Abstract— Breast cancer is an important deleterious disease. Mortality rate from this cancer is effectively high and rapidly increasing. The detection at the earlier state can help to reduce the mortality rate. In this paper, we apply the interval type-2 fuzzy system with automatic membership function generation using the Possibilistic C-Means (PCM) clustering algorithm. We utilize four features, i.e., B-descriptor, D-descriptor, average intensity of the inside boundary, and intensity difference between the inside and the outside boundaries. We also compare the result with the result from the interval type-2 fuzzy logic system with automatic membership function generation using the Fuzzy C-Means (FCM) clustering algorithm. The interval type-2 fuzzy system with PCM membership functions generation yields the best result, i.e., 89.47% correct classification with only 6 false positives per image.

I. INTRODUCTION t the present day, breast cancer causes a high mortality rate for women and keeps expanding in number. The

cause of this disease has not yet been discovered. The effective detection in the earlier stage is preferable because it will help reducing the mortality rate from the cancer.

An important indication of breast abnormalities that can cause breast cancer is microcalcification (a very small deposit of calcium) [1]. A radiologist has to look at several mammograms each day. His/her ability in detecting microcalcification will decrease over time. There have been several studies on utilizing computer techniques, i.e., image processing and pattern recognition, to help a radiologist detecting microcalcification. Some of these studies [2-6] achieved approximately 80% to 90% correct classification. However, the numbers of false positives are still high. In [7], they were able to achieve a good classification rate: 78% at a false positive of 2.09 per image. However, a region of interest (ROI) had to be selected manually. In [8], Salvado and Roque proposed a wavelet-based method to enhance the contrast of mammograms which achieved good result in low dense image. On the contrary, the parameters in high dense image detection must be defined to keep all significant

Suraphon Chumklin is with the Department of Computer Engineering, Faculty of Engineering, Chiang Mai University, Thailand 50200.

Sansanee Auephanwiriyakul is with the Department of Computer Engineering, Faculty of Engineering, and is with Biomedical Engineering Center, Chiang Mai University, Thailand 50200. (corresponding author* phone: 6653942024; fax: 6653942072 e-mail: sansanee@ieee.org).

Nipon Theera-Umpon is with the Department of Electrical Engineering, Faculty of Engineering, and is with Biomedical Engineering Center, Chiang Mai University, Thailand 50200 (e-mail: nipon@ieee.org).

information. Even though the detection technique provides a good result, ROI’s have to be selected manually.

One of the popular techniques in pattern recognition is fuzzy logic system [9]. There are two types of fuzzy logic system, i.e., type-1 and interval type-2 [10] fuzzy logic system. However, membership function generation is still a problem

There are several studies on the application of interval type-2 fuzzy system. Some of them involve with automatic membership functions generation from training data [11–16]. These methods utilize the Fuzzy C-Means clustering algorithm, histogram and genetic algorithm. There are some disadvantages with these methods though. For example, the interval type-2 membership function generated from FCM may not be convex since points that are far away from its cluster center may have high membership value instead of 0.

In this paper, we build a system that can detect microcalcification without selecting ROI. In particular, we implement the interval type-2 fuzzy logic system with automatic membership function generation using the Possibilistic C-Means [17, 18] and compare the results with that using the Fuzzy C-Means applying to the same set of features.

II. INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS

A. Interval Type-2 Fuzzy Set In interval type-2 fuzzy logic system [10], the crisps

inputs are mapped into type-2 fuzzy singleton shown in

figure 1. The upper ( f ) and lower ( f ) membership function is computed. Suppose there are M rules and

Fig.1 Interval singleton type-2 fuzzy logic system.

Microcalcification Detection in Mammograms Using Interval type-2 Fuzzy logic System with Automatic Membership Function

Generation Suraphon Chumklin, Sansanee Auephanwiriyakul, Senior Member, IEEE, and

Nipon Theera-Umpon, Senior Member, IEEE

A

978-1-4244-8126-2/10/$26.00 ©2010 IEEE

Fig.2 Inputs and antecedent operator for an interval single type-2 fuzzy

logic system. there are two inputs and one output for each rule. Each rule is in the form of: “If x1 is F1 and x2 is F2 then y is G” where F1, F2 and G are interval type-2 fuzzy sets in linguistic terms of antecedents and consequent, respectively. The minimum t-norm is used to combine the upper and lower membership values of antecedents as shown in figure 2 whereas the maximum t-conorm is utilized to aggregate outputs from all rules that are fired. In order to get crisp output, the type reduction and defuzzification is needed.

III. POSSIBILITIC C-MEANS CLUSTERING Let X = {x1, x2,..., xN} be a set of vectors, where each

vector is p-dimensional. Let C = {c1, c2, ..., cc} represents a C-tuple of prototypes, each of which is the center of a cluster and uji is the membership value of vector xj in cluster i. The objective function for PCM [17, 18] clustering algorithm is:

21 1

1 1

( )

(1 )

C N mPCM ji jii j

C N mi jii j

J u d

uη= =

= =

=

+ −

∑ ∑∑ ∑

(1)

with 1

0 , 1Njij

u N i C=

< < ≤ ≤∑ .

where m ∈ [1, ∞) is a fuzzifier. ηi determines the relative degree to which the second term in the objective function is important compared with the first and is defined as

2

1

1

η =

=

=∑∑

N mji jij

i N mjij

u d

u. (2)

The minimization of the objective function with respect to membership values leads to

12 ( 1)

1

1

jim

ji

i

udη

−

=⎛ ⎞

+ ⎜ ⎟⎜ ⎟⎝ ⎠

. (3)

The minimization of the objective function with respect to the center of each cluster yields:

1

1

=

=

=∑∑

N mji jj

i N mjij

u

u

xc . (4)

The PCM algorithm is as follows:

Fix number of the clusters C; Initial the center using the Fuzzy C-Means (FCM) algorithm [16] Estimate ηi using equation (2); Do { Update membership using equation (3); Update center using equation (4); } Until (center stabilize)

IV. FEATURE EXTRACTION The input to this system is an original image of the

mammogram and the image can be of any size. We scan 50×50 pixels window from top to bottom and left to right with 8 pixels shifting. We select this window size because it fits the largest microcalcification area. In this research, we use four features, i.e., B-descriptor, D-descriptor, average intensity inside boundary and intensity difference between inside and outside boundaries. We preprocess each sub-image with the Gaussian filter to reduce noise and Sobel edge detection to find edge. To get a binary image, we utilize an adaptive threshold, i.e. ,

max( ) min( )2

Intensity IntensityT += , (5)

where max(Intensity) and min(Intensity) are the maximum and minimum of gray levels in each sub-image, respectively.

Finally, we get a contour using clockwise direction and recording [19]. The coordinate sequence in this system is downsampled to produce k more or less equally space coordinate. In this work we used k = 64. An example of this preprocessing is shown in figure 3.

original contour

(a)

original contour

(b) Fig 3. original sub-image and its contour (a) microcalcification, and (b) non-microcalfication

The details of each feature are described as follow: 1) B-descriptor: The contour was previously

downsampled into k points of roughly equal arc length. For each index n, the coordinates of the contour point u(n) are (x(n), y(n)) and be treated as a complex number so that u(n) = x(n) + jy(n) for n = 0,1,...,k−1 [20, 21]. It is possible to express u as a complex Fourier series. The Fourier coefficients of period T are as follows:

0

1 2( ) expT

nj nta u t dt

T Tπ−⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ . (6)

A normalized Fourier feature, called B-descriptor, is calculated using Granlund’s method [20, 21], i.e. ,

1 121

.n nn

a ab

a+ −= , for each n = 0,1,…,k − 1, (7)

where bn values are symmetric around point at n = k/2. The B-descriptors (bn) are invariant of size, angular

orientation, position and starting position of the sequence contour.

2) D-descriptor: A normalized Fourier feature, called D-descriptor, is calculated using Granlund’s method [20, 21], i.e. ,

1 11

1

. nn

n n

a ad

a+

+= , for each n = 0,1,…,k−1. (8)

The D-descriptors (dn) are independent of translation and dilation. Examples of B-descriptors and D-descriptors are shown in figure 4.

3) Average intensity inside boundary: This feature is calculated by

1

1 ( )ninside t

AVG g in =

= ∑ , (9)

where n is the amount of pixels inside the boundary and g(i) is intensity value of pixel i.

4) Intensity difference between inside and outside boundary: This feature is computed by

inside outsideDIFF AVG AVG= − , (10) where AVGoutside is the average value of intensities outside the boundary.

(a) (b)

(c) (d)

Fig. 4. Fourier descriptors (a) B-descriptors of figure 3(a), (b) D-descriptors of figure 3(a), (c) B-descriptors of figure 3(b), and (d) D-descriptors of figure 3(b).

B. Design the membership function The membership functions are generated using the

Possibilic C-Means (PCM) clustering algorithm. Each cluster center is a vertex of the membership function (triangular), and associated with linguistic variables, e.g., “low”, “medium”, and “high”. 1) Upper Membership Function (UMF)

After clustering each feature, each data point will be assigned into cluster i if its membership value in cluster i is maximum. Figure 5(a) shows an example of upper membership function of DIFF plotted from membership values of each data point in each cluster. For the left-most upper membership function, the left boundary will be the left-most data point from the centroid. The membership value of the function is membership value in that cluster. We compute the same thing for the right boundary of the right-most upper membership function. The left and right boundaries of the upper membership function are the position of the first data point on the left and right of the centroid with a membership value equal to a predefined threshold.

(a)

(b)

(c)

(d)

Fig. 5. Upper membership function generation using PCM of DIFF feature, (a) membership values of each data point in each cluster, (b) defined left and right boundaries, and (c) adjusted boundary of each cluster. (d) membership value of the left-most and the right-most membership function.

(a)

(b)

(c)

Fig. 6. Upper membership function generation using FCM of DIFF feature, (a) membership values of each data point in each cluster, (b) defined left and right boundaries, and (c) membership value of the left-most and the right-most membership function.

If the threshold is in between 2 data points, the boundary will be interpolated from these 2 data points. An example of

this process is shown in figure 5(b). If the left or the right boundary is beyond the centroid of the closest cluster, that boundary will be the centroid of that cluster shown in figure 5(c). We call the system with UMF using this method as MF1. Since, we would like to build the left-most membership function to be a trapezoid membership function, we set the membership value of data point on the left side of the centroid to 1. We also make the right-most membership function to be a trapezoid with the similar scheme. An example of this scheme is shown in figure 5(d). We call the system with this type of UMFs as MF2.

Since we would like to compare our results with that using Fuzzy C-Means clustering algorithm, we build UMFs similar to MF1 and MF2. We call these systems as MF3 and MF4, respectively. Examples of these systems are shown in figures 6(b) and 6(c), respectively. These systems are built from the membership values of each data point in each cluster shown in figure 6(a). We also compare our results with the best manually picked membership function system called MF5. 2) Lower Membership Function (LMF)

The method of designing lower membership function used in this research is obtained from [12]. The sum of Euclidean distance of cluster j is computed by

1== −∑ n

j ij jid x v . (11)

where xij is the input feature i in cluster j, n is the number of data features in cluster j and vj is the cluster center j . Since the sum of all distances has to be equal to one, we normalize dj to dj′ as

'

1=

=∑

jj C

jJ

dd

d (12)

The height of lower membership function is

1 ′= −j jH d . (13)

Fig. 7. The left and the right end-point of the lower membership function.

The boundaries of the lower membership function are computed from β1j and β2j as shown in figure 7. These values are computed by '

1 1β γ= ×j j jd (14)

'2 2β γ= ×j j jd (15)

where γ1j is the distance from the core to the left end-point of the upper membership function, and γ2j is the distance from

the core to the right end-point of the upper membership function.

An example of MF1 with UMF from figure 5(c) is shown in figure 8(a), whereas figure 8(b) shows MF2 with UMF from figure 5(d). Examples of MF3 and MF4 with UMF from figures 6(b) and 6(c) are shown in figures 9(a) and 9(b) respectively.

(a)

(b)

Fig. 8. Interval type-2 membership function generation using PCM of DIFF feature, (a) Upper and lower membership function in MF1, and (b Upper and lower membership function in MF2.

(a)

(b)

Fig. 9. Interval type-2 membership function generation using FCM of DIFF feature, (a) upper and lower membership function in MF3, and (b) upper and lower membership function in MF4.

V. EXPERIMENTAL AND RESULTS The mammograms used in this study are obtained from

Maharaj Hospital, Chaing Mai University. There are 21 mammograms with different sizes, e.g., 2048×2560, 2560×3072, 512×2560, etc. The number of microcalcifications in each mammogram is shown in table I. We collect microcalcification and non-microcalcification sub-images (each with 50×50 window size) from 14 mammograms and use them as the training data set. There are 160 sub-images in total with 90 microcalcification and 70 non-microcalcification. We test our system with the seven remaining mammograms.

We utilize the fuzzifier (m) = 2 in both the PCM and FCM clustering algorithms. The number of clusters for B-descriptor, D-descriptor, AVGinside and DIFF are 2, 2, 3 and 3, respectively. After clustering each feature, we predefine threshold to 0.01 which is the best threshold for clustering the data points into clusters. Figure 10 shows interval type

1 'jd

jH

1 jβ 2 jβ1 jγ 2 jγ

membership functions used in MF1 to MF5. The output membership functions for all systems are shown in figure 11.

We use the same rule set with linguistic variables, “L”, “M” and “H” representing “Low”, “Medium”, and “High”, respectively, for all the systems as shown in table II.

(a)

(b)

(c)

(d)

(e)

Fig. 10. (a) – (e) The interval type-2 membership functions from experiment MF1 to MF5, respectively.

We also compare our system with the best result from the Mamdani fuzzy inference system. The best Mamdani fuzzy inference system with PCM membership function generation (called MF6) is the one that utilize UMF from MF2 as type-1 membership functions [22]. While the best Mamdani fuzzy inference system with FCM membership function generation (called MF7) is the one that utilize UMF from MF4 as type-1 membership functions [22]. We call the best manually pick membership function for the Mamdani fuzzy inference system as MF8.

Fig. 11. The interval type-2 membership functions of output.

From the result of the training data set shown in table III, we found that the best result is from MF2 with 91.25 % correct classification. Although MF1 is not the best one, its result (90% correct classification) is still higher than that using PCM in the Mamdani system, FCM in both the Mamdani and interval type-2 fuzzy logic system, or manually system in both the Mamdani and interval type-2 fuzzy logic system.

TABLE I MICROCALCIFICATION IN TRAINING MAMMOGRAMS

Mammogram No 1 2 3 4 5 6 7 Amount of

microcalcification 12 4 3 3 8 1 1

Mammogram No 8 9 10 11 12 13 14

Amount of microcalcification 11 12 1 19 1 4 10

MICROCALCIFICATION IN TESTING MAMMOGRAMS Mammogram 15 16 17 18 19 20 21

Amount of microcalcification 5 8 3 2 2 13 5

TABLE II

INFERENCE RULES FOR ALL EXPERIMENTS Features

Rules Input

Output B-DES D-DES AVG DIFF

1 L L L L L 2 L L L M L 3 L L L H L 4 L L M L L 5 L L M M M 6 L L M H M 7 L L H L M 8 L L H M M 9 L L H H H

10-36 H L,H L,M,H L,M,H L

TABLE III

RESULTS OF TRAINING DATA SET Membership

function Correct

detection False

positive* False

negative** Correct

(%) MF1 144 7 9 90 MF2 146 9 5 91.25 MF3 141 11 8 88.13 MF4 141 11 8 88.13 MF5 138 13 9 86.25 MF6 143 11 6 89.38 MF7 139 13 8 86.88 MF8 135 15 10 84.38

* False positive means classifying non-microcalcium as microcalcium. ** False negative means classifying microcalcium as non-microcalcium.

Examples of the results of the original mammogram in figure 12, where MF1 and MF2 do not produce false positives when the other systems do, are shown in figure 13. The firing rules in MF1 and MF2 are 5 and 8 where that in MF3 to MF5 are 5, 6, 8 and 9. This results in the misclassification.

(a) (b)

Fig. 12. Mammogram number 21 (a) original mammogram, and (b) the expert’s opinion (circle area).

(a) (b) (c)

(d) (e) Fig. 13. Crisp result and false positives from mammogram number 21 (at threshold = 0.5), (a) MF1, (b) MF2, (c) MF3, (d) MF4, and (e) MF5.

Fig. 14. ROC curve of the microcalcification detection in blind test data set.

The receiver operating characteristics (ROC) of the blind test is shown in Fig. 14. All methods in the interval type-2 fuzzy logic system yield 89.47% correct classification. However, the false positives per image of MF1 to MF5 are 6, 6, 8, 8 and 8.5, respectively. While all methods in the Mamdani fuzzy inference system yield 86.84% correct classification with the false positives per image of MF6 to MF8 are 6.6, 8.4 and 8.8, respectively.

Since PCM does not require that the summation of membership value of each data point in clusters have to be 1, this becomes an advantage of PCM over FCM. Because if

there is points that are far away from the cluster centroids, they will be discarded because of their small membership values in all clusters. If this happen in FCM case, their membership values will decrease in the cluster that are far away but at the same time their membership value in another cluster will also increase. Using PCM in generating interval type-2 membership function will help reducing the effect of the noisy data point.

VI. CONCLUSION In this work, we propose a microcalcification detection

system using the interval type-2 fuzzy logic system with membership function generation using the Possibilistic C-Means (PCM). We compare the result with the interval type-2 fuzzy logic system with membership function generation using the Fuzzy C-Means (FCM) and the Mamdani fuzzy inference system with membership function generation using both PCM and FCM. From the experiment, the best result in blind data set is from the interval type-2 fuzzy logic system using PCM generation, i.e., 89.47% correct classification with only 6 false positives per image. The outputs of the system are drawn from 7 mammograms with 38 microcalcification clusters. The advantage of PCM over the FCM is that if the data point is far away from the centroid, its membership value will be close to 0. This will decrease the effect of noise to membership function generation.

ACKNOWLEDGMENT The authors would like to thank Dr. Pim Handagoon for

providing us knowledge and ground truth.

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