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CHAPTER IV

COMPARATIVE STUDY ON FUZZY MEMBERSHIP FUNCTIONS

4.1 Introduction

Objective of this chapter is identification of a fitting fuzzy membership degree

function. It is an essential task to define a suitable function to measure the global or local

image fuzzyness or ultrafuzzyness. In this backdrop a comparative study on fuzzy

membership functions for image segmentation using ultrafuzziness is carried out. In this

work, Tizhoosh membership function which is totally supervised, Huang & Wang

membership function and S-function are considered. Each membership function has its

own advantages and challenges in the computation process. Using fuzzy logic concepts,

the problems involved in finding the minimum or maximum of an entropy criterion

function are avoided. An attempt is made to identify the better membership function

through experiments to assign the fuzzy membership grade to every pixel in the image,

for optimum image segmentation using ultrafuzziness. For low contrast images contrast

enhancement is assumed. Experimental results demonstrate that there is a quantitative

improvement with S-function over the other two functions.

In many computer vision and image processing applications the fundamental task

performed on image data is image segmentation, which is done in order to process the

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foreground objects and to explore the features. A fast and facile technique for image

segmentation is thresholding. The accuracy of segmentation depends upon the process

which is based on the gray level histogram. Fuzzy set theory provides better convergence

when compared with non-fuzzy methods. This thesis focuses mainly on an automated

method with fuzzy S-function and image ultrafuzziness as a fuzzy measure not

considering an entropic criterion function. In this scenario choosing better fuzzy

membership function is an essential task.

Fuzzy based Image Thresholding methods are introduced in literature to

overcome the grayness ambiguity problem. Fuzzy set theory is used in these methods to

handle grayness ambiguity or image vagueness during the process of threshold selection.

The gray level intensity value is selected as the optimum threshold at which the

ultrafuzzy index is minimized. The image is considered as a fuzzy set and the

membership distribution explains each pixel belongs to either objet set or background set

in the misclassification region of the histogram. Hamid R. Tizhoosh, introduced a new

fuzzy measure called ultrafuzziness and also a new a membership function.

4.2 Fuzzy Membership Functions

In this work, we considered three different fuzzy membership functions (MF)

such as:

Tizhoosh membership function.

S-function for membership grades.

Huang and Wang membership function.

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These are some of the important fuzzy membership functions. In the following

sections each one is explained briefly and results are compared in chapter VI.

4.2.1 Fuzzy membership function of Tizhoosh

To measure the image fuzziness Hamid R. Tizhoosh [26] defined a new

membership degree function as shown in Equation (1) which comprises of three

unknown quantities α ,β and T must be estimated from the image statistics.

μ(g) = (4.1)

In this e experiment we have considered α ,β both values are equal to 2.

Fuzzy sets of type I

The most common measure of fuzziness is the linear index of fuzziness. For an

M N image subset A ⊆ X with gray levels g ⊆ [0, L-1], the linear index of fuzziness

can be estimated as follows:

(A) = (g), 1- (g)] (4.2)

Where ( g) is obtained from Equation(1). So the optimal threshold can be obtained

though maximizing the linear index of fuzziness criterion function that is given by

t*= Arg min { (A: T )}, 0 ≤ T ≤ L-1 (4.3)

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Fuzzy sets of Type II

Definition: A type II fuzzy set à is defined by type II membership function X

where x∊ X and u ∊ ⊆ [0, 1]

à can be expressed in the notation of fuzzy set as

à = {(( , u), ( , u))| ∊ X, u ∊ ⊆ [0, 1]},

in which 0≤ (x,u) ≤ 1.

Figure 4.1 A possible way to construct type II fuzzy sets. The interval between lower/left

and upper/right membership values (bounded region) will capture the footprint of

uncertainty.

A type II fuzzy set can be defined from type I fuzzy set and assign upper and lower

membership degrees to each element to construct the footprint of uncertainty as shown in

Figure 1. A more suitable definition for a type II fuzzy set can be given as follow:

(4.4)

The upper and lower membership degrees and of initial membership

function µ can be defined by means of linguistic hedges like dilation and concentration:

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,

,

Hence, the upper and lower membership values can be defined as follows:

,

,

Where δ ∊ (1, ∞) but δ > 2 is usually not meaningful for image data.

Ultrafuzziness

If the degrees of membership can be defined without any uncertainty as type I

fuzzy sets, then the ultrafuzziness should be zero. When individual membership values

can be indicated as an interval, the amount of ultrafuzziness would increase. The

maximum ultrafuzziness is one when the information of membership degree values are

totally ignored. For a type II fuzzy set, the ultrafuzziness is defined as γ for an M N

image subset à ⊆ X with gray levels g ⊆ [0, L-1], histogram h(g) and membership

function can be defined as follow:

(Ã)= (4.5)

where

,

, δ ∊ (1, 2)

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4.2.2 S-Function

The existing method has several drawbacks in constructing the fuzzy membership

degree function. The three unknown quantities α, β and T are to be estimated from the

image statistical parameters of the image histogram. Since they vary from one image to

another it becomes difficult to automate the entire process of image thresholding. We

considered the standard S-function to compute membership degree of the fuzziness of the

given image. We derived a most convincing method to compute initial fuzzy seed subsets

as the S-function described in Figure 4.2.

Figure 4.2 Shape of the S-function.

The S- function is used for modeling the membership degrees for object pixels.

(x) = S (x; a, b, c) = (4.6)

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Figure 4.3 Multimodal image histogram and the characteristic functions for the seed

subsets.

Iinitial fuzzy seed subset values a, b and c are computed based on. . Let x(i, j)

be the gray level intensity of image at (i, j). I={ x(i, j)| i∊ [1, M], j ∊ [1, N]} is an image

of size M N, i.e., n. The gray level set {0,1,2,…..255}. The mean(µ) and standard

deviation( ) are calculated as follows

µ = (4.7)

= (4.8)

From Equations (4.7) and (4.8) fuzzy seed set values a, b and c as shown in figure 4.3,

are estimated as follows:

b = µ = (4.9 )

a = µ - (4.10)

c = µ + (4.11)

Kaufmann introduced an index of fuzziness first to measure the vagueness of a fuzzy set.

He also established four conditions that every measure of fuzziness should satisfy. This

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fuzzy measure, ultrafuzziness is also satisfying his four conditions. So the optimal

threshold can be obtained though minimizing the ultrafuzziness criterion function that is

given by

t*= Arg min { (Ã: T )}, 0 ≤ T ≤ L-1 (4.12)

4.2.3 Fuzzy membership Function of Huang and Wang

Let X denotes an image set of size M x N with L levels, and is the gray level

of a (m, n) pixel in X. Let denote the membership value which represents the

degree of possessing a certain property by the (m, n) pixel in X; that is, a fuzzy subset of

the image set X is a mapping μ from X into the interval[0,1]. In this notation of fuzzy set,

the image set X can be written as

X= {( ( ))}, ( ) where 0 ≤ ( ≤ 1, m=0, 1,… M-1 and n=0, 1,…,N-1.

Here, the membership function can be viewed as a characteristic function that

represents the fuzziness of a (m, n) pixel in X. For the purpose of image thresholding,

each pixel in the image should possess a relationship between the pixel and its belonging

region.

Let h(g) denote the number of occurrences at the gray level g in an input image. Given a

certain threshold value t, the average gray levels of the background and the object

can be, respectively, obtained as follows:

= / (4.13)

= / (4.14)

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The average gray levels, and , can be considered as the target values of the

background and the object for the given threshold value t. The relationship between a

pixel in X and its belonging region should intuitively depend on difference of its gray

level and the target value of its belonging region. Thus, let the relationship possess the

property that the smaller the absolute difference between the gray value of a pixel and its

target value is, the larger the membership value that pixel has. Hence, the membership

function which evaluates the above relationship for a (m, n) pixel can be defined as:

( ) = (4.15)

Where C0 and C1 are two constants such that 0 ≤ ( ≤ 1. For the given threshold t,

any pixel in the input image should belong to either the object or the background. Hence

it is expected that the membership value of any pixel should be no less than ½. The

membership function in ( ) really reflects the relationship of a pixel with its belonging

region.

4.3 General Thresholding algorithm for type II fuzzy sets with

ultrafuzziness

The general algorithm for image thresholding based on type II fuzzy sets and a

fuzzy measure called Ultrafuzziness can be formulated as follows:

1 Select

any one membership function.

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2 Comput

e membership degree for every pixel.

3 Calculat

e image histogram.

4 Initializ

e the position of the membership function.

5 Shift

the membership function along the gray-level range.

6 Calculat

e in each position the amount of ultrafuzziness from Equation(5).

7 Find

out the position with minimum ultrafuzziness.

8 Thresho

ld the image with t* = .

4.4 Experimental trials

The comparison work involved with lot of experimentation by considering a

suitable data set which comprises of various kinds of images. Actually, effectiveness and

challenges of each function is figured out in the upcoming chapter called results and

discussions. Here, it is tried to show some images and their thresholded images as a result

of using different membership functions.

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(a)

(b) (c)

(d) (e)

Figure 4.4 (a) Original image of Bacteria. (b) Its ground truth image. (c) Thresholded

image using Tzhoosh function. (d) Thresholded image using Wang function. (e)

Thresholded image using S- function.

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(a)

(b) (c)

(d) (e)

Figure 4.5 (a) Original image of Rhino. (b) Its ground truth image. (c) Thresholded image

using Tizhoosh function. (d) Thresholded image using Wang function. (e) Thresholded

image using S-function.

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(a)

(b) (c)

(d) (e)

Figure 4.6 (a) Original Ultrasound image (b) Its ground truth image (c) Thresholded

image using Tizhoosh function. (d) Thresholded image using Wang function. (e)

Thresholded image using S-function.

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4.5 Chapter summary

This chapter provides an overview about fuzzy membership functions and their

significance in fuzzy oriented image segmentation approaches. In this perspective we

considered S-function along with other membership functions. Tizhoosh introduced a

new fuzzy membership function with many parameters and needs a great supervision for

every image considered. Therefore the process is truly manual and hard to get the

threshold. Wang membership function may be simple compared to Tizhoosh function,

however, it involves more complexity. S-function is more simplistic and demands less

computational complexity with our simplification of linguistic hedges based on image

basic characteristics. But this method can be further improved and tested against other

fuzzy membership functions available or a still new suitable membership function can be

derived. Efficiency of threshold selection is demonstrated with experimental results. We

assume a reasonable contrast enhancement for low contrast images. Performance

evolution is carried out with the help of two popular approaches, misclassification error

and Jaccard Index on the proposed work in the chapter titled „results and discussions‟.

Owing to the computational limitations all the other fuzzy membership functions

available in the literature are not tested. The other available functions can also be tested

against popular S-function. With this work we conclude that the performance of S-

function is relatively good over the other two membership functions and so here

employed this function.