Medical Image Thresholding Scheme using ...

International Conference on Methods and Models in Computer Science, 2009

Medical Image Thresholding Scheme using Atanassov Intuitionistic Fuzzy Set Tamalika Chaira Centre/or Biomedical Engg. Indian Institute ofTechnology, Delhi Hauz Khas, Delhi, INDIA e-mail: [email protected]

Abstract-This paper addresses a novel image thresholding scheme using Atanassov intuitionistic fuzzy set. The intuitionistic fuzzy set theory takes into account the membership degree and another uncertainty term, the non- membership degree. Non membership degree is calculated from Sugeno type intuitionistic fuzzy generator. The use of two uncertainties in the Atanassov's intuitionistic fuzzy set helps in thresholding the image more appropriately. The effectiveness of the algorithm is demonstrated by performing experiment on different types of poorly illuminated pathological blood vessel images and comparing with other existing fuzzy and intuitionistic fuzzy thresholding algorithms Keywords: Intuitionistic fuzzy set, thresholding, gamma membership function, Sugeno intuitionistic fuzzy generator, hesitation degree.

fuzzy set theory is used by many authors in image thresholding that takes into account the ambiguity and vagueness. It fmds extensive application in image processing areas to describe vague concepts, which is represented in the form of membership function. It denotes the degree of belongingness of the pixel to that class. The membership values lies in the interval [0, 1] with 0 signifying no membership and 1 signifying full membership. This has led to an introduction of large number of algorithms suggested by different authors for optimum threshold selection. Many of these fuzzy algorithms are based on Shannon entropy [4], measures of fuzziness [4, 5, 6], fuzzy cluster based thresholding [7], fuzzy compactness [8] and so on. Again, if we explore deeply about the membership I. INTRODUCTION function in a fuzzy set theory, one can fmd that there is hresholding is partitioning the image pixels into another uncertainty or hesitation lies while defming the two levels as background and foreground. It is membership function. This uncertainty is due the lack of used in extracting objects from the picture. The problem knowledge or the personal error while defming the of thresholding is identifying an optimum threshold T membership degree and this has led to an introduction and segment the image into two regions - object and of higher order fuzzy set theory called intuitionistic background. It is a preferable method of segmentation fuzzy set (IFS) theory introduced by Atanassov in 1986. when object and background regions is clearly The non membership degree is not the complement of distinguishable. In such a case, the histogram is bimodal the membership degree as in the ordinary fuzzy set, and the threshold lies in the deep valley between the two rather, less than or equal to the complement of the peaks. Threshold selection may be global or local. membership degree due to the hesitation degree. There Global thresholding method segments the entire image is very little related work using IFS in literature. with a single threshold but local threshold method Vlachos and Sergiadis [9] used gamma membership partitions the image into sub regions and calculates the function to fmd the membership values of the pixels of threshold for each sub region. Global method is fast and an image and used cross entropy to fmd the optimum computationally complexity is less. The problem lies in threshold. Couto et al. [10] used a different approach in thresholding is the selection of optimum threshold such the calculation of hesitation degree and used that objects are clearly separated from the background. intuitionistic fuzzy entropy to fmd the optimum Different authors selected threshold such as Kapur's threshold. entropy [1], Tsai's moment [2], Otsu's class separability But thresholding of real time images such as medical [3] and so on. images is very difficult. Medical images are poorly But in this real world, uncertainties are almost illuminated and so it becomes very difficult to extract inherent in the real time mages and so the selection of the objects from the image. As for example, in human an optimum threshold becomes quiet difficult. For this,

T

blood vessel images, there are various types of uncertainties present in the form of noise, overlapping of gray levels, vagueness in class defmition; indistinguishable histograms etc. and so blood vessels are barely visible. So, intuitionistic fuzzy set theoretic approach is the best way to deal with these types of images, which takes into account two uncertainties membership and non-membership degree. In this paper, a novel global thresholding method on pathological images is introduced using intuitionistic fuzzy set theory where good results are obtained. This paper is organized as follows. In section II the preliminaries on intuitionistic fuzzy set is presented while Section III tells about the construction of intuitionistic fuzzy set using Sugeno type intuitionistic fuzzy generator. Section IV details the proposed intuitionistic fuzzy thresholding scheme. Section V displays the results and discussion. Conclusion is drawn in section VI.

III. CONSTRUCTION OF INTUITIONISTIC Fuzzy SET In order to construct Attanassov's intuitionistic fuzzy set (IFS), intuitionistic fuzzy generators are used. From the defmition of intuitionistic fuzzy generators by Bustince et al. [12]: A continuous, decreasing, and increasing function rp(x) : [0,1] will be called continuous, decreasing, and increasing intuitionistic fuzzy generator if: tp(x) ~ (1- x) for all x E [0,1] · tp(O) ~ 1 and tp(l) ~ 0 ·

Intuitionistic fuzzy generator or fuzzy complement is created from Sugeno's generating function [13]. Fuzzy complemental function is given as: N(p(x)) = g-l(g(I)_ g(p(x))

(2)

where g(.) is an increasing function and g:[O,I]~[O,I]

II. PRELIMINARIES A fuzzy set A in a fmite set X

= {Xl'X 2,X3 , ••• ,xn }

may

Then, Sugeno's intuitionistic fuzzy generator is written as: (3) N(p(x)) = (1- p(x)) / (1+ Ap(X)) , A > 0

be represented mathematically as: A = {(x, PA (x)lx EX},

where, the function JiA(X): X ~ [0,1] is the membership degree of an element X in the fmite set X . Thus, automatically the measure of non-belongingness is 1- P A (x) . Atanassov [11] introduced intuitionistic fuzzy set that takes into account the membership degree, u, and also the non membership degree, v, of the elements of a set. An intuitionistic fuzzy set A in a universal set X is given by : A = {x, JLA (x), VA (x) I x EX} where PA(x)~[O,I],vA(x)~[O,I] are the membership and non-membership degrees of an element x to the set A with the condition O~JLA(x)+vA(x)~1

When JLA(x)=I-JLA(x), every x in set A becomes a fuzzy set. The hesitation degree, 1iA (x), which arises due to lack of knowledge while defming the membership degree, for each element x in A and is given by:

Jl"A(X)

= 1- JLA(X)-VA(x)

Obviously 0

~ Jr A (x) ~

Using the function 1 g(p(x)) = ~ 10g(1 + A.p(X))

(1)

1.

Due to the hesitation degree, the membership values lie in the interval

where N(I) = 0, N(O) = 1. Non-membership values are calculated from Sugeno type intuitionistic fuzzy generator, N (p( x)). Thus, with the help of Sugeno type intuitionistic fuzzy generator, IFS becomes: AA,IFS = {x,JiA(x),(I- JiA(X)) / (1+ A.JiA(X)) I x EX} with hesitation degree 7[A(X) =1- PA(X)- (1- PA(X))/ (1 + A.PA(X))

(4)

This Sugeno intuitionistic fuzzy generator is used to create an intuitionistic fuzzy image.

IV. INTUITIONISTIC Fuzzy THRSHOLDING SCHEME It consists of two parts- A) Membership function and B) intuitionistic fuzzy divergence as a criterion function for threshold selection. A. Initially the membership function is calculated from Gamma membership function [5,6]. Gamma membership function is computed from Gamma distribution which is given as follows: The probability density function of the gamma distribution is given as

1

(

=

f(x)

B. Optimum threshold is selected using an intuitionistic fuzzy divergence measure. Fuzzy divergence is proposed by the author [5]. Using the intuitionistic fuzzy set theory, intuitionistic fuzzy divergence is suggested.

x _m)]r- exp (x-m)] ---

f3

f3

,

r(y)

(5)

with x ~ m;y,f3 > 0

r

where is the shape parameter, m is the location parameter, f3 is the scale parameter, and r is the Gamma function. Now if f3 = 1 , Y = 1 and m"* 0 then the gamma distribution in eq (5) becomes: f(x) = exp(-(x-m)) withr(I)=I, r(l)

So

I(x)

= exp( -(x -

m))

(6)

In the thresholding method, consider an image A of size M x M with maximum gray level Land ai) is the (i,j)th pixel of image with i.] = 1,2,3 ...,M . Given a certain threshold 't' that separates the objects and the background regions, the average gray levels of the object and background regions respectively are given by the relation:

t .count{f)/tj;;count{f) j;;f ~ = 'f f .count(f)/ 'f count({) f110 =

f=1+1

f=1+1

where count (f) denotes the number of occurrences of the gray level f in the image, L is the maximum gray level ofthe image A. Replacing m in eq. (6) by moandm 1 the membership function for the object and background regions become:

t for object aij > t for background

f.!A(aij) = exp( -c.laij - mol)'. if aij::;

= exp(-c.aij -mIl) If

(7) where t is the any chosen threshold and aij is the (i,j)th pixel in the image A. The constant 'c' is chosen to ensure the membership of the gray level representable in the range [0, 1] and is c= 1/ calculated as /(!max - !min)

, where frnin and t.; are the minimum and maximum value of gray level in the image respectively. After the membership function is calculated, nonmembership degree is computed for each pixel using Sugeno type intuitionistic fuzzy generator using eq. (3) and then the hesitation degree is calculated from the eq. (4).

Consider

two

intuitionistic

fuzzy

A = {(X,PA (x),vA(x))lxeX} and sets B = {(x, JiB (x),vB(x))lx eX}. If PA(Xi) and VA(Xi) are the membership and non membership values of a particular element, then due to the hesitation degree

Jl'A(Xi) , Jl'A(Xi) = 1- JiA(Xi) - VA(Xi)' membership value will lie in an interval range tUA(X;),PA(Xi) + Jl'A(X i)} . So the new membership value may be written as:

J.lnew A(Xi ) = PA(Xi)+Jl'A(X;)

(8)

Using the fuzzy divergence concept, the intuitionistic fuzzy divergence between the images A and B is given as: I _ Div(A, B) =

L L (2 - (1- Jinew A(Xij) + Jinew B(Xij))e,unewA(Xij)-,unewB(Xij)_ j

i

(1- u": B(Xij) + u": A(xij))e,unewB(Xij)_,unewA(Xij))

(9) In this work, the image is compared with an ideally thresholded image whose membership degrees are 1 and the hesitation degree is 0 [5, 6]. Ideally segmented image is that image whose background/object pixels completely belong to their respective background/object regions. So the membership degrees of the background/object pixels with respect to the background/object regions are 1, Le., J.lnew B(Xi)) = 1 . Substituting J.lnew B(xi)) = 1 in eq. (9), the intuitionistic fuzzy divergence reduces to:

I _Div(A,B) =LL(2-(2-~A(Xij))erA(Xij)-1j

;

(10) For all threshold gray levels, intuitionistic fuzzy divergence is calculated. The threshold level corresponding to the minimum divergence is selected as the optimum threshold.

v. RESULTS AND DISCUSSION Experiment is performed on several medical images. The value of A = 2 in eq. 3 is chosen in experimentation. Three sets on blood vessel images are

shown below. Some of the images are downloaded from the image gallery

for carrying out the research under the scheme "SERC Fast Track Scheme for Young Scientists".

http://www.polconsultant.com/conteduc/hematology/micro/.

Results are compared with Fuzzy method [5], intuitionistic fuzzy method by Bustince et al. [11]. Fig.1 (a) is a figure of poorly illuminated blood vessel of size 160 x 160. With intuitionistic fuzzy method by Bustince in Fig. 1(c), large amount of noise is present and the blood vessels are hardly segmented. But the fuzzy method in Fig. 1(b) and the proposed intuitionistic fuzzy method in Fig. 1(d) are similar and all the blood vessels are segmented clearly. Fig. 2(a) is an image of blood vessel of size 200 x 160 where the blood vessels are not visible properly. With fuzzy method in Fig. 2 (b), noise is present in the upper part of the image and in Bustince et al. intuitionistic fuzzy method in Fig. 2(c), there is a lot of noise present at the upper part of the image. In the proposed intuitionistic fuzzy method in Fig. 2(d), there is hardly any noise and all the blood vessels are thresholded clearly. Fig. 3 (a) is also a blood vessel image of size 130 x 180, where the blood vessels are hardly visible. With fuzzy method in Fig. 3(b), the blood vessels are somewhat segmented with little noise present but in Bustince's intuitionistic fuzzy method in Fig. 3(c), large amount of noise present is present especially in the upper part of the image. The proposed intuitionistic fuzzy method in Fig. 3(d) almost contains no noise.

VI. CONCLUSION

This paper provides a new approach to global image thresholding using intuitionistic fuzzy set theory on poorly illuminated pathological blood vessel images. The algorithm is tested on several medical images and the thresholded images using the proposed intuitionistic fuzzy method are better. The reason is that when the membership function is not accurately defmed due to the personal error or the hesitation degree, then intuitionistic fuzzy set method gives better result due to the incorporation of the hesitation. When the membership degree selection is adequate, the intuitionistic fuzzy method is similar than the ones using fuzzy method. ACKNOWLEDGMENT

The author would like to acknowledge the Department of Science and Technology, Govt. of India, New Delhi

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