J. Biomedical Science and Engineering, 2009, 2, 575-581
doi: 10.4236/jbise.2009.28083 Published Online December 2009 (http://www.SciRP.org/journal/jbise/
Published Online December 2009 in SciRes. http://www.scirp.org/journal/jbise
A novel vague set approach for selective contrast
enhancement of mammograms using multiresolution
Arpita Das1, Mahua Bhattacharya2*
1Institute of Radio Physics & Electronics, University of Calcutta, Kolkata, India;
2Indian Institute of Information Technology & Management, Gwalior, India.
Emali: arpita.rpe@caluniv.ac.in; *mb@iiitm.ac.in
Received 23 July 2009; revised 2 September 2009; accepted 3 September 2009
The proposed algorithm introduces a novel vague set
approach to develop a selective but robust, flexible
and intelligent contrast enhancement technique for
mammograms. Wavelet based filtering analysis can
produce Low Frequency (LF) and High Frequency
(HF) subbands of the original input images. The ex-
tremely small size microcalcifications become visible
under multiresolution techniques. LF subband is then
fuzzified by conventional fuzzy c-means clustering
(FCM) algorithm with justified number of clusters.
HF components, representing the narrow protrusions
and other fine details are also fuzzified by FCM with
justified number of clusters. Vague set approach
captures the hesitancies and uncertainties of truly
affected masses/other breast abnormalities with
normal glandular tissues. After highlighting the
masses/microcalcifications accurately, both LF and
HF subbands are transformed back to the original
resolution by inverse wavelet transform. The results
show that the proposed method can successfully en-
hance the selected regions of mammograms and pro-
vide better contrast images for visual interpretation.
Keywords: Multiresolution; Vague Set Approach; True-
membership Functions; False-Membership Functions
Breast cancer continues to be a significant public health
problem. Primary prevention seems to be impossible
since the causes of this disease still remain unknown.
Thus early detection is the key to improving breast can-
cer prognosis. Screen/Film mammography is one of the
most reliable, effective methods for early detection of
breast carcinomas in women [1]. Screening of asympto-
matic women using screen/film mammography has been
shown a significantly reduction of breast cancer death
Major advances in screen/film mammography have been
occurred in the past few decades which in turns improved
the image resolution and film contrast [2]. Despite of
these advantages, screen/film mammography based im-
age interpretation still remains very difficult. Breast ma-
mmograms are generally examined in presence of be-
nign/malignant masses and other indirect signs of ab-
normalities like microcalcifications, skin thickening. The
major reason of poor visibility is due to the slight dif-
ferences of X-ray attenuation between normal glandular
tissues and affected mass/microcalcification spots. The
extreme small size of microcalcifications is also a rea-
sonable cause of its low contrast appearance in mam-
mograms. These facts create problem in the detection of
breast cancer, especially in younger women with dense
breasts. For this purpose development of computerized
automated breast cancer diagnosis system attracts much
attention of the researchers.
As it is described that contrast enhancement of mam-
mographic features is critical but essential for breast can-
cer diagnosis. Many conventional contrast enhancement
techniques adopt a global approach to enhance the im-
ages. However, it is quite difficult to enhance all features
equally in the mammograms using those global app-
roaches, because many local contrast information and
details may be lost in the dark or bright regions of the
breast [3].
As a result local-contrast enhancement techniques are
developed to highlight the necessary local features.
Adaptive neighborhood contrast enhancement (ANCE)
method was implemented for improvement of medical
image quality [4]. ANCE method provides the advan-
tages of enhancing or preserving image contrast while
suppressing the noise. However, it has a drawback. The
performance of the ANCE method largely depends on
how to determine the parameters used in the processing
steps. In the article [5], contrast-to-noise ratio (CNR) of
the low-contrast lesions is improved relative to the back-
ground. Moreover, adaptive contrast enhancement (ACE)
algorithm is developed to adjust high frequency compo-
nents of the images using contrast gains [6]. Incorporat-
576 A. Das et al. / J. Biomedical Science and Engineering 2 (2009) 575-581
SciRes Copyright © 2009 JBiSE
ing non-linear function for computing ACE produces an
adequate contrast gains resulting in little noise over en-
hancement. Mammographic feature enhancement algo-
rithms are also key to the detection of breast abnormali-
ties [7]. Lai et al. [8] compared several image enhance-
ment methods for detecting circumscribed masses in ma-
mmograms. Authors compared an edge-preserving smoo-
thing function [9], a half-neighborhood method [10],
k-nearest neighborhood, directional smoothing [11], and
median filtering [12]. In addition, authors also proposed
an algorithm of selective median filtering. Among the
techniques implemented, they concluded that selective
median filtering with a particular size of mask performed
best for image enhancement.
The fuzzy set theory [13] provides a suitable algorithm
in analyzing complex systems and decision processes
when the pattern indeterminacy is due to the inherent
variability and vagueness. Image enhancement using
smoothing with fuzzy sets [14] is developed for improv-
ing contrast of the pixels. Adaptive fuzzy logic based
contrast enhancement method [15] is also developed to
enhance the important mammographic features. In this
technique, uncertain nature of mammograms is captured
by the fuzzy membership functions. The index of fuzzi-
ness along with entropy of an image also reflects a kind
of quantitative measure of its enhancement quality [16].
In this approach, fuzzy theory is adapted to the fre-
quency content of each coefficient block in the DCT
(Discrete Cosine Transform) encoded JPEG images. In
decision making problems, particularly in a case of me-
dical diagnosis, there is a fair chance of the existence of
a non-null hesitation part at each moment of evaluation
of any unknown object.
Vague Sets (VS) is intuitively straightforward exten-
sions of Zadeh’s fuzzy sets [17]. The drawback of using
single membership value in fuzzy set theory is that the
evidence for an element Uu
and the evidence
against are mixed together (U is a classical set
of objects, called the universe of discourse). The nota-
tion of VS, proposed by Gau et al. [18] allows more
generalized interval based membership values instead of
point based single membership as in fuzzy sets. To be
more precise, a basic assumption of fuzzy set theory is
that, if we specify the degree of membership of an ele-
ment in a fuzzy set as a real number from [0,1], say a,
then the degree of its non-membership is automatically
determined as 1-a, need not hold for vague sets. In VS
approach, it is assumed that non-membership should not
be more than 1-a. The difference expresses the hesitancy
concerning both membership and non-membership of an
element to a set. This is mainly due to the fact that VS is
more consistent with human behavior by reflecting the
hesitancy present in real life situations.
In the proposed algorithm, VS approach is imple-
mented to develop a selective but robust, flexible and
intelligent contrast enhancement method for mammo-
grams. Moreover, the hesitation of the breast mass boun-
daries and other abnormalities with the surrounding
dense breast tissues is captured by VS. Multiresolution
technique is incorporated to achieve more accurate en-
hancement of the detail features. Experimental results
successfully enhance the selected mass regions of mam-
mograms. Experimental results evaluate the proposed
technique with conventional fuzzy based contrast enhan-
cement methods.
The overall proposed selective contrast enhancement
scheme implemented on mammogram has been demon-
strated in Figure 1 stepwise and in subsequent sections.
The scheme involves a) decomposition of input images
by wavelet transform for multiresolution analysis of
coarse and fine objects b) fuzzification by standard FCM
with justified number of clusters c) implementation of
Vague Set approach for capturing the uncertainties be-
tween truly masses/microcalcifications from the sur-
rounding dense glandular breast tissues d) defuzzifica-
tion of membership functions to the spatial intensity
domain and finally e) reconstruction of selectively en-
hanced image by inverse wavelet transform.
2.1. Multiresolution Approach for Decomposition
of Input Image in Fine and Coarse Objects
In an image, if both fine and coarse objects or low and high
contrast objects are present simultaneously, it is advan-
Figure 1. Block diagram for selective
contrast enhancement scheme.
A. Das et al. / J. Biomedical Science and Engineering 2 (2009) 575-581 577
SciRes Copyright © 2009 JBiSE
tageous to study them at several resolutions. This is the
fundamental motivation for multiresolution processing.
In the mammograms, very fine shades of gray-level in-
tensities can be pointed out clearly by using multiresolu-
tion approach [18-19]. The mathematical concept behind
this approach is described below.
To determine the fine details of level j image, we first
interpolate the level j-1 image to produce one of the
same sizes as the level j image. This interpolated image
is also known as prediction of level j image.
Subtracting the prediction of level j image from the
original level j image produces level j residual image.
This residual image contains only the fine details or the
high frequency sub-band component of level j image.
The technique for producing level j-1 approximation and
level j prediction residual is the fundamental logic of
wavelet based multiresolution processing.
Image pyramid: An image pyramid is a collection of
decreasing resolution images arranged in the shape of a
pyramid as shown in Figure 2. The base of the pyramid
contains the highest resolution representation of the im-
age. Moving up the pyramid both size and resolution of
the images are decreased. The apex contains the low-
est-resolution approximation. The base level J is size N
N when N=2J, intermediate level j is size 2j2j, where
0 j J. Fully populated pyramids are composed of J+1
resolution levels from 2J2J to 2020, but in practice
most pyramids are truncated to P+1 levels, where J-p
jJ and 1 < P J, since a 11 pixel image is of little
In multiresolution analysis (MRA), scaling function is
used to create a series of approximations of an image,
each differing by a factor of 2 from its nearest neighbor-
ing approximations. Additional functions, called wave-
lets, are then used to encode the difference in informa-
tion between adjacent approximations.
In the multiresolution approach, input image is de-
composed into LF and HF subbands by forward wavelet
transform whereas after defuzzification, LF and HF
subbands are composed by inverse wavelet transform.
As shown in Figure 3, any image of level j can be ap-
proximated to level j-1 by applying HAAR wavelet
transform. It will contain only the gross structure of level
j. The size of level j-1 image is just half of level j image.
Figure 2. A pyramidal image structure.
Figure 3. System for constructing image pyramids.
2.2. Intensity Based Clustering of LF and HF
Details of Input Image Using Fuzzy C-Mean
Fuzzy c-means clustering is the most widely used algo-
rithm of fuzzy classification. While considering the
fuzzy set theory, the algorithm is developed based on k-
means clustering. In this algorithm, each pixel does not
belong exclusively to any single cluster but is represent-
ed by several memberships of each cluster. For a pixel,
membership of each cluster is [0,1] and sum of those
memberships is defined to be 1. The algorithm is per-
formed with an iterative optimization of minimizing a
fuzzy objective function.
In fuzzification step, the HF detail image is fuzzified
by 3 clusters whereas LF approximate image fuzzified
by 4 clusters. This particular choice of clusters repre-
sents the different gray shades of mammograms (clusters
due to dark, gray, semi bright sets of pixels) along with
the mass region (brighter sets of pixels).
Since FCM is intensity based clustering technique, it
groups the dark, gray, bright pixels into separate clusters.
2.3. Vague Sets for Capturing
Incompleteness/Hesitancy of Data
In this section the basic concepts related to Vague Sets
(VS) and Fuzzy Sets are described. It is also illustrated
that algebraic/graphical representation of VS is more
intuitive for capturing the hesitancy or incompleteness of
2.3.1. Basics
Let U be a classical set of objects, called the universe of
578 A. Das et al. / J. Biomedical Science and Engineering 2 (2009) 575-581
SciRes Copyright © 2009 JBiSE
discourse, where an element of U is denoted by u.
Fuzzy Set: A fuzzy set A = {< u, µA(u) > | u
U} in
a universe of discourse U is characterized by a member-
ship function, µA, as follows: µA : U [0,1].
Vague Set: A vague set V in a universe of discourse U
is characterized by a true membership function, αV , and
a false membership function, βV , as follows: αV: U
[0,1], βV: U [0,1], and αV(u)+βV(u) 1, where αV(u)
is a lower bound on the grade of membership of u de-
rived from the evidence for u, and βV(u) is a lower
bound on the grade of membership of the negation of u
derived from the evidence against u.
It can be seen that the difference between VS and FS
is due to the definition of membership values. In VS, the
boundary (1-βV) is able to indicate the possible existence
of a data value. This subtle difference gives rise to a
simpler but meaningful graphical view of datasets. Fig-
ure 4 and Figure 5 depict a VS and a FS respectively. It
can be seen that, the shaded part formed by the boundary
in a given VS in Figure 4 represents the possible “hesi-
tation region” corresponds to the intuition of represent-
ing vague data.
In order to compare vague values, it is required to in-
troduce two derived membership values for discussion.
The first is called the median membership, Mm = (αV +
1 βV)/2, which represents the overall evidence con-
tained in a vague value. The second is called the impre-
cision membership, Mi = (1 βV αV), which repre-
sents the overall imprecision/hesitation of a vague values.
Figure 4. Membership functions of a VS.
Figure 5. Membership functions of a FS.
2.4. Defuzzification and Reconstruction of
Images Using LF and HF Subbands
In fuzzification step, each pixel-intensity of LF and HF
subband images are transformed to intuitionistic fuzzy
membership domain. After highlighting the appropriate
cluster, both LF and HF subband images are transformed
back to the spatial gray-level intensity domain. Then LF
and HF subbands reconstruct the resulting image of
original resolution by inverse wavelet transform.
We have applied the proposed algorithm to a database
consisting of 100 images obtained from Mammographic
Image Analysis Society (MIAS), BIRADS and from
EKO X-Ray & Imaging Institute, Kolkata. In the fuzzi-
fication step, the number of clusters (c) chosen by 4 and
3 for approximate images (low- frequency subband) and
detail images (high frequency subband) respectively.
These numbers of clusters are appropriate to represents
the different gray shade intensities of the Mammo-
graphic features.
In approximate image, 4 clusters, denoted as A, B, C
& D is shown graphically in Figure 6. Cluster A indi-
cates the dark background of the images. Cluster B indi-
cates the fatty breast tissues. Cluster C & D represents
the diffused breast tissues and true mass regions respec-
Figure 6. Schematic of fuzzy membership partition
functions for approximate images.
Figure 7. Schematic of fuzzy membership partition
functions for detail images.
A. Das et al. / J. Biomedical Science and Engineering 2 (2009) 575-581 579
SciRes Copyright © 2009
In detail image, 3 clusters, denoted as A, B & C is
presented graphically in Figure 7. Cluster A indicates
the drk background of the images. Cluster B indicates
the irregular shades of breast due to dense parencymal
tissues and Cluster C represents the fine boundaries of
masses present in the mammograms.
After fuzzification with proper number of clusters,
the vague set approach has been introduced for cap-
turing the hesitancy and incompleteness of mammo-
graphic features. In the case of approximate images,
only cluster D represents the membership function in
evidence to true mass region (αV), and clusters A, B,
C represent the evidence against the true mass region
Thus the average membership function in evidence to
false mass region (βV) is calculated by (A+B+C)/3.
Similarly cluster C of detail images, represents the
evidence of true mass boundaries (αV) whereas cluster A
and B indicate the evidence against the true mass regions.
The average membership function in evidence to false
mass region (βV) is calculated by (A+B)/2.
The median membership value Mm = (αV+1−βV)/2,
is set for the boundary value between the membership
functions αV and 1−βV. In the present article, median
membership value is considered as the limit of affected
mass or other type of breast abnormalities.
The following mammograms are used to demonstrate
the improved contrast of masses/microcalfications using
VS approach in compare to the standard fuzzy set theory.
Figure 8(a) is the original mammogram to be en-
hanced. Figure 8(b) indicates the improved contrast
between true mass region and normal breast tissues us-
ing the proposed algorithm, whereas Figure 8(c) high-
light the same region using standard FCM. It is noted
that Figure 8(b) obtained by VS approach, is capable of
highlighting the true masses properly because of its effi-
ciency to handle the uncertainty /hesitancy between true
and false membership functions.
On the contrary, the enhanced result obtained by FCM
skips to highlight some of the true mass regions that ef-
fect severely in proper diagnosis. Figure 9(b) and Fig-
ure 10(b) also represent improved contrast mammo-
grams with circumscribed masses using the proposed
(a) (b) (c)
Figure 8. (a) Original Mammographic Image; (b) Enhanced image by VS; (c) Enhanced image by FCM.
(a) (b)
(a) (b)
Figure 9. (a) Original Mammographic Image; (b) Enhanced
image by proposed method.
Figure 10. (a) Original Mammographic Image; (b) Enhanced
image by proposed method.
580 A. Das et al. / J. Biomedical Science and Engineering 2 (2009) 575-581
SciRes Copyright © 2009 JBiSE
Figure 11(b) also shows the enhanced indistinct shap-
ed mass region properly. In Figure 12(b) the proposed
algorithm highlights comparatively large shaped mass
boundary clearly along with the presence of few micro-
calcification spots surrounding the mass.
Table 1 exhibits the median and imprecision mem-
bership values of VS approach for a particular mammo-
gram. The single membership grade of conventional
fuzzy set theory for same mammographic image also list-
ed in right most column. The larger values of median
membership function, obtained from VS approach is
capable of highlighting the true masses /microcalci fica-
tions with allowable hesitation margins (Mm-μA) in com-
pare to ordinary fuzzy set theory.
(a) (b)
Figure 11. (a) Original Mammographic Image; (b) Enhanced
image by proposed method.
During the past two decades, interval based intuitionistic
fuzzy sets have been used increasingly in the research
areas focused on fuzzy sets and fuzzy logic. Goal of this
paper is not to develop an interval based fuzzy set ap-
proach for handling the imprecise data but to apply more
(a) (b)
Figure 12. (a) Original Mammographic Image; (b) Enhanced
image by proposed method.
Table 1. Membership values of vague set theory and fuzzy set approach.
Elements of clusters
in Approx image
Median Membership Value (Mm)
according to VS Approach
Imprecision /hesitation
membership Value(Mi)
according to VS Approach
Membership Value μA
according to Fuzzy
Set Approach
1st element of cluster D 1.0000 0.0000 1.0000
2nd element of cluster D 0.9999 0.0000 0.9999
3rd element of cluster D 0.9995 0.0004 0.9991
4th element of cluster D 0.9993 0.0013 0.9980
5th element of cluster D 0.9984 0.0006 0.9978
6th element of cluster D 0.9982 0.0023 0.9959
7th element of cluster D 0.9967 0.0032 0.9935
8th element of cluster D 0.9948 0.0019 0.9929
9th element of cluster D. 0.9942 0.0036 0.9906
10th element of cluster D 0.9926 0.0036 0.9890
11th element of cluster D 0.9910 0.0037 0.9873
12th element of cluster D 0.9901 0.0059 0.9842
13th element of cluster D 0.9874 0.0038 0.9836
14th element of cluster D 0.9869 0.0056 0.9813
15th element of cluster D 0.9844 0.0050 0.9794
A. Das et al. / J. Biomedical Science and Engineering 2 (2009) 575-581 581
SciRes Copyright © 2009 JBiSE
robust Vague Set approach for uncer tainty management.
Introducing two parameters, like median and impreci-
sion membership values, VS approach becomes much
easier to interpret and to visualize the vague data objects.
Since medical diagnosis deals with the imprecise and
incomplete information, accurate detection of truly af-
fected region as well as degree of prognosis of the dis-
eases is a difficult task. VS approach is then appropriate
for the area concerning medical diagnosis. The authors
have presented VS approach, which is efficient than the
ordinary fuzzy sets for this purpose. The major advan-
tage of VS over conventional fuzzy sets is that, VS in-
cludes both positive and negative evidences of an ele-
ment in the universal set. As a result, the proposed
method has the advantages of modelling and analyzing
the uncertainties and hesitancies which are present in the
diagnostic system in a more flexible and intelligent man-
ner. Measurement of impreciseness in practice, it is
found from the experimental results that VS is more
natural than the conventional FS, especially for overlap-
ping membership function domain. Introduction of mul-
tiresolution analysis makes the methodology more robust
in that sense it is capable of enhancing very fine detail
features by processing the high frequency subband
components. For Mammographic images, ordinary con-
trast enhancement algorithms are unable to provide good
contrast information in the selected region of interests.
The concept of VS is found to be applied successfully to
the problems of selective contrast enhancement. The
resulted performance of the proposed algorithm shows
improvements over the ordinary fuzzy based operations.
The authors would like to thank to Dr. S. K. Sharma, Director, EKO
Imaging and X-Ray Institute, Kolkata.
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