application of mehmet emin yüksel and alper bas‚türk · 2016-09-17 · ii. the type-2 fuzzy...

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I. Introduction

one of the most common prob-lems encountered during the acquisition and/or transmission of digital images is the con-

tamination of the image with impulse noise. Due to this inevitable fact, the acquired or received digital image must usually be pre-processed by applying an appropriate filter ing operator to the image to suppress the contaminating noise. The performance of this pre-filtering operation is of critical importance as the

Digital Object Identifier 10.1109/MCI.2012.2200624Date of publication: 12 July 2012

Application of Type-2 Fuzzy Logic Filtering to Reduce Noise in Color Images

Abstract—In this paper, we present a novel application of type-2 fuzzy logic to the design of an image processing operator called an impulse detector. The type-2 fuzzy logic based impulse detector can be used to guide impulse noise removal filters to significantly improve their filtering performance and enhance their output images. The design of the proposed impulse detector is based on two 3-input 1-output first order Sugeno type interval type-2 fuzzy inference systems. The internal parameters of the type-2 fuzzy membership functions of the systems are determined by training. The performance of the impulse detector is evaluated by using it in combination with four popular impulse noise filters from the literature on four different popular test images under three different noise conditions. The results demonstrate that the type-2 fuzzy logic based impulse detector can be used as an efficient tool to effective-ly improve the performances of impulse noise filters and reduce the impulse noise undesirable distortion effects.

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Mehmet Emin Yüksel and Alper Bas‚türk Erciyes University, TURKEY

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26 ieee ComputAtionAl intelligenCe mAgAzine | August 2012

noise will dramatically decrease the performances of subse-quent image processing tasks that are to be performed on the image if it is not properly removed from the image. Hence, the research efforts for the development of methods and design of operators for the restoration and enhance-ment of digital images corrupted by impulse noise have received high interest in the last few decades, which conse-quently led to the development of many different methods and operators for identifying and canceling impulse noise from digital images.

The most simple and natural approach to design an impulse noise removal operator is to utilize the order statistics between the corrupted pixel, to which the operator is to be applied to obtain its restored value, and other pixels in its neighborhood. The simplest impulse noise removal operator, the standard medi-an filter [1], exploits the rank order information between the pixels in the filtering window and attempts to restore the pixel at the center of the window by replacing it with the median pixel of the window. This approach is computationally very efficient and provides an acceptable noise removal performance for low noise densities. However, both noise suppression and detail preservation performances radically decrease as the noise density increases. Moreover, the standard median filter consid-erably blurs the output image and distorts the texture and small details in the image. Various derivatives of the standard median filter modifying its simple filtering behavior have been pro-posed to avoid this problem. In order to improve the standard median filtering, some operators were proposed as presented in [2] where these operators usually operate by performing a cus-tom weighting on certain pixels in the filtering window, which results in some improvement in detail preservation perfor-mance but at the cost of some decrease in the noise suppres-sion performance.

All impulse noise removal operators more or less distort the small details in the image during noise filtering, which causes some blurring in the output image. One major cause of this undesirable effect is the simple and efficient (but inap-propriate) filtering behavior of applying the same filtering operation to all pixels of the input image. This negative effect can be significantly reduced by first observing whether the pixel under filtering is noisy or not and then filtering it only if it is noisy. This selective filtering behavior can be achieved by guiding the noise filter by an additional image processing

operator called impulse detector, which decides whether the pixel concerned needs filtering. In this way, the filter restores only the pixels that are identified by the detector as impulses and leaves the other pixels (i.e., the uncor-rupted pixels) unchanged. However, it is not difficult to see that the performance of the impulse detector guided filtering naturally depends on the performance of the impulse detector. Hence, use of methods from many different areas of signal and image processing that may provide information for classifying

image pixels has led to the development of many different types of impulse detectors exploiting median filters [3], cen-ter-weighted median filters [4], boolean filters [5], edge detec-tion kernels [6], homogeneity level information [7], statistical tests [8], classifier based methods [9], rule-based methods [10], level detection methods [11], pixel counting methods [12] and soft computing methods [13]–[14].

Even though a significant majority of the impulse noise removal operators are order statistics filters such as the median based filters and their enhanced derivatives, various types of mean filters have been employed for impulse noise removal from digital images too [15]–[17]. These filters more or less exhibit a similar performance as median based filters in terms of both noise removal and detail preservation.

A final category of impulse noise removal operators contain the filters based on nonlinear methods. These are usually more complicated than the ones mentioned above but offer better performance. The first group in this category include the oper-ators based on computational intelligence techniques [18]–[23]. These operators are computationally more complex than the others but yield very good noise suppression and detail preser-vation performance. The second and the last groups comprise nonlinear filters [24]–[30] which combine the desired proper-ties of the above mentioned filters. These filters again provide much better filtering performance at the cost of increased computational load.

In the last decade, type-2 fuzzy logic systems have been of high research interest and applied to many different problems in various areas of science and technology [31]–[44]. They have also been successfully utilized in image processing for a number of tasks including gray scale image thresholding [45], edge detection [46]–[48], noise filtering [49]–[52], corner and edge detection in color images [53], deinterlacing of video signals [54], and image enhancement [55].

In a recent paper [51], we presented a noise filtering opera-tor designed by employing type-2 fuzzy logic techniques and shown that it offers excellent noise suppression performance while effectively preserving small details and other useful infor-mation within the input image. In this paper, we propose another novel application of type-2 fuzzy logic techniques, in which a first order Sugeno type interval type-2 fuzzy inference system is trained to function as an impulse detector. The pro-posed detector can be used to guide any impulse noise filter to

One of the most common problems encountered during the acquisition and/or transmission of digital images is the contamination of the image with impulse noise. Due to this inevitable fact, the acquired or received digital image must usually be pre-processed by applying an appropriate filtering operator to the image to suppress the contaminating noise.

August 2012 | ieee ComputAtionAl intelligenCe mAgAzine 27

improve its performance. Both the impulse noise filter and the proposed impulse detector operate independently of each other. That is, the proposed detector does not require any prior assumption about the impulse noise fil-ter and its performance is independent of the impulse noise filter it guides.

The presented impulse detector is used with several popular impulse noise removal operators from the litera-ture in carefully designed filtering experiments and the performance improvements that it provides to the noise fil-ters are evaluated for different color test images and differ-ent noise densities.

It should be noted here that a very detailed study about a type-1 fuzzy logic based impulse detector, which may be con-sidered as the type-1 version of the type-2 detector presented in this paper, can be found in [13].

The rest of the paper is as follows: The construction, train-ing and implementation of the proposed detector as well as its application with a given impulse noise filter are explained in Section II. The results of the filtering experiments performed to evaluate the performance of the presented detector are reported and discussed in Section III. The conclusions and remarks are presented in Section IV.

II. The Type-2 Fuzzy Logic Based Impulse Detector

A. The Structure of the DetectorFigure 1a shows the general structure of the type-2 fuzzy logic based impulse detector and Figure 2 shows its combi-nation with an impulse noise filter to enhance its perfor-mance. The detector comprises two type-2 fuzzy logic based subdetectors, two defuzzifiers and a postprocessor. The detector generates an output using type-2 fuzzy inference by processing the pixels contained in its filtering window, which is shown in Figure 1b. Each subdetector in the structure pro-cesses a different neighborhood relationship between the center pixel of the analysis window and two neighboring

pixels. The upper subdetector evaluates the vertical neigh-borhood whereas the lower subdetector evaluates the hori-zontal neighborhood.

The two subdetectors employed in the structure of the proposed impulse detector are structurally identical to each other and each of them has the same number of internal parameters governing the operation of that subdetector. How-ever, the values of these internal parameters will take different values after training because of the fact that the upper subde-tector is trained to evaluate the vertical pixel neighborhood relationship while the lower subdetector is trained to evaluate the horizontal pixel neighborhood relationship during train-ing, which will be discussed in detail later.

(a)

Type-2 FuzzySubdetector V

Defuzzifier

Pos

tpro

cess

or

Defuzzifier

DV dV

DH dHp

Type-2 FuzzySubdetector HX [l ,c +1]

X [l ,c –1]X [l ,c ]

X [l ,c ]X [l +1,c ]

X [l –1,c ]

X [l +1,c +1]

X [l –1,c +1]

X [l ,c +1]

X [l +1,c –1]

X [l –1,c –1]

X [l ,c –1]

X [l +1,c ]

X [l –1,c ]

X [l ,c ]

(b)

Figure 1 (a) Structure of the type-2 fuzzy logic based impulse detector. (b) The filtering window of the detector. (Adapted from [51] with permission from the IEEE. © 2008 IEEE.)

NoiseFilter

p = 0

p = 1

p

R G B

R G B

Noisy ColorInput Image

Enhanced ColorOutput Image

Type-2 FuzzySubdetector V

Defuzzifier

DefuzzifierType-2 FuzzySubdetector H P

ostp

roce

ssor

Figure 2 The general setup for combining the impulse detector with a noise filter for enhancing its performance.

Selective filtering behavior can be achieved by guiding the noise filter by an additional image processing operator called impulse detector, which decides whether the pixel concerned needs filtering.


During operation, the center pixel of the filtering window and two of its appropriate neighbors (vertical or horizontal) are fed to the corresponding subdetector as the input. The subdetector processes this input information by employing type-2 fuzzy inference and generates an output, which is a type-1 interval fuzzy set. This fuzzy set actually represents the uncertainty interval (i.e., lower and upper bounds) for the restored value of the center pixel. The output fuzzy sets com-ing from the two subdetectors are then fed as the input to the corresponding defuzzifier blocks, which convert the input type-1 interval fuzzy set into a single scalar value by using centroid defuzzification. These two scalar values coming from the two defuzzifiers are finally evaluated by the postprocessor and converted into a single output value, which is also the output value of the proposed impulse detector.

B. The SubdetectorsEach subdetector employed in the structure of the presented impulse detector is a Sugeno type first order type-2 interval fuzzy inference system with 3-inputs and 1-output. As dis-cussed in the previous section, the internal structures of the two subdetectors are exactly the same. The input-output rela-tionship of any of the two subdetectors is as follows:

Let , ,X X X1 2 3 denote the inputs of a given subdetector (vertical or horizontal) and D (DV or DH , see Figure 1a) denote its output. These scalar input values are converted to fuzzy numbers by evaluating the membership degree of each input to its associated fuzzy sets. The membership degrees are determined by applying the membership functions of the input fuzzy sets to the scalar input values. Therefore, each combination of inputs and their associated membership functions correspond to a rule in the fuzzy rule-base of the subdetector. The rule-base contains a certain number of fuzzy rules corresponding to the number of fuzzy sets assigned for inputs. Hence, the structure of the rule-base is as follows: 1) i f ( )X M1 11! & X M2 12!^ h & ,X M3 13!^ h then R c X1 11 1= + c X c X c12 2 13 3 14+ +

2) i f X M1 21!^ h & X M2 22!^ h & ,X M3 23!^ h then R c X c X c X c2 21 1 22 2 23 3 24= + + +

3) if X M1 31!^ h & X M2 32!^ h & ,X M3 33!^ h then R c X c X c X c3 31 1 32 2 33 3 34= + + +

h h

i. i f X Mi1 1!^ h & X Mi2 2!^ h & ,X Mi3 3!^ h then R c X c X c X ci i i i i1 1 2 2 3 3 4= + + +

h h

N. if X MN1 1!^ h & X MN2 2!^ h & ,X MN3 3!^ h then R c X c X c X cN N N N N1 1 2 2 3 3 4= + + + ,

where N is the number of fuzzy rules in the rule-base (which is also the number of fuzzy sets assumed for each of the three inputs), Mij is an interval type-2 fuzzy membership function denoting the ith membership function of the j th input and Ri

denotes the output of the i th rule. It should be observed that the left hand side of each rule is a type-2 interval fuzzy number while the right hand side is a crisp number.

Here, type-2 interval Gaussian membership functions with uncertain mean are chosen as the antecedent member-ship functions:

m( ) [ , ]expM xx m

m21

ijij

ijij ij ij

2

!v

= --

mc m; E , (1)

with 1, ,i Nf= and 1,2,3j = .In the above definition, the mean and the standard deviation

of the type-2 interval Gaussian membership function Mij are represented with the parameters mij and ijv , respectively. Consequently, the lower and the upper bounds of the uncer-tainty in the mean are represented by the parameters ijm and mij , respectively. A sample type-2 interval Gaussian member-ship function and its associated footprint of uncertainty (FOU) are illustrated in Figure 3.

Since the membership functions Mij are interval type mem-bership functions, the boundaries of their FOU are character-ized by their lower and upper membership functions, which are defined as

ijm

m

m( )

exp

expM x

x

x

x

x

21

21

2

2

ij

ij

ij

ij

ij

ij

ij

2

2

2

#

v

v

=

--

--

+

+

m

m

m ijc

e

m

o

=

=

G

G

Z

[

\

]]]

]]

(2)

and

m

mm

( )

exp

exp

M x

x

x

x

m x

x

21

1

21

,ij

ij

ij

ij

ij

ij

ij ij

ij

2

2

1

2

# #v

v

=

--

--

mmc

e

m

o

=

=

G

G

Z

[

\

]]]

]]]

(3)

where ijM and M ij denote the lower and the upper member-ship functions of the type-2 interval membership function Mij .

The output of the subdetector is obtained by calculating the weighted average of the individual rule outputs:

Dw

w R

ii

N

ii

N

1

1=

=

=

i

/

/. (4)

0

1

x

Mij(x)

Mij(x)Mij(x)

Mij(x)

mij mij

Figure 3 A type-2 interval Gaussian membership function with uncertain mean. The shaded area is the footprint of uncertainty (FOU). (Adapted and reproduced from [51] with permission from the IEEE. © 2008 IEEE.)


This equation involves the computation of an interval weighted average, which is a spe-cial case of the linguistic weighted average. More information on this kind of calculations can be found in an excellent book by Mendel and Wu [56].

In the above expression, wi denotes the weighting factor of the i th rule and is calculat-ed by evaluating the membership expressions in the antecedent of the rule, which corresponds to the conversion of the scalar input values to fuzzy membership values by the use of the ante-cedent membership functions Mij and then application of the and operator to these fuzzy membership values. The and opera-tor is applied by multiplying the antecedent membership values:

( ) . ( ) . ( )w M X M X M Xi i i i1 1 2 2 3 3= . (5)

It must be observed that the weighting factor wi is a type-1 interval set, i.e., w[ , ]wi i i= w , as the membership functions Mij

in the antecedent of the ith rule are type-2 interval member-ship functions. Therefore, the lower and upper boundaries of the weighting factor wi , which may be denoted by iw and w i ( , , ,i N1 2 f= ) respectively, are determined by using the lower membership function ijM and the upper membership function Mij as follows:

i i3( ) . ( ) . ( )X X Xi i1 1 2 2 3=w MM M

( ) . ( ) . ( )w M X M X M Xi i i i1 1 2 2 3 3= . (6)

The output D of the subdetector can be found by calculat-ing the weighted average of the individual rule outputs by using (4) once all of the weighting factors are obtained. Since the wi s in the above equation are type-1 interval sets and Ri s are scalars, the output D is a type-1 interval set, i.e., [ , ]D D D= , whose lower and the upper boundaries can be determined by using the iterative procedure proposed by Karnik and Mendel [57].

C. The DefuzzifierThe type-1 interval fuzzy set D obtained at the output of a subdetector is fed to the corresponding defuzzifier as input and converted into a scalar value by performing centroid defuzzifi-cation. Since the input set D is a type-1 interval fuzzy set, i.e.,

[ , ]D D= D , its centroid is equal to the center of the interval:

dD

2=

+D. (7)

D. The PostprocessorThe two scalar values obtained at the outputs of the two defuzzifiers are given to the inputs of the postprocessor, which converts them into a single scalar output representing the out-put of the impulse detector. The postprocessor actually calcu-lates the average value of the two defuzzifier outputs and then suitably maps this value to either 0 or 1 by converting it with a

threshold corresponding to the half of the dynamic luminance range. Assuming that dV and dH denote the outputs of the defuzzifiers in the structure of the proposed detector as illus-trated in Figure 1a, the input-output relationship of the post-processor may be expressed as follows:

pd d0

12

128if

otherwise,

V H 1=

+* (8)

where p is the output of the postprocessor, which is also the output of the impulse detector.

E. Training of the SubdetectorsThe internal parameters of the proposed impulse detection operator are optimized by training. The training of the pro-posed operator is accomplished by training the individual type-2 subdetectors in its structure. Each subdetector in the structure is trained individually and independently of the other. The training setup is shown in Figure 4 and images used in training are shown in Figure 5.

The parameters of the subdetector under training are itera-tively adjusted in such a manner that its output converges to the output of the ideal impulse detector. The ideal impulse detector is a conceptual operator representing the relationship between the input and the target training images. It does not necessarily exist in reality. It is only the output of the ideal impulse detector that is necessary for training and this is repre-sented by a suitably designed target image.

Figure 5 shows the three training images, which are the original training image, the noisy training image and the

Type-2 FuzzySubdetector V Defuzzifier – +

– +Defuzzifier

IdealImpulseDetector

TargetTraining Image

InputTrainingImage

Type-2 FuzzySubdetector H

Figure 4 Setup for training the subdetectors in the structure of the type-2 fuzzy logic based impulse detector. (Adapted from [51] with permission from the IEEE. © 2008 IEEE.)

The internal parameters of the proposed impulse detection operator are optimized by training. The training of the proposed operator is accomplished by training the individual type-2 subdetectors in its structure.


noise detection image, from left to right. The first image in Figure 5a is a computer generated artificial image. Each square box in this image has a size of 4-by-4 pixels and the 16 pixels contained within each box have the same lumi-nance value, which is an 8-bit integer number uniformly dis-tributed between 0 and 255. The image in Figure 5b is obtained by corrupting the image in Figure 5a by impulse noise. The third image deserves a little explanation. This image is obtained from the difference between the original training image and the noisy training image. For this purpose, the noisy training image in Figure 5b is subtracted from the original training image in Figure 5a. The pixels which are the same in the two images (the uncorrupted pixels) convert to zero values while the pixels which are different (the pixels corrupted due to noise) in the two images convert to nonze-ro values. The noise detection image in Figure 5c is comput-ed by replacing the zeros with black pixels and the nonzeros with white pixels. Therefore, locations of the white pixels in this image indicate the locations of the noisy pixels that need filtering while the locations of the black pixels indicate the uncorrupted pixels that need to be left unfiltered. Hence, it is not difficult to see that the images in Figure 5c and Figure 5b are used as the target (desired) and the input images during training process, respectively.

It should be noted that the images used for training the pre-sented impulse detector are gray level intensity images whereas the presented detector is intended to be used with color imag-es. This is no problem because each of the three color bands (red, green and blue) of a color image, when processed individ-ually, is actually a gray level intensity image. As it will be explained in the next subsection of this section, the presented detector processes a color image by processing its three color bands individually. Hence, the detector is successfully used to detect the impulses in a color image even though it is trained with simple gray level images.

The parameters of the subdetector under training are tuned by using the Levenberg-Marquardt optimi-zation algorithm [58]–[60] so as to minimize the learning error. Once the training of the two subde-tectors is completed, the internal parameters of the subdetectors are frozen to their optimal values. The subdetectors are then combined with their corre-

sponding defuzzifiers and the postprocessor to construct the impulse detector, as illustrated in Figure 1a.

F. Processing of the Input ImageThe procedure for processing the noisy input image with a noise filter and applying the presented impulse detector for improving the output image of the noise filter is as follows: 1) A 3-by-3 pixel filtering window is slid over each color

band of the noisy input image one pixel at a time. The window is started from the upper-left corner of the color band and moved sideways and progressively downwards in a raster scanning fashion until the bottom right corner position is reached.

2) For each filtering window position, the appropriate pixels of the filtering window representing the appropriate neighborhoods of the center pixel are fed to both the noise filter and the impulse detector inputs.

3) If the output of the impulse detector for the filtering window under concern is 0, which means that the center pixel of the filtering window is uncorrupted and does not need restoration, the center pixel of the filtering window is directly copied to the output image.

4) If the output of the impulse detector for the filtering win-dow under concern is 1, which means that the center pixel of the filtering window is corrupted and needs resto-ration, the pixel value obtained at the output of the noise filter is copied to the output image as the restored value of the center pixel of the filtering window under concern.

5) This procedure is repeated until all pixels of the color band under analysis and all color bands of the noisy input image are covered.

A conceptual illustration of this procedure is given in Figure 2.

III. Results and DiscussionThe type-2 fuzzy logic based impulse detector described in the previous section is implemented and the enhancements that it provides to the performance of a noise filter are evaluat-ed by conducting a number of filtering experiments. The experiments are carefully designed to enable the reader to observe the behavior of the proposed impulse detector for dif-ferent noise filters, for different test images and under different noise conditions.

Four popular test images from the literature are included in the filtering experiments. These are the Baboon, House, Lena, and Peppers images, which are shown in Figure 6. These imag-es are chosen to include different image properties in the experiments. All of these images are 24-bit color images and they have the same size of 256-by-256 pixels.

(a) (b) (c)

Figure 5 Training images: (a) original training image, (b) noisy train-ing image (input), and (c) noise detection image (target). (Repro-duced and adapted from [51] with permission from the IEEE. © 2008 IEEE.)

The proposed detector does not require any prior assumption about the impulse noise filter and its performance is independent of the impulse noise filter it guides.


The noisy test images used in the experi-ments are obtained by contaminating a given test image with an impulse noise of given noise density. Three different noise densities are considered. These are 25%, 50% and 75% representing the low, medium and high noise densities, respectively.

The performance enhancements obtained by using the proposed impulse detector with a noise filter is evaluated by using it with four representative impulse noise filters from the literature. These are the standard median filter (SMF) [1], the edge-detecting median filter (EDMF) [6], the minimum-maximum exclusive mean filter (MMEMF) [16] and the alpha-trimmed mean-based filter (ATMBF) [17]. All of these impulse noise filters operate on a minimal filtering window, which has a size of 3-by-3 pixels.

The improvement contributed by the proposed detector to the performance of the noise filter is measured by using the mean squared error (MSE) criterion, which is defined as

1 [ , ] [ , ]LC

s l c y l cMSEc

C

l

L2

11

= -==

^ h// , (9)

where [ , ]s l c and [ , ]y l c represent the luminance value of the pixel at line l and column c of one of the three color bands of the original and the restored versions of a corrupted test image respectively.

It should be noted that this definition of MSE is valid for gray level images only. Since the test images used in the filter-ing experiments reported in this section are color images, this MSE computation is performed for three times, one for each of the three color bands (red, green and blue), and the three resulting MSE values are then averaged to obtain the represen-tative MSE value for that image.

From a different point of view, a pixel in a color image may be considered as a point in the three dimensional R-G-B color space. The contamination of this pixel by noise implies a change in at least one of the three color components (red, green or blue) of this pixel, which corresponds to a shift of the location of the point that represents this pixel in the R-G-B color space. Hence, the filtering operation performed on this pixel by a noise filter may be thought of as an attempt to move the point representing this pixel back to its original location in the R-G-B color space.

Based on these observations, we introduce a novel criterion termed as the mean pixel distance as an alternative performance measure that is better suited to color images. The MPD criteri-on can mathematically be defined as follows:

[ , ], [ , ]LC

l c l c1MPD s yc

C

l

L

11

===

// , (10)

where [ , ]l cs and [ , ]l cy are vectorial quantities in the R-G-B color space (i.e., s [ , ] { [ , ], [ , ], [ , ]}l c s l c s l c s l cR G B= ) and repre-sent the color value of the pixel at line l and column c of the

original and the restored versions of a corrupted color test image respectively.

Here [ , ], [ , ]l c l cs y is the vectorial (Euclidian) distance between two points in the R-G-B color space defined as

i[ , ], [ , ] [ , ] [ , ]l c l c s l c y l cs y i2

{ , , }i R G B

= -!

^ h/ . (11)

To the best of our knowledge, the MPD criterion described above has not been proposed before.

Each individual filtering experiment represents a combina-tion of a noise filter, a test image and a noise density. Hence, a total of 48 individual filtering experiments are performed since there are four different noise filters, four different test images and three different noise densities included in the experiments.

A pixel in a color image may be considered as a point in the three dimensional R-G-B color space. The contamination of this pixel by noise implies a change in at least one of the three color components (red, green, or blue) of this pixel, which corresponds to a shift of the location of the point that represents this pixel in the R-G-B color space. Hence, the filtering operation performed on this pixel by a noise filter may be thought of as an attempt to move the point representing this pixel back to its original location in the R-G-B color space.

Figure 6 The test images used in the experiments: (a) Baboon, (b) House, (c) Lena, and (d) Peppers.

(b)(a)

(d)(c)


In each of the individual filtering experiments performed for a given {noise filter / test image / noise density} combi-nation, the noise filter is combined with the detector, as

shown in Figure 2, and applied to the test image of that experiment corrupted with the noise density of that experiment. The perfor-mance of the noise filter is separately evaluated on the same noisy test image for the uses with-out and with the detector.

The same filtering procedure is followed in each of the individual filtering experiments. First, the noisy test image of the experiment is created as discussed before. Then, the noisy test image is filtered by the noise filter of that

Table 1 Comparison of the mean squared error (MSE) values calculated for the output images of the filters when used without and with the proposed impulse detector.

baboon House lena PePPers

25% 50% 75% 25% 50% 75% 25% 50% 75% 25% 50% 75%

beFore FilTering 4765.45 9550.69 14359.65 4517.09 9035.53 13678.13 5260.07 10400.14 15635.00 4970.96 9955.33 14839.99

WiT

Ho

uT

DeT

ecTo

r

SMF 805.52 2745.44 8952.80 196.54 1958.17 8109.13 288.89 2354.98 9320.32 300.06 2285.30 8691.36

EDMF 468.05 1222.69 5302.06 138.14 768.22 4653.65 200.77 924.58 5272.11 220.20 903.74 4913.40

MMEMF 450.51 549.23 769.86 72.56 93.52 151.53 72.70 114.71 224.32 183.65 246.72 443.52

ATMBF 699.06 2698.29 8947.56 174.88 1946.55 8107.50 267.12 2358.96 9334.60 270.34 2274.66 8692.86

WiT

H

DeT

ecTo

r

SMF 303.73 1841.83 7480.65 100.24 1392.49 6796.60 142.56 1643.21 7827.98 147.31 1622.88 7343.96

EDMF 260.91 1015.90 4864.27 93.41 670.25 4320.13 132.54 750.06 4728.53 135.94 750.48 4523.58

MMEMF 178.06 384.65 712.40 65.36 88.80 148.71 40.91 94.19 214.81 133.27 214.44 428.72

ATMBF 308.59 1854.82 7493.09 102.49 1398.06 6801.44 156.72 1674.60 7858.59 151.97 1636.75 7358.70

ga

in (

%) SMF 62.29 32.91 16.44 49.00 28.89 16.19 50.65 30.22 16.01 50.91 28.99 15.50

EDMF 44.26 16.91 8.26 32.38 12.75 7.17 33.98 18.88 10.31 38.27 16.96 7.93

MMEMF 60.48 29.97 7.46 9.92 5.05 1.86 43.73 17.89 4.24 27.43 13.08 3.34

ATMBF 55.86 31.26 16.26 41.39 28.18 16.11 41.33 29.01 15.81 43.79 28.04 15.35

Table 2 Comparison of the mean pixel distance (MPD) values calculated for the output images of the filters when used without and with the proposed impulse detector.


25% 50% 75% 25% 50% 75% 25% 50% 75% 25% 50% 75%

beFore FilTering 84.20 148.37 196.59 83.31 147.33 195.95 84.08 148.47 197.98 84.68 150.19 201.12

WiT

Ho

uT

D

eTec

Tor SMF 37.73 69.98 146.28 11.66 46.22 137.34 14.70 49.23 138.75 15.25 49.79 139.24

EDMF 22.46 43.22 104.36 7.78 25.16 92.07 9.12 26.21 91.49 9.94 27.21 93.10

MMEMF 25.11 31.21 39.97 5.08 7.85 11.94 6.77 10.86 16.49 9.53 14.19 22.05

ATMBF 30.05 67.66 146.19 8.04 45.43 137.45 10.25 48.18 138.93 10.52 48.48 139.33

WiT

H

DeT

ecTo

r SMF 15.94 49.79 128.21 5.33 33.42 119.93 6.60 35.53 121.18 6.88 36.23 122.36

EDMF 15.60 37.97 98.39 6.45 22.92 86.68 7.53 23.71 85.92 7.96 24.45 87.36

MMEMF 13.58 25.24 38.04 4.30 7.44 11.78 5.25 10.03 16.18 7.67 13.05 21.59

ATMBF 16.44 50.67 128.68 5.97 34.58 120.48 7.44 36.92 121.89 7.41 37.42 123.04

ga

in (

%) SMF 57.75 28.85 12.35 54.29 27.69 12.68 55.10 27.83 12.66 54.89 27.23 12.12

EDMF 30.54 12.15 5.72 17.10 8.90 5.85 17.43 9.54 6.09 19.92 10.14 6.17

MMEMF 45.92 19.13 4.83 15.35 5.22 1.34 22.45 7.64 1.88 19.52 8.03 2.09

ATMBF 45.29 25.11 11.98 25.75 23.88 12.35 27.41 23.37 12.27 29.56 22.81 11.69

The performance of the noise filter is separately evaluated on the same noisy test image for the uses without and with the detector.


The performance improvement is especially evident when the appearance of the hair around the mouth of the animal is compared in each image pair.

experiment. Next, the MSE and MPD values for the output image of the filter are calcu-lated. Following this, the noisy test image is processed by using the proposed impulse detector. After that, the blur-reduced final output image is constructed from the select-ed pixels from the noisy input image and restored output image of the noise filter. The selection process is guided by the noise detector as illustrated in Figure 2. Finally, the MSE and MPD values are calculated for the final output image for comparison with the previously calculated values.

Table 1 and Table 2 show the average MSE and MPD val-ues for the filtering experiments performed. Each of these tables comprises three sections, which are entitled “Without detector,” “With detector” and “Gain (%),” respectively. The MSE/MPD values listed in the first section of the tables rep-resent the results obtained by using the noise filters directly on the noisy test images without the proposed impulse detector, whereas the values in the second section represent the results obtained by using the noise filters with the proposed detector. Comparison of the values in these two sections, as seen in the third section, clearly demonstrates that the presented impulse detector significantly improves the performance of all noise filters regarding both the MSE and the MPD criteria inde-pendent of the test image and the noise density.

In order to allow the reader to make a visual evaluation of the performance improvement obtained by using the presented impulse detector, Figure 7 presents the output images of the four noise filters for the Baboon image corrupted by 25% impulse noise. For each image pair on each line in this figure, the image on the left is the output image of the noise filter obtained without the detector while the image on the right is that obtained with the detector. Here, the undesirable distor-tions and blurring caused by the noise filters can be observed in the images on the left and the restorations of these distortions by the use of the impulse detector can be observed in the images on the right. The performance improvement is especial-ly evident when the appearance of the hair around the mouth of the animal is compared in each image pair in this figure.

Figure 8 shows the output images of the same operator for different test images in order to provide an additional visual evaluation of the performance dependency of the impulse detector on image properties. Here, the SMF operator is cho-sen as the comparison filter because of its comparatively lower noise removal and detail preservation performance. As in Figure 7, the image on the left side of each image pair is the output image of the SMF obtained without the detector while the image on the right is that obtained with the detec-tor. The images in this figure also support the results present-ed in Table 1, Table 2 and Figure 7 by providing an additional visual evidence indicating that the presented impulse detector effectively improves the output images of the noise filters.

In addition to the filtering experiments reported so far, an additional set of experiments are performed to evaluate and compare the detection performance of the presented

detector with selected impulse detectors from the literature. These impulse detectors include the median based detector (MBD) used in [3], the adaptive impulse detector (AID)

Figure 7 Comparison of the output images of the noise filters for the Baboon image corrupted by 25% impulse noise. (a) SMF, (b) SMF with proposed detector, (c) EDMF, (d) EDMF with proposed detector, (e) MMEMF, (f) MMEMF with proposed detector, (g) ATMBF, and (h) ATMBF with proposed detector.

(a) (b)

(c) (d)

(e) (f)

(g) (h)


used in [4] and the edge detecting impulse detector (EDID) used in [6].

Table 3 presents the false classification ratio (FCR) values calculated for all impulse detectors. The FCR value is calculat-ed as follows:

NN 100FCR

T

F #= , (12)

where NF denotes the number of falsely classified pixels of the input image and NT denotes the total number of pixels. It is easily observed from this table that the presented impulse detector considerably outperforms the other four operators from the literature.

IV. ConclusionIn this paper, we have presented a novel impulse detector based on type-2 fuzzy logic techniques. The presented detector can be used as a very efficient tool for guiding an impulse noise fil-ter in restoring noisy color images. The detector effectively reduces the undesirable distortions and blurring effects of the impulse noise filter without interfering with its filtering behav-ior. Fundamental advantages of the proposed detector may be summarized as follows: 1) It consists of only two identical subdetectors, two simple

defuzzifiers and a simple postprocessor. 2) The internal parameters of its subdetectors can be deter-

mined by training, which is achieved by using simple arti-ficial images generated in computer.

3) Its operation does not interfere with the operation of the noise filter. Therefore, it can be used with virtually any spatial domain noise filter to improve its performance and reduce distortions.

4) Its operation is completely independent of the operation of the noise filter. Therefore, its performance is indepen-dent of the performance of the noise filter.

Based on the results presented in the previous section and the remarks listed above, we conclude that the presented impulse detector can be used as an efficient tool for improving the output performance of a noise filter.

V. AcknowledgmentPart of the material (text, equations, figures, etc.) from a previ-ously published work of the authors [51] copyrighted by the

Table 3 Comparison of the false classification ratio (FCR) values calculated for the impulse detectors.


25% 50% 75% 25% 50% 75% 25% 50% 75% 25% 50% 75%

MbD 8.27 16.43 37.76 2.71 12.03 35.99 3.98 13.70 37.15 4.73 15.09 37.70

aiD 6.52 17.71 41.49 1.75 13.03 39.01 3.04 14.98 40.29 3.30 16.22 41.44

eDiD 8.53 13.95 19.81 4.71 12.96 20.07 5.51 13.79 20.24 5.77 15.62 22.11

ProPoseD 0.00 0.00 0.01 0.08 0.09 0.12 0.01 0.01 0.01 2.53 3.04 3.55

Figure 8 Output images of the SMF operator for 25% corrupted Baboon, House, Lena, and Peppers images. For each image pair, the image on the left shows the output of the SMF operator used with-out the proposed detector and the image on the right shows the out-put of the SMF operator used with the proposed detector.

(a) (b)

(c) (d)

(e) (f)

(g) (h)


IEEE (© 2008 IEEE) has been adapted and/or reused in this paper. The authors wish to thank the IEEE for its kind permis-sion to reuse this material.

Part of the results presented in this paper have been obtained during two research projects funded by Erciyes Uni-versity Scientific and Technological Research Center (Project code: FBT-07-12) and Turkish Scientific and Technological Research Council (TÜBI·TAK) (Project code: 110E051). The authors also wish to thank Erciyes University and TÜBI·TAK for their kind support to this work.

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