Fuzzy Sets and Systems 157 (2006) 1126–1138www.elsevier.com/locate/fss

Parametric indices of fuzziness for automated image enhancementIoannis K. Vlachos, George D. Sergiadis∗

Department of Electrical & Computer Engineering, Telecommunications Laboratory, Faculty of Technology, Aristotle University of Thessaloniki,University Campus, GR-54124, Thessaloniki, Greece

Received 23 February 2005; received in revised form 2 August 2005; accepted 16 November 2005Available online 19 December 2005

Abstract

This paper presents an automated fuzziness-driven algorithm for image enhancement. A class of parametric indices of fuzziness isintroduced, which serves as the optimization criterion of the contrast-enhancement procedure. We show that the parametric class ofindices constitutes a one-parameter generalization of the linear index of fuzziness of a set. The modification of the membership valuesof image pixels in the fuzzy plane is performed by finding the optimal S-function, which maximizes the parametric index of fuzziness(PIF). The first-order fuzzy moment of the image is used for tuning the PIF. Experimental results demonstrate the efficiency of theproposed framework in enhancing even highly low-contrasted images and also its ability to improve existing contrast-enhancingalgorithms.© 2005 Elsevier B.V. All rights reserved.

Keywords: Image processing; Contrast enhancement; Measures of fuzziness; Parametric indices of fuzziness; Fuzzy complements

1. Introduction

Enhancement is a fundamental task in digital image processing and analysis, aiming to improve the appearance ofimages in terms of human brightness perception. Many algorithms for image enhancement have been proposed thatencompass a variety of operations such as noise removal, deblurring, gray-level dynamic range modification. Contrastenhancement is among them and is often part of image processing systems in the pre-processing and/or post-processingstage.

Fuzzy sets theory [12] has been successfully applied to many image processing and pattern recognition problems.Fuzzy techniques offer a non-linear and knowledge-based framework for coping with the ambiguity and vaguenessoften present in digital images, associated mostly with the definition of various elements such as edges, regions orboundaries. In this sense, the contrast of an image may be considered fuzzy as well. Despite the fact that contrastenhancement is a usual task in digital image processing, the very concept of image contrast lacks of a precise definition

∗ Corresponding author. Tel.: +30 2310 996314; fax: +30 2310 996312.E-mail addresses: ivlachos@mri.ee.auth.gr (I.K. Vlachos), sergiadi@auth.gr (G.D. Sergiadis)URL: http://mri.ee.auth.gr.

0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2005.11.016

I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138 1127

that can serve as an optimization criterion. Exploiting this weakness in defining contrast, which is fuzzy in nature, weinterpret the term “contrast” as a qualitative rather than a quantitative measure of the image. Fuzzy sets theory is thesuitable tool to deal with such qualitative measures.

The concepts of fuzziness and fuzzy entropy have been extensively used in the field of image processing as anoptimization criterion. However, both concepts are characterized by the limitation of their maximum being committedto the fixed point 1

2 ; i.e. the midpoint of the unit interval. This can be considered as a drawback when such measures areto be applied in real-world images, as argued in [14], since this point with inherent maximum fuzziness is a fixed pointand thus it is not application-dependent. In other words, maximum fuzziness is characterized by maximum ambiguitythat may vary between images, and not by the midpoint of the [0, 1] interval. Therefore, using optimization criteriabased on this point in order to enhance the image contrast by adjusting gray-level transformation functions, may resultin enhanced images that are nevertheless sub-optimal.

In this paper, we present an automated algorithm for contrast enhancement, based on a class of parametric indicesof fuzziness. This class of indices has the ability of adjusting the location of the maximum in any point of the (0, 1)

interval and thus modelling the ambiguity and vagueness of images in a more adaptive way. In order to derive classesof parametric indices of fuzziness, parameter-dependent complements were used. Motivated by [11] and extending theapproach of maximizing the index of fuzziness of an image also found in [11], the modification of the membershipvalues of image pixels in the fuzzy plane is performed according to an S-function, which maximizes the appropriateparametric index of fuzziness (PIF). The selection of the optimal parameter of the PIF is carried out by exploiting thestatistics of the image in the fuzzy domain, using its first-order fuzzy moment.

This paper is organized as follows. In Section 2, we briefly describe an image in terms of fuzzy sets theory and weintroduce a class of parametric indices of fuzziness derived from Sugeno fuzzy complement. In Section 3, the proposedmethod is presented in full detail. Furthermore, the description of the family of functions applied for modifying themembership values of the gray levels is carried out and the definition of the first-order fuzzy moment of the image isgiven. In Section 4, the experimental results obtained using the proposed approach are presented and a comparison tothose derived using a number of different contrast enhancement techniques is carried out. Moreover, an improvementof the algorithm proposed in [11] is demonstrated. Discussion and conclusions are made in Section 5.

2. Fuzzy sets and parametric indices of fuzziness

2.1. Image representation in the setting of fuzzy sets theory

Let us consider an image A of size M ×N pixels, having L gray levels g ranging from 0 to L−1. The image A can beviewed as an array of fuzzy singletons [7–10]. Each element of the array is the membership value �A(gij ) of the graylevel gij , corresponding to the (i, j)th pixel, regarding to an image property such as brightness, edginess, homogeneity,etc. The selection of the specific image property depends on the requirements of the particular application and thecorresponding expert knowledge. For the task of image enhancement we consider the property “brightness” of the graylevels. Using the fuzzy sets notation image A can be represented as:

A = {�A(gij )/gij |i = 0, 1, . . . , M − 1, j = 0, 1, . . . , N − 1}. (1)

2.2. Parametric indices of fuzziness

Contrast enhancement is a common task in digital image processing systems. Several methods have been proposedfor measuring the grayness ambiguity of an image and therefore providing a criterion for the enhancement procedure.

In general, a measure of fuzziness is a function

f :P(X) → R+, (2)

where P(X) denotes the fuzzy power set of the finite universe X. If we employ a fuzzy complement, other than thestandard one, with equilibrium ec, f must satisfy the next axiomatic requirements for any fuzzy sets A, B ∈ P(X), in

1128 I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138

order to be qualified as a sensible measure of fuzziness [3,4]. These are a generalization of the axioms for nonprobabilisticfuzzy entropy stated in [6].F1. f (A) = 0 if and only if A is a crisp set.F2. f (A) assumes its maximum if and only if A is maximally fuzzy; that is �A(xi) = ec for all xi ∈ X.F3. f (A)�f (B), if and only if A ≺ B; that is A is “sharper than” B.In the last requirement, the relation “sharper than” defined on P(X) means that

�A(xi)��B(xi) when �B(xi)�ec (3)

and

�A(xi)��B(xi) when �B(xi)�ec (4)

for all xi ∈ X.One of the most common measures of fuzziness is the “linear index of fuzziness”, which for a set A is defined as

�l (A) = 2

|X||X|∑i=1

min{�A(xi), 1 − �A(xi)}, (5)

where |X| denotes the cardinality of the finite universe X. In (5) the standard fuzzy complement �cs

A (xi) = 1 − �A(xi)

was used as the complementation operator. For an image A defined as in (1) the linear index of fuzziness of (5) can berewritten as

�l (A) = 2

MN

L−1∑g=0

hA(g) min{�A(g), 1 − �A(g)}, (6)

where hA(g) denotes the frequency of occurrence of the gray level g. It should be mentioned that the linear indexof fuzziness of (6) considers the intersection between fuzzy sets and their complements, using a t-norm such as minoperator. In [11] the algebraic product was selected as the intersection operator and a modified version of (6) wasintroduced as

�′l (A) = 4

MN

L−1∑g=0

hA(g)�A(g)�cs

A (g). (7)

In the proposed method, we use a class of “parametric indices of fuzziness”, based on the Sugeno class of involutivefuzzy complements and algebraic product t-norm operator. Sugeno class of involutive fuzzy complements is describedby

�c�A (g) = 1 − �A(g)

1 + ��A(g), (8)

where � ∈ (−1, ∞). Applying the Sugeno class of fuzzy complements in (7), instead of the standard one, a new classof parametric indices of fuzziness is derived given by the following formula:

�∗�(A) = K(�)

L−1∑g=0

hA(g)�A(g)�c�A (g), (9)

for � ∈ (−1, ∞) \ {0}. K(�) is a normalization factor, in order for the PIF to lie in the unit interval, and is described by

K(�) = �2√1 + �

MN(−1 + √(1 + �))(1 + � + √

(1 + �)). (10)

I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138 1129

�(x)

γλ

(�(x

))

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0.8

1.0

0.0

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0.8

1.0

1.0

Equ

ilibr

ium

ecλ

(λ)

-1 0 1 3 4 52λ(a) (b)

Fig. 1. (a) Curves of the parametric index of fuzziness �� for � = 0 (ec� = 12 ), depicted with solid line, and � = −0.9375 (ec� = 0.8) shown with

dashed line. (b) The equilibrium ec� of the Sugeno fuzzy complement as a function of �.

For the special case of � → 0 the Sugeno fuzzy complement identifies with the standard fuzzy complement and thecorresponding PIF coincides with the modified linear index of fuzziness of (7); that is

lim�→0

�∗�(A) = �′

l (A). (11)

Therefore, the proposed class of parametric indices of fuzziness can be considered as a generalization of the linearindex of fuzziness of a set and can be formulated as

��(A) ={

�∗�(A) if � ∈ (−1, ∞) \ {0},

�′l (A) if � = 0.

(12)

It is easy to verify that (12) satisfies the set of axiomatic requirements F1–F3 for any � ∈ (−1, ∞). Fig. 1(a) illustratesthe curves of Sugeno PIF �� for � = 0 (ec� = 1

2 ) and � = −0.9375 (ec� = 0.8). From Fig. 1(a) it is apparent thatthe Sugeno class of parametric indices of fuzziness has the property of assigning different weights as the membershipmoves away from the two values 0 and 1, since it has the advantage of centering the maximum of the PIF at any point ofthe (0, 1) interval. This property of the proposed parametric indices of fuzziness overcomes the intrinsic limitation thatthe linear index of fuzziness imposes, where the location of the maximum is committed to the fixed point 1

2 . Selectingdifferent parameters � results in varying the equilibrium ec� of the Sugeno fuzzy complement and consequently thepoint where the PIF attains a maximum. The equilibrium ec� as a function of � is given by

ec�(�) =

⎧⎪⎨⎪⎩

−1 + √1 + �

�if � ∈ (−1, ∞) \ {0},

1

2if � = 0.

(13)

Fig. 1(b) illustrates the curve of the equilibrium ec� of the Sugeno fuzzy complement as a function of �.

3. Proposed method

Fuzzy image processing consists of three sequential stages: (a) fuzzification, (b) suitable membership values mod-ification, and (c) defuzzification. In the fuzzification stage of the proposed algorithm, the membership function �A isinitialized according to

�A(g) = g − gmin

gmax − gmin, (14)

where gmin and gmax are the minimum and maximum gray levels of the image respectively. Using (14) for initializingthe membership values has the advantage of stretching out the membership function over the unit interval. In the secondstage, an S-function is used for modifying the membership values.

1130 I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138

�(g)

S f(�

(g);

P)

0.00.0

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0.2

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0.8

1.0

1.0First-order fuzzy moment m1

λ opt

(m1)

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80

100

(a) (b)

Fig. 2. (a) Flexible S-function Sf with P=(0.1, 0.4, 0.9). (b) The curve of the optimal parameter �opt as a function of the first-order fuzzymoment m1.

3.1. S-function

A function commonly used to describe fuzzy sets is the S-function, which is suitable for representing the “degree ofbrightness” of gray levels in an image. It is given as [13]

S(x; a, b, c) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0 if x�a,

2

(x − a

c − a

)2

if a < x�b,

1 − 2

(x − a

c − a

)2

if b < x�c,

1 if x > c,

(15)

where x is the independent variable. In this work, the S-function is not used to transfer the image to the fuzzy domain,but to transform the membership values of its gray levels. Therefore, we consider the independent variable to be themembership function �A. The parameters a, b, and c define a parameter set P = (a, b, c), which determines the shapeof the S-function. The parameters satisfy the condition of 0�a < b < c�1. Defining the S-function using (15) has thelimitation of b being committed to the midpoint of the [a, c] interval. Moreover, b always corresponds to the crossoverpoint of the membership function. Therefore, a more “flexible” S-function is used [5], in which b can be any point inthe (a, c) interval

Sf (�A(g); P) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 if �A(g)�a,(�A(g) − a

)2

(b − a)(c − a)if a < �A(g)�b,

1 −(�A(g) − c

)2

(c − b)(c − a)if b < �A(g)�c,

1 if �A(g) > c,

(16)

as depicted in Fig. 2(a) for P = (0.1, 0.4, 0.9).

3.2. Contrast enhancement by maximizing the parametric index of fuzziness

For an image A defined as in (1), different images can be generated by modifying its membership values using theflexible S-function of (16) for different values of a, b, and c. Therefore, the PIF of the resulting image can be regardedas a function of the parameter set P = (a, b, c) and can be written as

��(A; P) = K(�)

L−1∑g=0

hA(g)Sf (�A(g); P)�c�A (Sf (�A(g); P)), (17)

I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138 1131

Fig. 3. (a) Initial highly low-contrasted image and images obtained using the maximization of the (b) linear index of fuzziness (ec� = 12 ) and (c) the

PIF for ec� = 0.9.

where hereinafter hA denotes the histogram of the image after fuzzification using (14). For a specific combinationPopt = (aopt, bopt, copt) of the S-function’s parameters, which is considered optimal, the PIF of the generated image ismaximized; that is

Popt = argP max{��(A; P)}. (18)

Images possessing maximal grayness ambiguity are expected to be more suitable in terms of human brightness per-ception [11]. Therefore, the modification of the membership values for performing contrast enhancement is carried outaccording to

�newA (g) = Sf (�A(g); Popt), (19)

where �newA (g) denotes the new membership value corresponding to the gray level g. However, in order to utilize the

class of parametric indices of fuzziness as an optimization criterion, the choice of � should first be made.Typically, low-contrasted images are characterized by a large concentration of pixels in a small area of the histogram.

The aim of the contrast enhancement algorithms, and especially of histogram equalization, is to find a suitable inten-sity transformation function, to uniformly distribute the histogram over the entire gray-level range. Such images areconsidered to be more adapted to the human visual system.

Let us consider a fuzzified dark image, which has the majority of its pixels located at the lower intensitylevels. Maximizing the linear index of fuzziness of the image with respect to the S-function Sf , results in select-ing a parameter set Popt such that low membership values move towards the point 1

2 , i.e. the equilibrium of thestandard complement, where the index equals unity. This transformation of membership values practically stretchesthe histogram of the dark image towards higher intensity values. Now consider the case of maximizing a PIF that hasits maximum positioned at even higher membership values. This will result in stretching further out the histogram.This procedure can be considered as a “biased histogram modification”. The inverse procedure is true for brightimages.

In order to further demonstrate the effects of the aforementioned “biased histogram modification”, we consideredthe low-contrasted image of Fig. 3(a). Figs. 3(b) and (c) illustrate images produced after applying the linear indexof fuzziness of (7) and the PIF for ec� = 0.9, respectively. The image obtained using the PIF reveals hidden detailsboth in the background and the foreground. This can also be verified by examining the corresponding normalizedhistograms of Fig. 4. Using the PIF with its maximum located at higher membership values, since the image is dark,results in a more uniformly distributed histogram than the one obtained using the linear index of fuzziness. In thisexample, the fuzzification was performed simply by gray-level normalization. The inverse procedure was used fordefuzzification.

It should be mentioned that there is a tradeoff between enhancing highly low-contrasted images and overexposure orunderexposure. However, as we will immediately discuss, in the proposed algorithm the location of the maximum ofthe PIF is calculated based on the first-order statistics of the image in the fuzzy domain, ensuring, as much as possible,that the enhanced image will not be saturated towards dark or bright gray levels.

1132 I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138

Gray level g Gray level g Gray level g

0.0

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1.0

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0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250

h A (g

)

h A (g

)

h A(g

)

(a) (b) (c)

Fig. 4. Corresponding normalized histograms for the images of Fig. 3.

In order to estimate the location where the concentration of the pixels in the histogram of an arbitrary image occurs,we propose to use the first-order fuzzy moment of the image itself. Since the image is considered in the fuzzy domain,we call the moment a fuzzy moment.

Definition 1. For an image A defined as in (1), the first-order fuzzy moment m1 is defined as

m1 =L−1∑g=0

�A(g)h̃A(g), (20)

where h̃A(·) is the normalized histogram of the image, given by

h̃A(g) = hA(g)

maxg {hA(g)} . (21)

Therefore, the “optimal” equilibrium point ec�optis selected as the symmetric of the first-order fuzzy moment m1

with respect to the midpoint of the unit interval; that is

ec�opt= 1 − m1. (22)

Evaluating (13) for ec�optyields

1 − m1 = −1 + √1 + �opt

�opt, (23)

where �opt denotes the parameter for which the Sugeno fuzzy complement has an equilibrium ec�opt. Solving (23) for

�opt delivers

�opt = 2m1 − 1

(1 − m1)2 . (24)

Since for images m1 ∈ (0, 1) we obtain that

limm1→0

�opt = −1 (25)

and

limm1→1

�opt = ∞. (26)

Therefore, for any m1 ∈ (0, 1) we ensure that the constraint of � ∈ (−1, ∞) is satisfied. Fig. 2(b) illustrates �opt as afunction of the first-order fuzzy moment m1.

Finally, in the defuzzification stage of the proposed algorithm and after having obtained the optimal transformationparameters Popt, the contrast enhanced image is derived by scaling the membership values up to the range of [0, L− 1]

I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138 1133

using the following formula

g′ = (L − 1)�newA (g) = (L − 1)Sf (�A(g); Popt), (27)

where g and g′ are the original and the transformed gray levels, respectively, and L the number of intensity values.It should be mentioned that searching exhaustively the state space of the Sf -function’s parameters is computationally

expensive, since the search space is about 2.7 × 106. Therefore, an optimization method can be used to significantlyreduce the computational cost of the algorithm, in order to be applicable in real-time applications. Simulated annealingis a numerical optimization scheme based on the principles of thermodynamics. It was introduced in [2] and is a tool forlarge scale optimization problems, especially in cases where the global extremum co-exists with many local extrema.Adopting the proposed optimization method, experimental results have demonstrated that the optimal parameter setPopt can be obtained in approximately 3000 iterations, thus reducing the computational cost to the 1

900 of the cost ofthe exhaustive search [1].

4. Experimental results

The proposed algorithm has been tested using various images of different types that had undergone extreme contrastdegradation. In order to evaluate the performance of the proposed method, we compared our results to those derivedusing the histogram equalization method, the maximum entropy principle of fuzzy events (MEP–FE) [1], and the�-enhancement method based on optimization of image fuzziness [11]. For our simulations we used gray-scale imagesof size 256 × 256 pixels with 8 bits-per-pixel gray-tone resolution.

Comparing the images of Figs. 5 and 7, one can observe the efficiency of the proposed method in enhancing highlylow-contrasted images. Fig. 5(a) illustrates a highly low-contrasted image with the majority of its pixels concentrated at

Fig. 5. Comparison of the proposed method with different contrast-enhancement approaches. (a) Original low-contrasted image. Images processedby (b) histogram equalization, (c) MEP–FE, and (d) proposed method.

1134 I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138

Gray level g Gray level g

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0 50 100 150 200 250 0 50 100 150 200 250

�A(g)0.0 0.2 0.4 0.6 0.8 1.0

�A(g)

γ λ(�A

(g))

S f(�

A(g

);P

opt)

h A (g

)

h A (g

)

(a) (b)

(c) (d)

Fig. 6. Normalized histograms of the (a) initial low-contrasted image of Fig. 5(a) and (b) image of Fig. 5(d) processed by the proposed method. (c)PIF curve used for maximization with �opt = −0.881642 (ec�opt

= 0.744030) and (d) membership modification function Sf (�A(g); Popt) with

Popt = (0, 1255 , 160

255 ).

Fig. 7. Comparison of the proposed method with different contrast-enhancement approaches. (a) Original low-contrasted image. Images processedby (b) histogram equalization, (c) MEP–FE, and (d) proposed method.

I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138 1135

Gray level g Gray level g

0.0

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�A(g)0.0 0.2 0.4 0.6 0.8 1.0

�A(g)

γ λ(�A

(g))

S f(�

A(g

);P

opt)

(a) (b)

(c) (d)

h A (g

)

h A (g

)

Fig. 8. Normalized histograms of the (a) initial low-contrasted image of Fig. 7(a) and (b) image of Fig. 7(d) processed by the proposed method. (c)PIF curve used for maximization with �opt = 25.148430 (ec�opt

= 0.163571) and (d) membership modification function Sf (�A(g); Popt) with

Popt = ( 151255 , 254

255 , 1).

the lower intensity levels as can be seen from Fig. 6(a), which shows the normalized histogram of the initial image. Theimages of Figs. 5(b) and (c) are obtained using the histogram equalization and MEP–FE methods respectively. Bothimages, although enhanced, still possess a large number of pixels in the lower and higher bins of the histogram. Thisover-enhancement results in images that appear somewhat saturated. On the contrary, the image of Fig. 5(d), derivedusing the proposed PIF approach, reveals even hidden details, such as the interior of the bacteria. The histogram ofthe enhanced image, depicted in Fig. 6(b), is more uniformly distributed over the total gray-level range. Fig. 6(c)illustrates the curve of the PIF used for obtaining the optimal parameter set Popt = (0, 1

255 , 160255 ). The corresponding

S-function is shown in Fig. 6(d). The maximum of the PIF has been automatically tuned at higher membership values(ec�opt

= 0.744030).Another demonstration of the proposed method is given in Fig. 7. Figs. 7(b) and (c) are enhanced versions of the

degraded image of Fig. 7(a), using the histogram equalization and the MEP–FE methods, respectively. The imageobtained using the proposed approach is shown in Fig. 7(d), where we can see that high-frequency edges and differentimage regions have been considerably enhanced. The normalized histograms of the initial and the PIF-enhanced imagesare depicted in Figs. 8(a) and (b), respectively, from which the effect of the “biased histogram modification” is clearlydemonstrated. The corresponding PIF curve is given in Fig. 8(c) with its maximum this time located at lower membershipvalues (ec�opt

= 0.163571), since the initial image is bright. The S-function used for modifying the membership values

is depicted in Fig. 8(d) for Popt = ( 151255 , 254

255 , 1).Finally, in order to further justify the efficiency of the proposed class of parametric indices of fuzziness, we applied

them in the �-enhancement method of [11]. Figs. 9(a) and (b) show a highly low-contrasted image and its enhancedversion using histogram equalization. Images of Figs. 9(c)–(e) were obtained respectively using the MEP–FE method,the �-enhancement algorithm and its PIF generalization. One can observe that using the concepts presented in thispaper, the enhancing procedure delivers better images in terms of contrast. In the image of Fig. 9(e), significant detailsfrom the background have been revealed. Another example is illustrated in Fig. 10.

It should be mentioned that equivalent results are obtained if instead of the Sugeno class of parametric indices offuzziness, we utilize parametric indices of fuzziness based on theYager class of involutive fuzzy complements. Finally,due to the fact that positioning ec�opt

to very low (high) membership values may lead to over-exposure (under-exposure),one can automatically select the point of maximum fuzziness, according to m1, to be located not in the (0, 1) interval,but instead for example in [0.2, 0.8] to avoid over-enhancement.

1136 I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138

Fig. 9. (a) Initial low-contrasted image and images obtained using (b) histogram equalization, (c) MEP–FE, (d) �-enhancement, and (e) generalized�-enhancement using parametric indices of fuzziness.

5. Discussion and conclusions

In this paper, we presented an automated fuzzy approach to contrast enhancement of digital images. The algorithmextends the principle of optimization of image fuzziness for contrast enhancement presented in [11], by considering aclass of parametric indices of fuzziness and the first-order fuzzy moment of the image. We have shown that the proposedclass of parametric indices constitutes a generalization of the linear index of fuzziness of a set.

I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138 1137

Fig. 10. (a) Initial low-contrasted image and images obtained using (b) histogram equalization, (c) MEP–FE, (d) �-enhancement, and (e) generalized�-enhancement using parametric indices of fuzziness.

1138 I.K. Vlachos, G.D. Sergiadis / Fuzzy Sets and Systems 157 (2006) 1126 –1138

Experimental results have demonstrated that the proposed method delivers contrast-enhanced images well adapted tothe human brightness perception. The ability of the presented algorithm to automatically enhance highly low-contrastedimages is mainly attributed to the use of the first-order fuzzy moment of the image, which tunes the PIF. We refer tothis procedure as “biased histogram modification”. Finally, we demonstrated that the application of the concepts ofparametric indices of fuzziness to existing contrast-enhancement algorithms, successfully improves their performance.Therefore, the proposed class of parametric indices provides a flexible framework for dealing with image processingproblems.

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