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Fuzzy Filters for Noise Reduction in Color Images

Om Prakash Verma1, Madasu Hanmandlu2, Anil Singh Parihar3 and Vamsi Krishna Madasu4

Delhi College of Engineering1, India, Indian Institute of Technology Delhi, India2,3, University of Queensland, Australia4 [email protected], [email protected], [email protected], [email protected]

Abstract This paper presents two fuzzy filters for the removal of both Impulse and Gaussian noise in color images. A combination of these two filters also helps in eliminating a mixture of these two noises. For dealing with the Impulse noise, an algorithm is developed to search for a set of uncorrupted pixels in the neighbourhood of the pixel of interest (i.e. the central pixel) and to compute the median of this set. A fuzzy filter consisting of two sub filters is proposed to cancel out the impulse noise. The first sub filter detects the noisy pixel along with the amount of noise in it by utilizing three fuzzy membership functions, defined for this purpose. The corrupted pixels are then corrected using the median of the noise free pixels. The second sub filter makes use of the relation between different color components of a pixel to remove the residual noise in the color image. For the removal of the Gaussian noise, we propose an adaptive distance between color component pairs of the central pixel and the neighbourhood pixel so as to determine the weight of the latter. The weighted average of the weights of all the neighbourhood pixels helps compute the correction term for the Gaussian filter. A cascade combination of the two fuzzy filters in which the output of the Impulse filter is fed to the input of the Gaussian filter is shown to effectively eliminate the mixture of Gaussian and Impulse noises from color images. The results on several color images prove the efficacy of the proposed fuzzy filters. Keywords: Gaussian noise, Impulse noise, Correction term, Adaptive distance, Fuzzy filters 1. Introduction Generally, noise in signal processing is interpreted as ‘unwanted signals’. However, in the context of image processing, noise is termed as the displacement of the signal intensities from their original values. A fundamental problem of image analysis is to effectively remove noise from an image while keeping its fundamental structure constituting of edges, corners, etc., intact. The nature of the noise removal problem depends on the type of the noise corrupting the image. The two most commonly occurring types of noise are: Additive noise (e.g. Gaussian and

Impulse noise) and Multiplicative noise (e.g. Speckle noise). The focus of this work is on additive noise removal. Image filters exist in three domains: spatial, frequency and fuzzy domain. This study deals with fuzzy filters which offer several advantages over classical filters even as they preserve the image structure. Moreover, fuzzy filters are easy to realize by means of simple fuzzy rules that characterize a particular noise. A brief review of well known fuzzy filters is presented in the following paragraphs. In the field of image noise reduction, several linear and nonlinear filtering methods have been proposed. Linear filters are not able to effectively eliminate impulse noise as they have a tendency to blur the edges of an image. On the other hand, nonlinear filters like median filters are better suited for dealing with impulse noise. Several non-linear filters based on classical and fuzzy techniques have emerged in the past few years. Recent progress in fuzzy logic allows different possibilities for developing new image noise reduction methods. The fuzzy median filter [24-25] is a modification to the classical median filter. The Fuzzy Inference Rules by Else action (FIRE) filters [5, 14, 15] are a family of non-linear operators that adopt fuzzy rules to remove impulse noise from images. Russo introduced a multi-pass fuzzy filter consisting of three cascaded blocks in [32]. Each block is hooked to a fuzzy operator that attempts to cancel the noise while preserving the image structure. The fuzzy multilevel median filter introduced by Jiu [21] is manifestation of the multilevel median filter in the fuzzy domain. It includes fuzzy rules for the elimination of impulse noise. The histogram adaptive filter by Wang and Chu [40] belongs to a class of filters which employs the histogram for reducing noise. Androutsos et al. [3] designed a new class of filters called Fuzzy vector rank filters based on a combination of different distance measures, fuzzy membership values and α -trimmed functions. Khriji and Gabbouj [27] developed a multi channel filter by combining fuzzy rational and median functions. This filter preserves the edges and chromaticity of the image. Wenbin [41] presented a novel idea of alpha trimmed mean and the similarity of pixels for the detection of impulse noise. A new algorithm is proposed in [31] for the detection of impulse noise based on a large difference between the

ICGST-GVIP Journal, Volume 9, Issue 5, September 2009, ISSN: 1687-398X

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noisy pixel and the noise free pixel. In [8] rank–order mean (ROM) is modified to devise a Least- Mean Square Design for filtering out the fixed and non linear impulse noise. Efforts are made in [23, 30] to reduce blurring at the edges due to linear filtering. A signal adaptive median filtering algorithm is proposed in [16] for the removal of impulse noise in which the notion of homogeneity level is defined for pixel values based on their global and local statistical properties. The co-occurrence matrices are used to represent the correlations between a pixel and its neighbors, and to derive the upper and lower bounds of the homogeneity level. The use of non linear filters for both noise correction and image preservation is also suggested in [9-11]. Median based filters are modified for detail-preservation images in [39]. Mansoor et al. [38] introduce an iterative edge preserving filtering technique using the blur metric. Noisy pixels have been categorized into edge and non edge pixels and different filtering schemes are applied. Stefan et al. [34] presented a fuzzy two-step color filter for the reduction of impulse noise. This filter utilizes the fuzzy gradient values and fuzzy reasoning for the detection of noisy pixels. Mansoor et al. [37] developed a recursive filter images which are highly corrupted by impulse noise. This filter estimates the noise level which is required to obtain the filter parameters. All these filters were specifically designed to reduce impulse noise. The performance of the median filter in removing Gaussian noise is inadequate. This drawback is overcome with some success by employing another nonlinear filter technique which makes use of moving average filters (MAV). The concept behind a standard moving average filter is to replace its central pixel by the average value of its predefined neighborhood. One of the major issues in removing Gaussian noise is to differentiate between noise and edges. Various attempts have been made in the past to solve this problem. Fuzzy derivatives are used for this task in [6]. The GOA filter [2] was designed for reducing Gaussian type noise by estimating a fuzzy gradient in each direction so as to distinguish the local variation due to noise from that of an image structure. Stefan et al. [35] consider the fuzzy distance between color pairs as a weight to perform the weighted average filtering for the removal of the Gaussian noise in color images. Russo [12] proposes a method for Gaussian noise filtering that combines a nonlinear algorithm for detail preserving and smoothing of noisy data, and a technique for automatic parameter tuning based on noise estimation. A new filtering architecture adopting multi parameter piecewise linear (PWL) functions is devised in [13] for the restoration of the images corrupted by the Gaussian noise [13]. Xiaofen and Qigang [42] made use of perceptual classification rules to separate noise from other relevant features of image. Shin et al. [7] designed a block based noise estimation algorithm using the Gaussian filter. Coefficients of the Gaussian filter are selected as a function of standard deviation of Gaussian noise in the input image. Bilateral filtering smoothes the images while preserving edges by means of a nonlinear combination of nearby pixel values [4]. Choi and Krishnapuram [43] designed a three fuzzy filter system for dealing with Gaussian and Impulse

noises. A particular system is selected based on the compatibility function at the pixel of interest. This compatibility function is clustered into low, medium and high membership functions and these are multiplied with the three filter outputs of [43] to get the combined output for dealing with any noise in Hanmandlu et al. [28]. Using the concepts of both [43] and [28], three sigma and Pi filters are developed in [19]. In yet another interesting work [20], fuzzy smoothing of images for Gaussian as well as Impulse noise is achieved by combining the output of several filters termed as hybrid filters. Most of the methods presented in literature deal with gray scale images. It is possible to extend these techniques to color images component wise i.e. each component R, G, and B can be passed through a filter separately. Barring a few, most papers on noise removal have dealt with the individual component of RGB color space separately. In the case of Gaussian noise, where differentiating between noise and edges (i.e. finer details of image) is difficult, dealing with each component introduces artifacts in the de-noised image. Only a few research studies have actually explored the interaction between the color components. For instance Shulte et al. [35] examined the fuzzy distances between the color components to sense the possible interaction to determine the correction term using these distances. The outcome of an interaction between the color values of each component during filtering should not affect the color composition of the image. The second sub filter presented in [35] uses the average of R, G, B component differences between the central pixel and its neighboring pixels, which disturbs the original color of the central pixel. In the present work, we attempt to remove Gaussian and Impulse noises in color images. We will present two fuzzy filters for this objective. For the Gaussian noise filter, adaptive fuzzy distances between color pairs (e.g. red-green, red-blue and green-blue) are computed to determine the weights of neighboring pixels for calculating the corrected value of the central pixel. For removing Impulse noise, two sub filters are used in cascade. An algorithm is developed to find the median value of noise free pixels in the neighborhood of the pixel of interest. The degree of noise in a pixel is computed using fuzzy rules and correction is made using the median of the noise free pixels. Organization of the paper is as follows: Section 2 illustrates the design of the impulse fuzzy filter by introducing the relevant concepts. Section 3 presents the Gaussian fuzzy filter. Results are detailed in Section 4 followed by a discussion on mixed noise in Section 5. Finally, conclusions are drawn in Section 6. 2. Fuzzy filter for Impulse noise A color image can be represented via several color models such as RGB, CMY, CMYK, HSI, HSV and CIE L a* b*. The most well known of these is the RGB model which is based on cartesian coordinate system. Images presented in the RGB color model consists of three component images, one for each primary color (Red, Green and Blue). Consider a color image represented in the x-y plane, then the third coordinate z =1,2,3 will represent the color component of


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the image pixel at ( , )x y . Let f be the image function then ( , ,1)f x y will represnt the Red component of pixel at ( , )x y .

Similarly, ( , , 2)f x y and ( , ,3)f x y represent the Green and Blue components respectivily. This notation is followed throught out this work Impulse noise in color images Images corrupted with impulse noise contain pixels affected by some probability. This implies that some of the pixels may not have a trace of any noise at all. Moreover, a pixel can have either all or one or two of its components corrupted with impulse noise. Mathematical modeling of impulse noise in color images is as follows:

( , , )( , , )

( , , ) (1 )z

z

N x y z with probability pf x y z

I x y z with probability p⎧

= ⎨ −⎩ (1)

Where, z =1,2,3 represents red, green & blue components. The probabilities zp ’s can have equal or unequal values. In Equation (1), f represents the final corrupted image, while N and I are the numbers of corrupted and uncorrupted pixels respectively. 2.1 Algorithm for Median of noise-free pixels An algorithm to determine the median of noise-free pixels in the neighborhood of a pixel under interest is now presented. The median of the noise free pixels is utilized to modify the pixel corrupted with impulse noise. This median is computed separately for each color component in the following steps: Step1: Take a window of size w w× centered on

the pixel of interest in the corrupted image. Step2: Arrange all the pixels of the window as a vector.

Sort the vector in an increasing order and compute the median of the sorted vector.

Step3: Calculate the difference between each window pixel and the median of the vector.

Step4: Arrange all the window pixels having the differences less than or equal to a parameter

1δ in a vector. Step5: Sort the new vector and obtain the median med of

the sorted vector. The procedure for the computation of the median med is

illustrated below.

1p 2p 3p

4p 5p 6p

7p 8p 9p The current pixel ( 5p ) with its neighborhood pixels ( 1p , 2p , 3p , 4p , 6p , 7p , 8p , and 9p ).

Sorted vector (increasing order) of window elements shown

above Calculate the median ( M ) of the above vector Calculate the difference between M and each pixel value of

window, herei id M p= − , 1, 2,3,4,5,6,7,8,9.i =

1d 2d 3d

4d 5d 6d

7d 8d 9d Arrange all pixels of window that have

1id δ≤ in a new vector and calculate the median ( med ) of it.

Figure 1: A scheme for the computation of Median of noise-free

pixel

The above median ( med ) is used to find the correction term for each pixel in the noisy image. 2.2. Structure of Impulse Filter The proposed filter is designed for the reduction of impulse noise in color images by treating each color component separately. Interactions among these color components are used to determine the similarity of the central pixel vis-à-vis the neighboring pixels. The nature of impulse noise is random in the sense that it corrupts some pixels while leaving others untouched. So our objective is to identify the noisy pixels along with the amount of noise present. It may be noted that the impulse noise bears similarity with the high frequency content of images like edges and fine details because both reflect sudden changes in pixel values. Three different membership functions, viz., Large, Unlike and Extreme are used to differentiate the noisy pixels from the high frequency contents. The proposed impulse filter consists of two sub filters in cascade. A. The First Sub filter The primary task of the first sub filter is to recognize the noisy pixel along with the amount of noise present, and modify the corrupted pixel value with the median ( med ) of the noise-free pixels present in the neighborhood. The three above mentioned membership functions are framed subsequently to identify the noisy pixels. The difference between the central pixel and the median of the noise-free pixels in the neighborhood of a window is denoted as ( , ,1) ( , ,1) ( , ,1)mD x y f x y med x y= − (2)

4p 2p 1p 7p 9p 3p 8p 5p 6p

Noisy Image


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The difference equations for the other two color components are obtained by replacing 1 in Equation (2) by 2 and 3 respectively. This representation is used through out the paper. We now devise a membership function, lμ to represent a fuzzy set “Large” that indicates how large the difference is. A pixel with higher noise will have a larger difference with the median value. This is defined by the membership function, lμ as

2

11 2

2 1

1

1, ( , ,1)( , ,1)( ( , ,1)) , ( , ,1)

0, ( , ,1)

m

ml m m

m

D x yD x yD x y D x y

D x y

ααμ α α

α αα

⎧ ≥⎪

−⎪= ≤ <⎨ −⎪⎪ <⎩

(3)

The parameters 1α and 2α in Equation (3) are obtained from experimentation. The membership function represented by Equation (3) is depicted in Figure 2.

Figure 2: Membership Function for “Large” Degree of similarity of a pixel with neighborhood pixels The degree of similarity of a pixel with respect to its neighborhood pixels gives a clue to its nature, viz., whether it is noisy or not. To decide whether a pixel is similar to a neighborhood pixel, a similarity criterion is devised. For the red component The differences between red and green component and that between red and blue components are computed as follows:

( , ) ( , ,1) ( , , 2)

( , ) ( , ,1) ( , , 3)rg

rb

d x y f x y f x y

d x y f x y f x y

= −

= − (4)

Similarly the differences between red and green components and that between red and blue components of the neighboring pixels at ( , )x i y j+ + are calculated as:

( , ) ( , ,1) ( , , 2)

( , ) ( , ,1) ( , ,3)rg

rb

d x i y j f x i y j f x i y j

d x i y j f x i y j f x i y j

+ + = + + − + +

+ + = + + − + + (5)

The second differences of the above pair-wise differences in (5) are computed from the following equation:

( , ) ( , ) ( , )

( , ) ( , ) ( , )

rg rg rg

rb rb rb

x i y j d x i y j d x y

x i y j d x i y j d x y

Δ + + = + + −

Δ + + = + + − (6)

We also need the differences between the neighboring pixels and the central pixel of the same color component (say, red) in the window given by:

( , ) ( , ,1) ( , ,1)r x i y j f x i y j f x yΔ + + = + + − (7)

Similarity Criterion: The red component of a pixel at ( , )x y is considered similar to that at ( , )x i y j+ + if the differences, ( , )r x i y jΔ + + , ( , )rg x i y jΔ + + and ( , )rb x i y jΔ + + are less than the parameter 2δ , which is experimentally determined. Thus, we need to check not only the red component value but also the interactions with other color components to ascertain the extent of similarity. A second membership function

uμ is devised to measure the degree of similarity of the central pixel to the neighboring pixels. This membership function describes the fuzzy set called Unlike over the discrete universe of discourse N ={0,1,2,3,4,5,6,7,8}. Let, N be the number of similar pixels (excluding the central pixel) in the window of size w w× . The number N is decided based on the differences calculated in Equations (6) and (7) and the similarity criterion. Considering a 3 3× window, the membership function is now defined as:

2

2

0, 4( , ) 0.5, 3

1,

m

u m m

N and DN D N and D

otherwise

δμ δ

≥ <⎧⎪= = <⎨⎪⎩

(8)

Note that

mD in Equation (8) is defined in Equation (2) and the parameter 2δ is the same as used in the similarity criterion. Therefore if a pixel has more than half pixels similar in the window and its value is close to the median, then it can be considered as a noise-free pixel. The membership function for Unlike is shown in figure 3.

Figure 3: Membership function for “Unlike”

The third membership function is characterized as follows. If we arrange pixels of the window in a vector V and sort them in an increasing order, we will obtain two extreme pixel values in the window, viz., minV and

maxV . The closer the value of a pixel is to these extremes, the higher is the possibility of the pixel being noisy. This concept is used in

1α 1α

2α

1

0

2mD δ>

2mD δ<

lμ

mD

uμ

N


32

obtaining Fuzzy set “Extreme”. The membership function for the fuzzy set “Extreme” applicable to each color component is given as:

min minmin

max maxmax

1 ; 0.11 exp(100( 0.1))

1( ) ; 0.11 exp( 100( 0.1))

0 ;

e

V VV

V VV

otherwise

ττ

μ τ ττ

⎧ ≤ ≤ +⎪ + − −⎪⎪

= − ≤ ≤⎨ + − − +⎪⎪⎪⎩

(9)

Where, τ represents the pixel value of each color component. The membership function for the fuzzy set Extreme is shown below.

Figure 4: Membership function for “Extreme” The degree of noise present in a pixel is ascertained from the following fuzzy rule: Fuzzy Rule

IF ( , ,1)mD x y is Large AND ( , ,1)f x y is Unlike neighborhood AND the central pixel is Extreme THEN this pixel is noisy. (10)

In this rule, noise designated by ( , ,1)dN x y is a fuzzy variable. This rule is of the Mamdani fuzzy model type. Note that we are not using any membership function for the fuzzy set formed by the variables. Correction Term for Impulse noise Let the membership functions Large, Unlike and Extreme for the red component be denoted by ,lr urμ μ and

erμ respectively. Then the degree of noise in the red component of a pixel is evaluated as:

( , ,1)min{ ( ( , ,1)), ( , ( , ,1)), ( ( , ,1))}

d

lr m ur m er

N x yD x y N D x y f x yμ μ μ

= (11)

Equation (11) is obtained using the fuzzy rule in (10). Antecedents in the fuzzy rule are combined using the fuzzy operator AND which is implemented as “minimum” operation. The correction term for the red component is computed as:

( , ,1) ( , ,1) ( ( , ,1) ( , ,1))df x y N x y med x y f x yΔ = × − (12)

As ( , ,1)dN x y in Equation (11) gives the degree of noise present in the Red component of the pixel at the location ( , )x y ; it will be zero for noise free pixel and will have some value between 0 and 1 for the noisy pixel. The correction term ( , ,1)f x yΔ will become zero if the pixel is noise free and its value is equal to the difference between the median ( med ) of noise free pixels in the neighborhood and the value of the pixel itself in the case of extremely corrupted pixels. Now the modified pixels arising out of the first sub filter (i.e. the output) are obtained as

1( , , ) ( , , ) ( , , )Ff x y z f x y z f x y z= + Δ (13) The extremely corrupted color components are replaced with the median ( med ) of the noise free color components (of the same color) of the neighborhood while the noise free components are left untouched. Pixels having noise in between are treated according to the amount of noise present in it. It can be observed that the above modified pixels are immediately put to use to correct the subsequent pixels. A scheme for the fist sub filter is shown in figure 5.

figure 5: A scheme for the first sub filter.

Noisy image

Blue componentRed component

Compute med as per section 2.1 for currently processing window.

Compute ( , ,1) ( , ,1) ( , ,1)mD x y f x y med x y= −

and obtain ( ( , ,1))lr mD x yμ using (3).

Calculate no. of pixels similar to central pixels ‘ N ’ using “similarity criteria”.

Compute the degree of similarity of ( , ,1)f x y using (8) i.e. ( , ( ( , ,1)))ur mN D x yμ .

Determine the membership value of pixel in the fuzzy set Extreme using (9) i.e.

( ( , ,1))er f x yμ .

Compute the correction term using (12) and add it to noisy value to get denoised value.

Output of first sub filter i.e. denoiosed image

1Ff .

Green component

eμ

τ0

1

minV maxV


33

(Note: The dotted line represents the steps for the green and blue components) B. The Second Sub Filter The output from the first sub filter serves as the input to the second sub filter. This filter invokes the interactions among the color components to remove the impulse noise. Differences between the color pairs are given as:

1 1

1 1

1 1

( , ) ( , ,1) ( , , 2)

( , ) ( , ,1) ( , , 3)

( , ) ( , , 2) ( , , 3)

rg F F

rb F F

gb F F

d x y f x y f x y

d x y f x y f x y

d x y f x y f x y

= −

= −

= −

(14)

A fuzzy rule is framed to express the degree of noise present in the color component of a pixel as part of this sub filter. For the Red component IF ( , )rgd x y is Large AND ( , )rbd x y is Large THEN the red component of the pixel is noisy. (15) This rule does not hold always. Suppose there is a red color region then the above differences will be large even without any noise but in that case the median of the region is again red. Hence this situation doesn’t affect the performance of our filter. Similar fuzzy rules are coined for other color components. To express the degree to which the differences ( , )rgd x y , ( , )rbd x y and ( , )gbd x y are Large, we fuzzify these adaptive differences to evolve the membership function lμ with two parameters 1β and 2β . Note that

1β and 2β will replace the original 1α and 2α in the function Large so as to have different shapes. Correction terms for this filter are computed in similar lines as in the first sub filter. The degree of noise present in a pixel is: ( , ,1) min{ ( , ), ( , )}d lrg lrbn x y x y x yμ μ= (16) Where,

lrgμ and lrbμ are the membership functions of Large sets of color pairs, red-green and red-blue respectively. The correction term is given by:

1 1( , ,1) ( , ,1) ( ( , ,1) ( , ,1))F d Ff x y n x y med x y f x yΔ = × − (17) Median ( med ) values are calculated again as in the first sub filter whose output is 1Ff . The final output of the impulse filter is a set of modified pixels given by:

2 1 1( , , ) ( , , ) ( , , )F F Ff x y z f x y z f x y z= + Δ (18)

The modified pixels from the second sub filter are immediately employed to correct the subsequent pixels. A scheme for the second sub filter is shown in figure 6. 2.3 Algorithm for Impulse Filter The individual steps of the algorithm for Impulse filter are as follows:

Step 1: Consider one color component, say red component of noisy image. Compute the median ( med ) as per algorithm in 2.1.

Step 2: Compute ( , ,1) ( , ,1) ( , ,1)mD x y f x y med x y= − and obtain ( ( , ,1))lr mD x yμ using (3).

Step 3: Calculate no. of pixels similar to central pixels ‘ N ’ using “similarity criteria”.

Step 4: Compute the degree of similarity of ( , ,1)f x y using (8) i.e. ( , ( ( , ,1)))ur mN D x yμ .

Step 5: Determine the membership value of pixel in the fuzzy set Extreme using (9) i.e. ( ( , ,1))er f x yμ .

Step 6: Calculate the correction term using (12) and add it to original value to obtain denoised value. Repeat the steps for other color components blue and green. Apply the process for the whole image pixel by pixel.

Step 7: For the image obtain in Step 6. Compute the differences: ( , )rgd x y , ( , )rbd x y and ( , )gbd x y as per (14) and fuzzify them using membership function large with parameter 1β and 2β .

Step 8: Use (16) to calculate correction term for red component and for other similar equations are used.

Step 9: Final output of impulse filter is obtained using (18).

Figure 6: A scheme for the second sub filter

A scheme for the Impulse filter is illustrated below.

Figure 7: Block diagram of Impulse Filter

Noisy Image

The First sub filter

Final denoised image

The Second sub filter

Image1Ff , output of first

sub filter.

Compute the differences: ( , )rgd x y , ( , )rbd x y and ( , )gbd x y as per (14) and

fuzzify them using membership function large with parameter

1β and2β .

Use (16) to calculate correction term for red component and for other similar equations are used

Compute the correction term using (12) and add it to noisy value to get denoised value.


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3. Fuzzy Filter for Gaussian Noise Removal As Gaussian noise is additive, a color pixel in RGB color space with co-ordinates (x, y, z) degraded by random noise is expressed as: ( , , ) ( , , ) ( , , )f x y z I x y z x y zη= + (19) Where ( , , )f x y z is the noisy color image, ( , , )I x y z is original color image both defined in RGB color space and

( , , )x y zη represents the signal independent additive random noise in the same color space. The methods for the reduction of Gaussian noise adopt the weighted average of neighborhood pixel values of the central pixel value. The key point here is to select the weights to the neighborhood pixels in such a way as to obtain the corrected value. Most of the methods in the literature use only the same color component of the neighborhood pixels to obtain the average value for the central pixel without considering the effect of other components. While processing RGB components separately we have used an adaptive fuzzy distance between color pairs (red-green, red-blue and green-blue) to compute the weights to the neighborhood pixels. The use of color pairs to assign weights to the neighborhood pixels leads to a reduction in the ensuing artifacts. Adaptive fuzzy distance between the color pairs gives similarity between the central pixel and the neighborhood. This distance helps preserve edges by way of giving less weight to the noisy pixels and more weight to the similar pixels during the computation of the weighted average. Therefore, if the adaptive fuzzy distance is more then less weight is given and vice-versa. 3.1 Adaptive Fuzzy Distances Adaptive fuzzy distance is found between each color pair of the central pixel and that of the neighbourhood pixels. Color pairs are denoted in terms of the image function ‘ f ’ as follows:

Red-Green ( ( , ,1), ( , , 2))f x y f x y Red-Blue ( ( , ,1), ( , ,3))f x y f x y Green-Blue ( ( , , 2), ( , ,3))f x y f x y

Adaptive distance between a color pair of central pixel \and that of neighbourhood pixel, say between red-green pairs is found from:

( , ) ( , ) ( , ) ( , )rg rg r gD x i y j d x i y j T x i y j T x i y j+ + = + + − + + − + + (20)

Where ( , )rgd x i y j+ + is the Euclidean distance given by

2 2

( , )

( ( , ,1) ( , ,1)) ( ( , , 2) ( , , 2)) ,

rgd x i y j

f i j f x y f i j f x y

+ + =

− + −

Note: ( , , )f i j z represents ( , , )f x i y j z+ + . The terms rT and

gT in Equation (2) are expressed as:

21( , ) ,

( , ,1) ( , ,1)r bT x i y j

f x i y j f x ya

+ + =+ + −⎛ ⎞

⎜ ⎟⎝ ⎠

(21)

21( , ) ;

( , , 2) ( , , 2)g bT x i y j

f x i y j f x ya

+ + =+ + −⎛ ⎞

⎜ ⎟⎝ ⎠

(22)

Similarly other fuzzy distances can be obtained. The terms rT and

gT make distances adaptive, for example, if the red components have almost the same value but the green components have a large difference then the Euclidean distance will be large. This large distance will give less weight to the red component of the neighbourhood but it should be large as two have similar values for the red component. Subtracting the term rT from the normal distance takes care of the above problem and the same is true for the term,

gT . The Adaptive Fuzzy distance between the color pairs is obtained by fuzzifying the adaptive distance using the membership function Small to be introduced later. 3.2 The Filter Structure In the proposed method the weighted average of the neighbouring pixels in the window of interest is calculated. The weights to the neighbouring pixels are determined according to the following fuzzy rules.

Fuzzy rule for the Red component IF ( , )rgD x i y j+ + is Small AND ( , )rbD x i y j+ + is small THEN weight ( , ,1)w x i y j+ + is a large value. (23) Fuzzy rule for the Green component IF ( , )rgD x i y j+ + is Small AND ( , )gbD x i y j+ + is Small THEN weight ( , , 2)w x i y j+ + is a large value. (24) Fuzzy rule for Blue component IF ( , )rbD x i y j+ + is Small AND ( , )gbD x i y j+ + is Small THEN weight ( , ,3)w x i y j+ + is a large value. (25) To express the degree to which an adaptive distance is Small, the adaptive distances are fuzzified using the membership function Small, defined as:

exp ,( )

0 ,t

for tt

for t

λ λμ λ

λ

⎧ ⎛ ⎞− ≤⎪ ⎜ ⎟= ⎝ ⎠⎨⎪ >⎩

(26)

This membership function for the set “Small” is shown in Figure 8. Parameter ‘ t ’ is the maximum distance between a color pair of a central pixel and that of the neighborhood in a window. Parameter ‘ t ’ for different color pairs are given as:


35

( , ) max( ( , ))

( , ) max( ( , ))( , ) max( ( , ))

rg rg

rb rb

gb gb

t x y d x i y j

t x y d x i y jt x y d x i y j

= + +

= + +

= + +

(27)

Figure 8: The membership Function for “Small”

The above fuzzy rules are implemented by calculating the adaptive fuzzy distances using the membership function “Small”. For example, fuzzy adaptive distance between the red-green color pairs of a pixel at ( , )x y and a neighboring pixel at ( , )x i y j+ + is represented as:

( , ) ( ( , ))rg x y rgD x i y jμ + + where, rgμ is the membership

function of the red-green color pair. The weights for a neighboring pixel at the location ( , )x i y j+ + corresponding to red, green and blue components are derived from the three fuzzy rules (Equations 23-25) as:

( , ,1) min{ ( ( , )), ( ( , ))}

( , , 2) min{ ( ( , )), ( ( , ))}

( , ,3) min{ ( ( , )), ( ( , ))}

rg rg rb rb

rg rg gb gb

gb gb rb rb

w x i y j D x i y j D x i y j



μ μ

μ μ

μ μ

+ + = + + + +

+ + = + + + +

+ + = + + + +

(28) The weights for red, green and blue components follow similarly. The final corrected value of a pixel at location (x, y) for the red component is given by

( , ,1) ( , ,1)( , ,1)

( , ,1)

k k

i k j kk k

i k j k

w x i y j f x i y jI x y

w x i y j

=− =−

=− =−

+ + × + +=

+ +

∑ ∑

∑ ∑

(29) Similarly we can find ( , , 2)I x i y j+ + for the green component and ( , ,3)I x i y j+ + for the blue component. 3.3 The algorithm for the Gaussian Filter Step1: Take a window of size w w× centered on the pixel

of interest in the noisy image. Step2: Pick a neighborhood pixel within the window and

compute adaptive distances for the three color pairs (red-green, red-blue and green-blue) from the central pixel to this pixel via Equation (20).

Step3: Fuzzify each adaptive distance using the membership function small defined in Equation (26), and parameter ‘t’ using Equation (27).

Step4: Calculate weight for each color component of the neighborhood pixel using Equation (28).

Step 5: Repeat Steps 2 to 4 for all neighborhood pixels in the window.

Step 6: Obtain the final corrected value for the central pixel using Equation (29).

Figure 9: A scheme for the Gaussian filter 4. Results and Comparative Analysis A color image consisting of an 3M N× × array of pixels at locations ( , )x y may be viewed as a “stack” of three gray scale images corresponding to RGB components. The data class of the component images determines their range of values. If an image is of class double, the range of values is [0, 1]. Similarly, the range of values is [0, 255] or [0, 65535] for RGB images of class uint8 or uint16, respectively. The color images “Lena”, “Fish” and “bird” of size 256 256× impregnated with the Gaussian noise and the impulse noise are considered as test images. The original images are shown in Figure 10. For assessing the relative performance, the proposed method is compared with the recent methods in the literature. The mean square error (MSE) is selected as the measure of performance.

( ) ( )3 2

1 1 1, , , ,

( , )3

N M

z x yI x y z f x y z

MSE f IN M

= = =

−⎡ ⎤⎣ ⎦=

× ×

∑∑∑ (30)

Where I is the original color image, f is the noisy image or the filtered image of size N M× . Mean Square Error (MSE) gives a similarity measure between two images.

Noisy Image

For pixel at ( , )x y , compute adaptive distances ( , )rgD x i y j+ + , ( , )rbD x i y j+ + and ( , )gbD x i y j+ + with

neighborhood window pixel at ( , )x i y j+ + using (20).

Parameters ( , )rgt x y , ( , )rbt x y and ( , )gbt x y are calculated using (27).

fuzzify above distances using membership function small with these parameters.

Calculate weight for each color component of the neighborhood pixel using (28).

Compute weight similarly for all other pixels in the window

Final corrected value for pixel is obtained using (29).


36

(a) (b)

(c)

Figure 10. Original colored Images used of size 256 256× (a) Lena (b) Fish, and (c) Bird

Another similarity measure is PSNR (peak signal to noise ratio) which is related to MSE as in the following: 1( , ) 10log

( , )PSNR f I

MSE f I⎛ ⎞

= ⎜ ⎟⎝ ⎠

(31)

In Equation (31), MSE values are used in the normalized form. For example, if the image class is uint 8 then it has to be divided by 255*255. Higher the value of PSNR better is the similarity between two images. 4.1 Results of the Impulse noise filter Three window sizes of 3×3, 5×5 and 7×7 are experimented. A plot between PSNR and percentage of impulse noise for these window sizes is drawn in Figure 11. The best results For higher percentages of the impulse noise, a larger widow seems to be more appropriate but this filter is less suitable for a high level of noise as there is a loss of image details. As the window size of 3×3 produces better results up to 20% impulse noise, this filter is meant to deal with low and middle percentages of the impulse noise. This level of noise is usually found in many practical applications. The performance of this filter is illustrated through a set of color images with the impulse noise of densities 10%, 15and 20%. The optimal values for the parameters 1 2 1 2 1, , , ,δ δ α α β and 2β , discussed in Section 2.2 are experimentally determined to be 0.3, 0.1, 0.078, 0.15, 0.5 and 0.6 respectively. are obtained using a window size of 3×3 for the lower and the middle percentage of the impulse noise.

Figure 11: PSNR vs. Window sizes for Impulse noise

A comparative analysis of the proposed techniques is carried out with respect to two recent approaches in the literature, namely, SMDE method proposed by Pei-Eng Ng et al. [24] and Luo’s EDPA [25]. A sample set of the original images used in the experimentation are displayed in Figure 10. The values of MSE and PSNR enumerated in Table 1 for different experiments indicate that the proposed method is able to reduce more noise from the images while preserving almost all image details. The results are better than those reported in the literature as demonstrated by a higher value of PSNR in most of the images analyses. It can also be observed visually that the proposed filters are quite effective in noise reduction. The results of denoising obtained by a few existing methods in the literature are shown in Figure 12 including the results achieved by the proposed impulse filter for comparative purposes. while input to each filter have the same level of noise (a) Lena Image with Impulse noise of density 15%, (b) Lena with FNRC, (c) Lena with NRFF, (d) Lena with Proposed, (e) Fish Image with Impulse noise of density 15%, (f) Fish with FNRC, (g) Fish with NRFF, (h) Fish with Proposed, (i) Bird Image with Impulse noise of density 15%, (j) Bird with FNRC, (k) Bird with NRFF, and (l) Bird with Proposed method. A plot (See Figure 13) between PSNR and percentage of Impulse noise for different methods proves this point for the Lena image. 4.2 Results of the Gaussian noise filter Experiments are performed using different sizes of windows and the results for these experiments are shown in the form of a plot between PSNR and Gaussian noise level (σ ) in Figure 14. The window sizes considered for the Lena image are: 3×3, 5×5 and 7×7. It can be seen from Figure 14 that the window size of 3×3 is the most suitable one for the noise level up to σ = 30 for which this Gaussian filter is best suited.


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Table 1: Comparison of Performance (Impulse Noise)

Image Noisy SMDE EDPA Proposed MSE PSNR MSE PSNR MSE PSNR MSE PSNR Lena 10% 12.40 37.20 5.18 40.99 1.85 45.46 1.47 46.46 15% 19.05 35.33 5.94 40.39 2.85 43.58 2.24 44.63 20% 25.28 34.10 7.27 39.52 3.89 42.23 3.12 43.19 Fish 10% 12.78 37.06 10.04 38.11 4.23 41.87 4.02 42.09 15% 19.12 35.32 10.85 37.78 6.03 40.33 5.53 40.70 20% 25.61 34.05 11.66 37.46 7.84 39.19 7.04 39.66 Bird 10% 12.83 37.05 9.39 38.40 3.28 42.97 2.71 43.80 15% 19.45 35.24 10.79 37.80 4.99 41.15 4.26 41.84 20% 25.80 34.02 12.41 37.19 7.11 39.61 5.99 40.36

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

Figure 12: Denoised Images obtained with different Filters

Figure 13: A comparative analysis of impulse noise reduction

For higher levels of noise a larger window size seems to work better. But at the same time the larger window sizes have a drawback of over smoothing the edges resulting in the pronounced burring effect.


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The performance of the Gaussian noise filter is evaluated over the three test color images with σ = 10, 20 and 30. The parameters a = 3.5 and b = 5 are found to be effective in the elimination of the noise. The results of the proposed approach are compared in terms of MSE with methods developed by Stefan Schulte et al. (FNRC) [4] and Dimitri Van et al. (NRFF) [1]. Comparison of MSE and PSNR values resulting from the application of the proposed filter and other methods in Table 2 shows the superiority of the proposed filter over the others in the reduction of Gaussian noise. The results of comparison are depicted in Figure15. The results of denoising of the test images are illustrated in Figure 16.

Figure 14: PSNR vs. Window sizes for Gaussian noise

Table 2: Comparison of Performance (Gaussian) Image Noisy FNRC NRFF Proposed

MSE PSNR MSE PSNR MSE PSNR MSE PSNR Lena

σ =10 38.96 33.22 25.22 34.11 21.94 34.72 16.58 35.94 σ =20 74.15 29.42 55.24 30.71 45.08 31.59 40.22 32.09 σ =30 89.79 28.60 75.18 29.37 73.33 29.48 68.02 29.81

Fish σ =10 39.53 32.16 32.10 33.07 26.77 33.85 22.12 34.68 σ =20 75.56 29.35 75.50 29.35 46.02 31.50 40.12 32.50 σ =30 90.71 28.55 86.34 28.77 60.26 30.33 56.87 30.58

Bird σ =10 39.30 32.19 31.91 33.09 21.21 34.86 17.27 35.76 σ =20 74.04 29.44 58.01 30.50 39.61 32.15 38.01 32.33 σ =30 91.01 28.54 73.77 29.45 55.14 30.72 54.21 30.74

Figure 15: A comparative analysis of impulse noise reduction

(a) (b)

(c) (d)

(e) (f)


39

(g) (h)

(i) (j)

(k) (l) Figure 16: Denoised images obtained with different filters while the input to each filter has the same level of noise (a) Lena Image with Gaussian noise σ =20, (b) Lena with FNRC, (c) Lena with NRFF, (d) Lena with the proposed method, (e) Fish Image with Gaussian noise σ =20, (f) Fish with FNRC, (g) Fish with NRFF, (h) Fish with the proposed method, (i) Bird image with the Gaussian noise σ =20, (j) Bird with FNRC, (k) Bird with NRFF, and (l) Bird with the proposed method The undesirable effect of a Gaussian filter is that it blurs the edges. This is because the value of each pixel in the image is replaced by the weighted average of the gray levels in the neighborhood defined by the filter mask (window). This process results in an image with the reduced “sharp” transitions in the gray levels. Hence the reduction of Gaussian noise is accompanied by the loss of finer details. 5. Mixed Noise In the case of mixed noise, two separate filters are applied in cascade. The first is the impulse filter with its output serving as the input to the Gaussian filter. As per our approach, the impulse filter will not be able to replace noisy pixels with the median of the noise free pixels since other pixels are corrupted with the Gaussian noise. In effect, the impulse filter will replace the impulse noise with the Gaussian noise to be dealt with later by the Gaussian filter. It is observed

that in the overall filtering process, blurring becomes pronounced as compared to the removal of separate noises. Results for mixed noise are shown in Figure 17. Our future effort will be directed towards reducing this blurring.To assess the performance of the cascaded filter in the presence of two noises that we are dealing with, the MSE obtained for the cases of Lena and Fish are given in Table 3. The denoising operation results in the reduction of MSE of the noisy images substantially as can be seen in the table.

Table 3: MSE values for mixed noise Impulse noise density

Gaussian noise σ

MSE value for noisy image

MSE value for denoised image

Lena image 10% 10 49.05 20.28 10% 20 81.36 32.49 15% 10 53.17 19.61 Fish image 10% 10 49.07 33.56 10% 20 82.42 43.16 15% 10 53.76 33.86

(a) (b)

(c) (d)

(e) (f)


40

(g) (h)

(g) (j)

(k) (l)

Figure 17: Noisy and deniosed images are shown side by side (a) Lena Image with Impulse noise of density 10% and Gaussian noise σ =10, (b) denoised image of image (a), (c) Lena image with Impulse noise of density 10% and Gaussian noise σ =20, (d) denoised image of image (c), (e) Lena image with Impulse noise of density 15% and Gaussian noise σ =10, (f) denoised image of image (e), (g) Fish with Impulse noise of density 10% and Gaussian noise σ =10, (h) denoised image of image (g), (i) fish image with Impulse noise of density 10% and Gaussian noiseσ =20, (j) denoised image of image (i), (k) Fish image with Impulse noise of density 15% and Gaussian noise σ =10, and (l) denoised image of image (k). 6. Conclusions Two fuzzy filters: one for reducing impulse noise and the other for Gaussian noise are presented. The impulse noise is detected using three fuzzy membership functions: Large, Unlike and Extreme which represent the varied degrees of noise present in a pixel. A proper detection and correction of the noisy pixels helps in preserving the details of the image. A new algorithm is developed for finding the noise free pixels in the neighborhood of a noisy pixel. The noisy pixel is corrected using the median of the noise free pixel. The impulse filter is suitable for lower and middle ranges of impulse noise percentage up to 20% as this range is of practical significance.

The removal of Gaussian noise is accomplished via fuzzy adaptive distances between the color components of a pixel of interest and the neighborhood pixel and represented by the membership function Small. The interactions between the color components are accounted for by taking pairs of color components instead of individual color components. The Adoptive fuzzy distance between color pairs approach produces a denoised image with all the significant details preserved. It was shown that this filer is capable of reducing Gaussian noise up to 30σ = . Fuzzy rules are constructed for both fuzzy filters to ascertain the extent of noise. The filters are applied separately to remove the respective noise type. The proposed filters are compared with two existing filters and the performance evaluated in terms of MSE and PSNR on several test images. The proposed filters justify their superiority in suppressing both types of noise on the most complex of images. For the removal of mixed noise, the impulse filter is applied first followed by the Gaussian noise filter. It is noticed that the filtered image is somewhat blurred and the blurring is tolerable up to the impulse noise density of 15% and Gaussian noise generated with 10σ = . Future work will seek to reduce the blurring exceeding this level by way of designing a deblurring stage to the cascaded filter. The existing deblurring methods are not applicable as they are only designed for globally degraded images whereas we are concerned with locally degraded images caused by smoothening of the Gaussian noise. The contributions of the paper merit a mention to be cognizant of the significance and relevance of the work for the noise removal. Firstly the design of the impulse filter differs from the existing methodology in terms of the usage of the fuzzy rules. Secondly, the use of the adaptive fuzzy distance between color components in the RGB space is new. Thirdly, the treatment of the mixed noise by way of a cascaded filter made up of the impulse and Gaussian filters is done comprehensively. References [1] A. Taguchi, H. Takashima, and Y. Murata, “A Fuzzy Filter for

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Om Prakash Verma received his B.E. degree in Electronics and Communication Engineering from Malaviya National Institute of Technology, Jaipur, India in1991 and M. Tech. degree in Communication and Radar Engineering from Indian Institute of Technology (IIT), Delhi, India, in 1996. He is shortly submitting PhD

thesis in the area of image processing using fuzzy technique From 1992 to 1998 he was a Lecturer in Department of Electronics & Communication Engineering, at Malaviya National Institute of Technology, Jaipur, India. He joined Department of Electronics & Communication Engineering, Delhi College of Engineering, Delhi, India, as Assistant Professor in 1998. Currently, he is Head of Department of Information Technology at Delhi College of Engineering, Delhi since November 2007. He has authored a book on Digital Signal Processing in 2003. His research interest includes Computer Vision and Image Processing, Application of Soft Computing techniques in Image Processing, Artificial Intelligent, Optimization techniques, Digital Signal Processing etc. He has published number of research papers in International Journal and conferences. He has guided 15 M. Tach. Students He is Principal investigator of “Information Security Education Awareness” Project at Delhi College of Engineering Delhi. This project is sponsored by Department of Information Technology, Ministry of MHRD, Govt. of India.

Madasu Hanmandlu (M’02) received the B.E. degree in Electrical Engineering from Osmania University, Hyderabad, India, in 1973, the M.Tech. degree in power systemsf rom R.E.C. Warangal, Jawaharlal Nehru Technological University (JNTU), India, in 1976, and the Ph.D. degree in control systems from Indian Institute of Technology, Delhi, India, in

1981. From 1980 to 1982, he was a Senior Scientific Officer in Applied Systems Research Program (ASRP) of the Department of Electrical Engineering, IIT Delhi. He joined the EE department as a lecturer in 1982 and became Assistant Professor in 1990, an Associate Professor in 1995 and finally a Professor in 1997. He was with Machine Vision Group, City University, London, from April –November, 1988, and Robotics Research Group, Oxford University, Oxford from March-June, 1993, as part of the Indo-UK research collaboration. He was a Visiting Professor with the Faculty of Engineering (FOE), Multimedia University, Malaysia from March 2001 to March 2003. He worked in the areas of Power Systems, Control, Robotics and Computer Vision, before shifting to fuzzy theory. His current research interests mainly include Fuzzy Modeling for Dynamic Systems and applications of Fuzzy logic to Image

Processing, Document Processing, Medical Imaging, Multimodal Biometrics, Surveillance and Intelligent Control. He has authored a book on Computer Graphics in 2005 under PBP publications and also has well over 185 publications in both conferences and journals to his credit. He has guided 15 Ph.Ds and 90 M.Tech students. He has handled several sponsored projects. He is presently an Associate Editor of both Pattern Recognition Journal and IEEE Transactions on Fuzzy Systems and a reviewer to other journals such as Pattern Recognition Letters, IEEE Transactions on Image Processing and Systems, Man and Cybernetics. He is a Senior member of IEEE and is listed in Reference Asia; Asia’s who’s who of Men and Women of achievement; 5000 Personalities of the World (1998), American Biographical Instituteoo

Anil Singh Parihar received his B.Tech. degree in Electronics and Communication Engineering from Priyadarshini College of Computer Sciences, Noida, India, in 2005 and the M.E. degree in Electronics and Communication Engineering from Delhi College of Engineering, New Delhi, India, in 2008. Presently he is working on multimodal biometric at IIT Delhi,

New Delhi. India

VAMSI KRISHNA MADASU obtained Bachelor of Technology degree in Electronics & Communication Engineering with distinction from Jawaharlal Nehru Technological University, India in 2002 and PhD in Electrical Engineering from the University of Queensland, Australia in 2006. From 2006-2008, he was a

Research Associate in the School of Engineering Systems at Queensland University of Technology where he developed innovative Image Processing and Fuzzy Logic based technologies for diverse industrial applications. Currently, he is a Senior Research Officer at TetraQ, University of Queensland, Australia working in the field of Medical Image Analysis. Vamsi is a member of IEEE, Computer Society and is listed in Who’s Who in the World.


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