Improved Median Filtering Algorithm With Average Filtering

Prepared byMukul ChauhanME(Regular 2010) ECE

Submitted to:Dr. Swapna Devi Professor(ECE)

Objective• IEEE Paper• Abstract• Median Filtering• Improved Median Filtering

– Improvement of the filter mask– Improvement of the median algorithm

• Analysis of the algorithm complexity• Simulation Experiments• Conclusions• References

IEEE Paper

Zhu Youlian, Huang Cheng,“An Improved Median Filtering Algorithm Combined with Average Filtering,” Third International Conference on Measuring Technology and Mechatronics Automation, 2011

Abstract

• To solve the contradiction between the noise reducing effect and the time complexity of the standard median filter algorithm, the paper proposed an improved median filter algorithm combined with average filtering. According to the correlation of the image, the algorithm adaptively resizes the filter mask according to noise levels of the mask. According to the sorting results of the selected pixel values in the neighborhood, the algorithm uses the median to replace the original pixel.

Median Filtering:

• The median filtering is a non linear signal processing technology based on statistics.

• Median filter is Used to reduce impulse noise without blurring edges.

• It has neighborhood operation and its noise reducing effect depends on the

size and shape of the filter mask. method to find the median.

• The noisy value of a digital image or a sequence is replaced by the median of the filter mask.

• The pixel in the mask are ranked in the order of their gray value and the median of the mask is stored to replace the noisy value in the mask.

The median filter output is

Where f(x,y) is original image. g(x,y) is the output image. W is a two dimensional mask.it nxn where n is

commonly odd such as 3x3,5x5 etc.The mask shape may be square,circular cross etc.

f1

f2

fN

.

.

.Order

samples

f(1)

f(2)

f(n)

.

.

.select f(m)

median

)()3()2()1( nffff

21

n m

n is odd

},),,({),( Wjijyixfmedyxg

Example of working:• The median is calculated by first sorting all the pixel values from the surrounding

neighborhood into numerical order

replacing the pixel being considered with the middle pixel value

fig.1:The central pixel value of 150 is rather unrepresentative of the surrounding pixels

and is replaced with the median value: 124. A 3×3 square neighborhood is used here, larger neighborhoods will produce more severe smoothing.

• For an image with zero mean noise under normal distribution the noise varience of the median filtering is approximately[1]

• Where is the input noise power(variance)• n is the size of the median filter mask.• is the noise density function. Noise variance for average filtering.

Noise reducing performance of the median filter

)1....(..........21

2)(4

12

2

2

mod

nnnf

i

2

in

)(

nf

)2...(....................2

2

0 ni

• It is clear from the previous two equations that median filter depends on two things size of the mask and the distribution of the noise.

• The median filter performance of random impulse noise is better than the average filter performance.

• The speed of the searching median people proposed some fast algorithms based on the dividing –conquering strategy and simplified the algorithms complexity of the standard median filter from O(n2) to O(nln n) given in [4] and [5].

• Further simplified the algorithm complexity to O{n(1+ln n)} in [6] but in this case the size of the mask was fixed so noise reducing effect may be affected when noise have same approximated same size as the filter mask.

IMPROVED MEDIAN FILTER ALGORITHM

• Author proposed an improved median filter algorithm in which the mask be adaptively resized according to noise density of the mask to improve the noise reducing effect.

• The median filter performance should be improved if the median filter algorithm combined with the average filter algorithm.

• It includes three things: (1)Improvement of the filter mask (2)Improvement of the median algorithm. (3)Analysis of the algorithm complexity

Improvement of filter mask

• The filter mask is mainly a square mask(nxn) or a cross mask. Considering of the symmetry of the mask ,n is commonly odd. The mask size and the noise-reducing effect are a contradiction.

• To solve contradiction ,the self adaptive filter algorithm is introduced.

• In the filter process take the peak signal to noise ratio(PSNR) as the target function and adaptively resize the filter mask.

• Supposing an image f(i,j) of size MxN, its output image is fout (i,j),where i=1,2,3,…,M, j=1,2,3,..N,the mean –square

deviation (MSE) is given by the expression:

PSNR may be defined as

Where amax =2k-1,k denotes the number of a pixel binary bit.

)3.......(....................),(,11 1

2

M N

out jifjifNM

MSE

)4.....(....................))(log(10 2max dBMSEaPSNR

• Adaptively resizing the filter mask:(1)use the median algorithm, filter the original image, get the

output image fn,and calculate PSNRn of fn;

(2)Let n=n+2,use the median algorithm, filter the output image fn of step(1),get new output image fn+1 and calculate PSNRn+1 of fn+1;

(3)If PSNRn>PSNRn+1,then maintain the size of the filter mask; if not, then turn to step (2),and enlarge the filter mask.

Improvement of the median algorithm

• For a natural image ,neighboring pixel has strong correlation. The gray level of each pixel is quit close to neighboring pixel value.

• If a pixel value is greater or less than the value in neighborhood, it signifies that the pixel is contaminated by the noise.

• The noise reducing process consists simply of moving the filter mask from pixel to pixel in an image.

• If the value of a pixel is greater than the average in the mask, then the pixel is contaminated by the noise and replace it with the median of the mask; if not maintain the original value of the pixel unchanged.

• The improved algorithm reduce the calculation time,but also retain the details of the image as far as possible.

• The original value of the pixel is replaced with the median in the mask, and the next calculating process of the average may make full use of the new pixel value.

• Due to above it also improves the noise reducing effect better.

Noise reducing Process• Supposing a 3x3 mask is

• If mask centre f(i, j) is greater than the average,then we rank elements in the neighborhood to obtain the median f’(i, j).

• According to the standard median algorithm,average and median of the pixel(i, j+1) are given as

)5......(....................9/)}2,1(...)1,(),1({ jifjifjifAverage

)6........(..........)}........2,1(...)1,(),1({ jifjifjifmed

• According to (5) calculate average, if f(i, j)>average,then filter the image. Suppose filtered value is f’(i, j).then changes in to 3x3 matrix.

• Now take new mask to and new average and median are respectively given as

)6......(..........9/)}2,1(()1,(),(...),1({ ' jifjifjifjifAverage

)7........(..........)}........2,1(),...,(),,1({ ' jifjifjifmed

• If f(i,j+1) >average using (7,then rank to obtain the median based on new matrix:

• If f(i,j+1) <average ,then maintain the pixel value and process the next pixel following the updating of part pixel values, average of the mask is updating also .

• Where f is the original value,f’ is the updating value.

)9........(..........)}........2,1(...),(),1({ ' jifjifjifmed

• Supposing {fi j (i,j),iƐ I2 } is the gray value of each pixel in a digital image, the filter mask is nxn , f’ij is the median of fij in the mask, then

• So Steps of improved algorithm are below:1. The mask slides over the image ,overlap the centre of the mask

with the pixel on the image to search the centre element f ij.

2. read the value in the mask, i.e.3. Compute average of the mask.4. Compare the value of each pixel with average :if value of pixel

greater than average, then rank to get the median from equation (8).at the same time the value of the pixel whose value is less than average, if the value of the pixel is equal to zero, then assign the median to pixel.

5. Repeat the step (4) for new pixel with equation (9).6. Repeat the steps (4) and (5),until i=j=n

)8..(..........}.........,......,...,{' ))(())(( sjriijsjriij ffff

)9(..........}.........,......,...,{' )1)(('

)1)(()1( sjriijsjriji ffff

},......,...,{ ))(())(( sjriijsjri fff

Analysis of algorithm complexity• Improved algorithm are the same with the conventional quick

shorting algorithm. The quick shorting algorithm takes the first as a bound to discriminate elements of the set, so the original set is separated in to two new sets and sorting operations are respectively operated in to two new sets.

• The improved algorithm takes as the average as the bound to sort the set ,the next repeating sort operation always aims at the sorted set in which the element number is greater than N/2 and the another set in which the element number is less is left out. The time complexity of the improved algorithm is

)11........().........1()()( jTNTNT passf

Simulation results• Comparative experiment among these standard median filter

algorithm ,the fast median filter algorithm based on average and the improved algorithm in the paper.10%,30% and 45% density impulse noise are respectively added to the original image of the camera man. with VC++6.0,results of comparative are shown in fig.1.

• low signal to noise ratio experiment. 60%,70%, and 80% density impulse noise are respectively added to the original image of flower. Results of the improved algorithm in the paper are shown as figure 2.Experimental results show that the performance of the improved algorithm in the paper is better than the standard median filter algorithm and the fast median filter algorithm based on mean in [6]. Especially in low signal to noise ratio condition, the improved algorithm has obvious advantages.

conclusion• The paper proposed an improved median filter algorithm for

image noise reduction. It takes the peak signal to noise ratio (PSNR) as the target function and adaptively resizes the filter mask according to noise levels in the mask. Combined the median filtering with the average filtering, the improved algorithm can reduce the noise and maintain image details better. Experimental results show that the improved algorithm can well do with the contradiction between the noise-reducing effect and the time complexity of the algorithm. Therefore, it has a good application prospect in image processing.

References:[1] Guohong LIU, Wenming GUO, "Application of improved arithmetic of median filtering

denoising, Computer Engineering and Applications," 2010, vol.46, no.10, pp.187-189. (in Chinese)

[2] Xiaokai WANG, Feng LI, "Improved adaptive median filtering,"Computer Engineering and Applications, 2010, vol.46, no. 3, pp. 175-176. (in Chinese)

[3] Chao WANG, Zhongfu YE, "Salt-and-pepper noise removal by adaptive median filter and TV inpainting," Journal of University of Science and Technology of China, 2008, vol.38, no. 3, pp. 282-287

[4] Qinghua HUANG, Heqin ZHOU, Huanqing FENG, "A fast and effective algorithm of pulse noise filtering for imaging data, "Computer Engineering and Applications, 2002, no. 17, pp. 113- 114,210. (in Chinese)

[5] Huang T S, Tang G T, "A fast two- dimensional median filtering algorithm," IEEE Trans Acoustics, Speech, and Signal Processing 1979, vol.27, no. 1, pp. 13- 18.

[6] Chubin WU, "The fast algorithm of medium filtering based on mean," Journal of Jiangsu Teachers University of Technology, 2006, vol.12, no.6, pp. 102-106. (in Chinese)

[7] Tingbiao CHEN, Liangzheng XIA, "Digital image processing,"Beijing: Posts & Telecommunications Press, 1994. (in Chinese)

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