High Density Impulse Noise Removal By Fuzzy Mean Linear Aliasing Window Kernel

Fitri Utaminingrum, Keiichi Uchimura, Gou Koutaki Human and Environmental Informatics Department

Computer Science and Electrical Engineering, Kumamoto University Kumamoto, Japan

f3-ningrum@navi.cs.kumamoto-u.ac.jp

Abstract—Fuzzy Mean Linear Aliasing Window Kernel

(FMLAWK) filter method proposed to reducing the high-density impulse noise interference and generating the smooth image performance. FMLAWK filter is a spatial filter, which combined from fuzzy method and Linear Aliasing Filter (LAF). The initial step is finding the degree of membership function (μ) value of each matrix element on the corrupted image which use the fuzzy method. Furthermore, the μ value of the corrupted image processed by LAF method which using 3x3 window. The reducing of 3x3 windows on LAF process will be obtain one pixel data based on Linear method. Our research also provides kernel algorithms. Preprocessing Kernel algorithm used for checking of each element matrix on the 3x3 window. If the matrix element contaminated by impulse noise, so the matrix element replaced with a new element data. Our simulation result shows the image filtering better and smoother quality than the comparison method.

Keywords-- impulse noise removal; linear aliasing filter; fuzzy method

I. INTRODUCTION Noise defined as an impairment, which converts the

original information signal. In addition, the image noise is the result of an error in image acquisition or transmission process that produces the pixel value does not reflect the actual intensity of the real pixel of digital data.

Important to know the characteristics from impulse noise to be used for evaluating and designing the new filter methods. Impulse noise (or salt-and-pepper noise) is caused by a sharp disorder in the image signal which is the appearance of randomly dispersed white or black spots (or both), that superimposed on the top of the image [1]. Impulse Noise happens during imperfect image acquisition, due to switching, sensor temperature, during image transmission.

In our research paper propose a new design method for reduce high-density impulse noise interference in the image. We compare the quality of Fuzzy Mean Linear Aliasing Window Kernel (FMLAWK) with another method, which are Median Filter, Median Relaxed (MR), Progressive Switching Median Filter (PSMF) and Decision Based Algorithm (DBA).

In generally to remove impulse noise, researchers often use the median method. Median Filter is a non-linear filter, which

particularly effective for reducing impulse noise. Median Filter has strong ability to refine noise. The filtering method also adopted from Median Filter, that conducted by Bogdan Smolka [2]. His research based on the evaluation of the statistical properties of the sort accumulation distance, which used for the calculation of the vector median. Median filter based on the vector concept has been conducted by Gaihua Wang [3], His research used two-phase noise detector, the adaptive vector median filter detection method to identify pixels that are likely to have been corrupted by noise and using for one-dimensional Laplacian operators, which allow edge pixels to be preserved. Vector Median (VM) implemented by Rastislav Lukac [4]. He used the trimmed aggregated distance minimization concept and robust vector order statistics to enhance edges and image details while retaining the noise removal characteristics of the standard VM operator.

Median Relaxed to reduce impulse noise proposed by Abdessamad ben hamza [5], the filter obtained by relaxing the order statistic for pixel substitution. Noise attenuation properties as well as edge and line preservation analyzed statistically. The output distribution of the Relaxed Median Filter is given in a more generalized form [6]. Performance of the Relaxed Median Filter should be described by using some statistics of the output. The Relaxed Median Filters show has better quality than the standard Median Filter on the impulse noise removal. However, in the Median Relaxed filtering result still has any spots of impulse noise, it is mean that not good enough to reduce impulse noise. After that, the merger between the Relaxed Median Filter method and the Mean Filter theory has been developed by A. Ben Hamza et al [7]. His research has shown that mean-relaxed median filter preserve more image details compared to other nonlinear filtering techniques. The Mean-Relaxed Median Filter yields better detail preservation at a cost of a reduction in its noise reduction performance [8]. However, the filtering result performance of this method is not optimal in high impulse noise content.

Progressive Switching Median Filter (PSMF) also used for comparison on our research. Research on this theme has been done by Zhou Wang et al. The algorithm has been developed by the two main points. The first is switching scheme an

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impulse detection algorithm used before filtering process, thus only a proportion of all the pixels will be filtered and the second progressive method applied through several iterations [9]. The combination of Progressive Switching Median Filter (PSMF) and Iterative Selective methods has been developed by Abdullah.A, et al [10]. His research is also to reduce impulse noise.

The last comparison method is Decision Based Algorithm (DBA) presented by Nair, M.S et al [11]. His method processed the neighboring pixel values to get better image quality.

The advantages of our new method divided into:

• Our method has maximal filtering to reduce the highly corrupted impulse noise content.

• Our method can removes only on the noisy pixel, without losing important information on digital image.

• Our simulation filtering result is more refined than the comparison method.

Peak Signal to Noise Ratio (PSNR) used as the quantitative measurement parameter on quality of the filtering process result. That measurement as an objective standard evaluation which measures the quality of the digital image refers to the original image.

This paper is organized as follows. Introduction, in the section 1 has provided a brief description of our method and also the comparison method has been proposed by previous researchers. Section 2 describes about the characteristic of impulse noise. Section 3, the proposed method is present. Meanwhile in this section divided into two subsections. The first is Flowchart Fuzzy Mean Linear Aliasing Window Kernel Filter (FMLAWK) and The second is the stages of Linear Aliasing Window Kernel. Moreover, that subsection is also consists of the three important points. The first is the Membership Function. The second is Window and Kernel Algorithms and the last is Linear Aliasing Filter (LAF). Section 4 presents the result performance of FMLAWK Filter is to assess the proposed filter, which compare with other filters. In this section divided into three subsections. Which, among other are Peak Signal to Noise Ratio (PSNR), the computation time result and the visual analysis of simulation result. Finally conclusions are drawn in Section 5.

II. IMPULSE NOISE Impulse noise sometimes called salt-and-pepper noise,

spike noise or binary noise, which causes wrong gray values for a few random distribution pixels [12]. Impulse noise is always independent, which un-correlate to the random image pixels. The impulse noise ratio is the ratio between the number of pixels corrupted by the impulse noise and the total number of pixels in the image [13].

The salt-and-pepper noise is disturbance information of pixels, it has dark pixels in bright regions and bright pixels in dark regions. In fact form, which emerging as black and white spots spread on image with random location, which has extremely large or small pixel values. Meanwhile, the salt

digital data value for gray level is 225 and the pepper for gray level is 0.

Wherein p denotes as the density of total interference noise and then p/2 represented as noise density. The mathematical formulation can be written by eq. (1). In the case of salt and pepper noise the image pixels are randomly corrupted by either 0 or 255 [13].

⎩⎨⎧

=-pfor x

for pny

ij

ijij 1

(1)

Denotes yij is noisy image, p is the total noise density and xij is the un-corruption image pixel. While, nij is the gray level value of noisy.

III. PROPOSED METHOD

A. Flowchart Fuzzy Mean Linear Aliasing Window Kernel Filter Figure 1 shows the working system of the Fuzzy Mean

Linear Aliasing Window Kernel Filter (FMLAWK). The input of our system is the image corrupted by impulse noise. The system will process the corrupted image, pixel by pixel. Next on the block system "Impulse Noise Detectors" will analyze and predict whether a pixel in the center area of the window, it containing noise or not. If the pixel does not contain of noise so is not needed to be done the filtering process. However vice versa, if the pixel indicates noise then the system will automatically conduct the filtering process using FMLAWK. Because of, the type of noise is salt-and-pepper. It has dark pixels in bright regions and bright pixels in dark regions in order words as black and white spots spread on image. So, if the pixel has the maximum value (255) or minimum value (0), it means the pixel indicated as salt- and-pepper noise.

FMLAWK Filter using 3x3 window size, and more details will be explained in the next section. Filtering system will process noisy image pixel by pixel. That will be continued until the last pixel of the image fn(N, M). While N and M respectively are maximum value of row and column in the corrupted image.

The flowchart system is "closed loop". The filtering result used feedback to the input. In this case, fn (x, y) is equal to fo'(x, y), which fo' (x, y) is pixel filtering results at the same time. The end of the process is fo (x, y) could be written mathematical formulation as eq. (2):

⎩⎨⎧ =

= other y)fn(x,

0 or 1 y)fn(x, if FMLAWK yxfo ),( (2)

B. The stages of Linear Aliasing Window Kernel 1) The Membership Function (μ). The first step of

FMLAWK method is finding the Membership function (μ) value for all pixels in the corrupted image (fn(x, y)). On the linear membership function (MF), xmin is the minimum value of the pixel data, it is equal to 0. Moreover, xmax is the maximum value of the pixel data, it is equal to 255. We can find the value of μ(x) using the following eq. (3) :

712

⎪⎪⎩

⎪⎪⎨

⎧

≥

<<−

−≤

=

xx 1

xxx xx

xx x x 0

μ(x)

max

maxminminmax

minmin

(3)

Membership function in eq. (3) obtained from the μ matrix on corrupted image fn(x, y). The highest degree membership of each μfnxy matrix element shown in eq. (4).

NMNNN

MM

M

fnxy

μμμμ

μμμμμμ

μμ

μμμμ

μ

..:..

..

..::::..

....:..

::

..

......

321

33332312

1

23

13

2221

1211

= (4)

Figure 1. Flowchart Fuzzy Mean Linear Aliasing Window Kernel

2) Window and Kernel Algorithm. Window Kernel in our research used 3x3 window sizes as in eq. (5)

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

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

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

+++−+

+−

+−−−−=

yxyxyx

yxyxyx

yxyxyx

fnxyw

μμμμμμ

μμμ

μ (5)

w3μfnxy is a 3x3 window of μ value from the corrupted image fn(x,y). While, maximum value of μ is 1 and minimum value of μ is 0.

In our research made subroutine of the program to modify the window w3μfnxy , we call "Kernel". Furthermore, by using Kernel, the value of each w3μfnxy window elements will be checked. If the matrix element has a maximum or minimum value, it considered to be noise. After that, the Kernel added in the every matrix element of the window, except for the highest and lowest values. Furthermore, the summing total of it, that taken from the average value. The average value used to replace the maximum and minimum value of matrix elements. Illustration of the windows kernel presented as follows:

maxa=max(max(wm)); mina=min(min(wm)); for x=1:size(wm,1) for y=1:size(wm,2) if a(x,y)==maxa w(x,y) = 1; elseif a(x,y)==mina w(x,y) =0; else n=n+1; w(x,y)=a(x,y); b = a(x,y)+b; end end end c = max(0,(b/n)); for x=1:size(a,1) for y=1:size(a,2) if w(x,y) == 1 ||w(x,y) ==0 ; w(x,y) = c; else w(x,y) = w(x,y); end end end

3) Linear Aliasing Filter. We use the Linear Aliasing Filter (LAF) to obtain one data pixel. LAF adopted from Aliasing Filter method [14]. Distinguishes between LAF and Aliasing Filter method is the LAF process taking the linear value based on the mean value of 3x3 window size that has been updated (w3μfnxy). Meanwhile, The Aliasing Filter method on each process only uses the average value directly from corrupted image.

The first process of the LAF method is finding the mean

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value of window w3μfnxy, it symbolized by μn. The μm obtained from eq. (6). Linearity aliasing process used in our research. The aliasing filter process is taking four coordinate points on the 3x3 window size in each process. On 3x3 window aliasing process will be resulted four digital data on 2x2 window size. This process continues until produced one digital data. Steps of aliasing process for obtaining one digital data shown in Fig. 2.

)( 3

xyfnmw wmean μμ = (6)

The symbol "L" is a linear relationship. For example, 1AμL is a searchable data using linear calculation of the four

points on the window 1Aμ as shown in eq.(7) and illustrated in Fig. 2.

If likened to the matrix of the window 1Aμ is:

2221

12111

aa

aaμμμμ

μ =A (7)

1AμL value is the linear value of each element 1Aμ matrix, which gotten from eqs.(8) until (11):

L1= μa11 + ½ (μmw-μa11) (8)

L2= μa12 + ½ (μmw-μa12) (9)

L3= μa21 + ½ (μmw-μa21) (10)

L4= μa22 + ½ (μmw-μa22) (11)

The linearity process should be continued. The first, finding L12 is a linear value between L1 and L2. The second L34 is linearity value between L3 and L4. The last linear stage, we seek 1AμL linear value between L12 and L34. We illustrated this process as in Fig. 3.

Based on the Fig. 3 could be calculated L12 and L34 values consecutive using eqs.(12) and (13) :

L12 = L1 + ½ (L2 – L1) (12)

L34 = L3 + ½ (L4 – L3) (13)

To seek the final linear value 1AμL between L12 and L 34 is presented eq. (14)

1AμL = L12 + ½ (L34 - L12 ) (14)

By performing the same steps as described in the previous explanation, then we can also seek the value of 2AμL , 3AμL

and 4AμL . Which is the matrix element of 21Aμ . The same

process steps also use in calculating the final value of 21Aμ by

using eqs.(8) to eq.(14).

Figure 2. The Step of Linear Aliasing Filter (LAF) process

Figure 3. Linear process to get L12 and L34 and 1AμL

2

1211

AAA

μμμμ LLL −+= (15)

2

3432

AAA

μμμμ LLL −+= (16)

)()2

( 121

21 fnMaxAFMLWK

μμμμ −+== L (17)

The final FMLAWK obtained from eq.(17). By previously calculating on the μ1 value which presented on eq.(15) and μ2 value presented on eq.(16). μ1 is a linear value between

1AμL and 2AμL .Whereas, μ2 is a linear value between 3AμL and 4AμL . While, Max(fn) is Maximum pixel data on corrupted image (fn).

IV. THE RESULT PERFORMANCE OF FUZZY MEAN LINEAR ALIASING WINDOW KERNEL FILTER

A. Peak Signal to Noise Ratio Analysis Peak Signal to Noise Ratio (PSNR) is generally which

used to analyze quality of filtering image. PSNR calculation of two images, one original and an altered image aims to measure how far two images are equal. PSNR as illustrated on eq.(18).

∑∑= =

−

×= M

y

N

x

yxfyxf

NMLogPSNR

1 1

2

2

10

)),('),((

25510 (18)

Wherein, M is rows and N is columns of the image matrix. The f(x,y) is an original image, and f’(x,y) is the image filtering result. The bigger PSNR value, the better image quality and vice versa.

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Figure 4. PSNR value result on Lena gray image

Figure 5. Computation time results on Lena gray image

(512x512) in different methods

Briefly, the simulation process of our experiment for the Lena image (512x512) with the filtering process using Fuzzy Mean Linear Aliasing Window Kernel method basically has significant good filtering results. This proven in the FMLAWK method has the highest PSNR value than the four comparison methods. PSNR values are graphically illustrated in Fig. 4

B. The Computation Time Result The quality filtering results of FMLAWK method have

superior performance in many impulse noise density contains compared to the comparison method. Our research has been tested on Lena Image 8 bit Gray color (512x512).

The result of computation times in the different method is shown in Fig. 5. The contamination by impulse noise density (p ≤ 20%), our method have a shorter process time compared to Decision-Base Algorithm (DBA). The computation time process on impulse noise density (p ≤ 20%) started with the short time process go to the long time process successively are Median Filter, Median Relaxed (MR), Progressive Switching Median Filter (PSMF) and Fuzzy Mean Linear Aliasing Window Kernel (FMLAWK), Decision-Base Algorithm (DBA).

Meanwhile, the large impulse noise density (p ≥ 30%), the computation time in our method has weaknesses. This is due to the computation simulation process on our method needed relatively long times which compared to other comparable methods. The long time computation process caused our method only focuses on the process in the pixel containing of noise. So that automatically needed long time process for reduction in the large impulse noise density.

(a) p = 30 % (b) p = 60 % (c) p = 90 %

(d) Median (e) Median (f) Median For p=30% for p = 60% for p =90%

(g) MR (h) MR (i) MR For p=30% for p = 60% for p =90%

(j) PSMF (k) PSMF (l) PSMF for p=30% for p = 60% for p =90%

(m) DBA (n) DBA (o) DBA for p=30% for p = 60% for p =90%

(p) FMLAWK (q) FMLAWK (r) FMLAWK for p=30% for p = 60% for p = 90%

Figure 6. Filtering processes Lena gray image color in the various methods.

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The computation time process on impulse noise density (p ≥ 30%) started with the short time process go to the long time process successively are Median Filter, MR, PSMF and DBA.

C. The Visual Analusis of Simulation Result On the FMLAWK method, we have been testing on

different impulse noise density (p) from 10% up to 90%. This has been done on Lena image 8-bit. In Fig. 6, the simulation results of our filtering process used three sample impulse noise densities. There are 30%, 60% and 90%.

Our FMLAWK methods compared with the Median Filter, MR, PSMF and DBA. Median method, MR Method and PSMF simulation result of small impulse noise density (30%) still has blotches impulse noise remaining in the image. However, three methods have a poor outcome in a large impulse noise density of 60% and 90%. The results filtering for 90% impulse noise degraded image, showing the FMLAWK and DBA method has better performance compared to Median Filter, MR and also PSMF. However, if the quality of the filtering DBA compared to FMLAWK, the quality of the filtering results using DBA is lower than FMLAWK. Meanwhile, the comparative method DBA is optimal for reducing impulse noise density less than 50%.

Visualization resulting of DBA method on the impulse noise density 90% has quality result that is very rough. The FMLAWK filtering result has the quality of filtering much better and smoother than the four comparison methods.

V. CONCLUSION Fuzzy Mean Linear Aliasing Window Kernel (FMLAWK)

Filter could be helpful in many image processing applications to reducing impulse noise. By combining Kernel Algorithm and Linear Aliasing Filter (LAF) methods have been able to reduce the impulse noise with percentage impulse noise densities up to the highest. The FMLAWK has been tested on corrupted image by impulse noise from the noise density of 10% up to 90%.

Performance simulation results are presented in two forms. The first is presents the qualitative results of filtering by visual appearance. The second is quantitatively through the calculation of Peak Signal to Noise Ratio (PSNR). The test results indicate FMLAWK has the best filtering performance compared to the Median Filter, Median Relaxed (MR), Progressive Switching Median Filter (PSMF) and Decision Based Algorithm (DBA).

Future research, we have plan to develop FMLAWK especially focuses on computation time process. We will design new method which used short time simulation process, however still produce a good quality filtering result

ACKNOWLEDGMENT The work of the first author is supported by Directorate of

Higher Education (DGHE) of Indonesia.

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