chatterji obtained BTech (Hons) (1965) and Phd (1970) in Electronics and
Electrical Communication Engineering of IIT, Kharagpur. He did Post Doctoral work at
University of Erlangen-Nurenberg, Germany during 1972 – 73. Worked with Telerad
Pvt Ltd, Bombay (1965), Central Electronics Research Institute, Pilani (1966) and IIT,
Kharagpur as faculty member during 1967 – 2005. He was Professor during 1980 –
2005, Head of the Department during 1987 – 1991, Dean Academic Affairs during 1994
– 1997 and Member of Board of Governors of IIT, Kharagpur during 1998 – 2000. He
has published about 150 journal papers, 200 conference papers and four books. He was Chairman of four
International Conferences and ten National Conferences. He has coordinated 25 short-term courses and was
the chief investigator of 24 Sponsored Projects. He is the Fellow/Life Member/Member of eight
Professional Societies. He has received ten National Awards on the basis of his Academic/Research
contributions. His areas of interests are Pattern Recognition, Image Processing, Signal Processing, Parallel
Processing and Control Systems.

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