Adaptive Window Length Recursive Weighted Median Filter ... · Adaptive Window Length Recursive...

Adaptive Window Length Recursive Weighted Median Filter for Removing Impulse Noise in Images with Details Preservation 73

Adaptive Window Length Recursive WeightedMedian Filter for Removing Impulse Noise in

Images with Details Preservation

V.R.Vijay Kumar 1 , S.Manikandan 2 , P.T.Vanathi 3 ,

P.Kanagasabapathy 4 , and D.Ebenezer 5 , Non-members

ABSTRACT

An adaptive window length Recursive WeightedMedian filter [ARWMF] for removing the impulsenoise with better edge and fine detail preservation ispresented. Larger window size may blur the imagesand the lower window size does not remove the noiseat high density. To overcome this, the window size ofthe RWMF is adaptive based on the presence of noisedensity. Median controlled algorithm is used to calcu-late the weights for the RWMF. In median controlledalgorithm, the filter gives the smallest weight for theimpulse. However, for many weight functions, includ-ing the exponential one, this weight is non-zero. Thusthe impulse has an effect on the output and the mag-nitude of the impulse is reduced. The computationalcomplexity for the weight calculation is simple and itis very efficient. The window size of the RWMF isadaptive based on the presence of noise density. Theproposed algorithm produces better edge and fine de-tails preservations and reduces blurring at the highdensity impulse noise.The performance of the pro-posed algorithm is given in terms of mean square error(MSE), mean absolute error (MAE) and peak signalto noise ratio (PSNR) and it is compared with Stan-dard Median filters, Weighted Median filters, CenterWeighted Median filters, Recursive Weighted Medianfilters and Lins Adaptive length Recursive weightedmedian filters using Median Controlled Algorithm.

Keywords: Adaptive Window size, High ImpulseNoise suppression, Less computation, Median con-trolled Algorithm, Recursive weighted median filter.

1. INTRODUCTION

I mpulse noise is caused by malfunctioning pix-els in camera sensors, faulty memory locations inhardware, or transmission in a noisy channel. Two

Manuscript received on May 11, 2007 ; revised on October13, 2007.

1,3 The authors are with Department of ECE, PSG Collegeof Technology, India, E-mail: vr [email protected]

2,5 The authors are with Department of ECE, SKCET, India,E-mail: [email protected]

4 The author is with 4Dean, Madras Institute of Technology,India, E-mail:

common types of impulse noise are the salt-and-pepper noise and the random-valued noise. For im-ages corrupted by salt-and-pepper noise (respectivelyrandom-valued noise), the noisy pixels can take onlythe maximum and the minimum values (respectivelyany random value) in the dynamic range. There aremany types of impulse noise. Let Yij be the graylevel of a true image Y at pixel location (i, j) and[nmin ,nmax] be the dynamic range of Y. Let Xij bethe gray level of the noisy image X at pixel (i, j), thenthe impulse noise model is given as equation 1

Xij = Rij ; with probability r;Yij ; with probability 1− r; (1)

where Rij ∈ [nmin; nmax] are random numbers andr is the noise ratio. For example, for fixed-valued(salt-and-pepper) impulse noise, noisy pixels Xij takeeither nmin or nmax. There are many works on therestoration of images corrupted by impulse noise. Themedian filter was once the most popular nonlinear fil-ter for removing impulse noise, because of it’s gooddenoising power and computational efficiency. Overthe last two decades, there is a significant improve-ment in the development of median filters. Weightedmedian filter (WMF) [3], RWM[4] filter are examples.However, when the noise level is over 50%, some de-tails and edges of the original image are smeared bythe filter. Different remedies of the median filter havebeen proposed, e.g. [1],[2],[12]the adaptive median fil-ter, the multi-state median filter, or the median filterbased on homogeneity information. These so-called“decision-based” or “switching” filters [9][10][11] firstidentify possible noisy pixels and then replace themby using the median filter or its variants, while leav-ing all other pixels unchanged. These filters are goodat detecting noise even at a high noise level. Theirmain drawback is that the noisy pixels are replacedby some median value in their vicinity without takinginto account local features such as the possible pres-ence of edges. Hence details and edges are not recov-ered satisfactorily, especially when the noise level ishigh. It has been proved that RWM [4] filter producesbetter result when compared to other median type fil-ter. The median type filters exhibit blurring for fixedwindow sizes and insufficient noise suppression for

74 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.6, NO.1 February 2008

small window sizes. In this paper an adaptive windowsize RWM filter algorithm using median controlledalgorithm is proposed, which achieve a high degreeof noise suppression and preserve image sharpness.Lin’s, & Huang proposed adaptive length median fil-ters for removal of impulse noise in images. Adaptivelength Recursive weighted median filter using Lin’salgorithm produces less efficient output and the algo-rithm has high complexity. In case of adaptive RWMfilter [4], the weights are chosen in accordance withwindow length. In some windows the signal may benoise free. However attenuation of the amplitude ofthe signal causes blurring. Window lengths are se-lected based on the amount of noises present in theinput signal. After calculating the window length,the RWM operation is performed. The weight for theproposed adaptive RWM filter is calculated by usingthe median controlled algorithm. This algorithm issimple and has less computation and less complexitycompared to the other weight calculating algorithms.And also the proposed algorithm is simple and pro-duces better PSNR, MSE and MAE compared to theother standard algorithms.

2. RECURSIVE WEIGHTED MEDIAN FIL-TER

The success of the median filters in the image pro-cessing is based on two intrinsic properties: edgepreservation and efficient attenuation of the impulsivenoise properties not shared by traditional filters. Theapplication of the weighted median filters, however,has not significantly spread beyond image processingapplications. When a median type filter filters a sig-nal, some characteristic(s) will change. But impulsenoise will be reduced significantly. In general, changesare more profound nearer edges than homogeneousregions. Thus the median filter can be understoodas a simple detector of impulses and edges. It is ahighly data dependent filter, by which weights havebeen given to the samples according to the changesby the low pass filter. The recursive weighted medianfilter detects and remove the impulses in the images.The general structure of linear IIR filters is denotedby the equation 2

Y (n) =N∑

l=1

AlY (n− 1) +M2∑

k=−M1

BkX(n− k) (2)

where the output is formed not only from the input,but also from previously computed outputs. The fil-ter weights consist of two sets: the feedback coeffi-cients {Al}, and the feed-forward coefficients (Bk).N +M1 +M2 + 1 coefficient are needed to denotethe recursive difference equation. For WM filters, thesummation operation is replaced with the median op-eration, and the multiplication weighting is replacedby signed replication as in equation 3

Y (n) = Median(|Al| ¦ sgn(Al)Y (n− l)|NL=1

|Bk| ¦ (Bk)Xn−k|M2k=−M1

) (3)

2.1 Recursive Weighted Median Filters

Given a set of N real-valued feed-back coefficientsAi |Ni=1 and a set of M+1 real-valued feed-forwardcoefficients Bi |Mi=0, the M+N+1 recursive WM filteroutput is defined as equation 4.

Y (n) = Median(|AN | ¦ sgn(AN )Y (n−N)|, . . .|Al| ¦ sgn(AN )Y (n− 1), |B0| ¦ sgn(B0)Xn,

. . . |BM | ¦ sgn(BM )X(n + M)) (4)

where < (AN,. . .A1,B0,B1,. . .BM) > are the coeffi-cients of recursive weighted median filter.

2.2 Adaptive window size Selection

Generally in the fixed small window size filters, theamount of noise density filtered will be very less, forfiltering high density noise the window size of the fil-ter may increase. This may lead to blurring in theoutput images. In order to overcome this, the adap-tive window length filters are designed for filteringhigh density noises. The proposed algorithm is sim-ple and has less computation. Lin [1] and Huang [12]proposed some adaptive algorithms for filtering im-pulse noise. But these algorithms are more complexand the results are not better compared to the pro-posed adaptive algorithm. In case of Lin’s algorithm,based on the threshold values the window size is se-lected but in the proposed algorithm there is no needto calculate the threshold values. The proposed adap-tive technique uses the intensity value of the pixels,to determine whether the pixel is corrupted or uncor-rupted pixel. And based on the amount of corruptedpixels, the window size is increased or decreased. Dueto this, the unwanted filtering of uncorrupted pixelsis reduced. And hence blurring is reduced even athigh density noise.

2.3 Median Controlled Algorithm

The weight calculation for the Recursive WM fil-ter is performed by threshold decomposition tech-nique, optimal weights by MAE technique for realweight calculation, Complex weights calculation andnegative weights calculation. The above method iscomplex and has high computation. Mean square er-ror and mean absolute error are used for calculatingthe weights for the RWM filters in optimization tech-niques. The original image is needed for the calcula-tion of MSE and MAE. And also, the weights calcu-lated by the optimization technique may be zero andnegative. In case of median controlled algorithm, theselection of weights is simple and also the filter gives


small weights for the impulse. For example, for eachwindow, those input samples which are closer to theoutput of the first filtering operation can be exponen-tially weighted more. Let the difference of the sampleXi and the result of the low pass filtering X’i at thesame position be |Xi-X’i|. Weight values can be ob-tained by using the equation 5

Weight(i,j)=exp{−α |original(i,j)−reference(i,j)|}(5)

Where α > 0. The output of the first itera-tion of the median controlled filter is obtained as aweighted sum of the samples inside the moving win-dow of the filter. This moving window need notbe the same window that is used in the calcula-tion of weights. The general weighted median fil-ter structure [3] with weights as a=(a1,a2,a3. . .ai)and the inputs x=(X1,X2,X3. . .Xi) is given by WeightMed(X1,X2,X3. . .Xi)=MED{(a1¦X1,a2¦X2,a3¦X3. . .ai

Xi)} Where ¦ is the replication operator defined asai¦Xi=(ai, ai. . .ai) Xi times[3] Selecting the outputof the first iteration to be the reference signal, com-puting the new weights by comparing the new ref-erence signal to the original signal, and computingthe output again using the new weights can continuethe procedure. This is repeated until the number ofthe iterations is reached. Thus the Median controlledRecursive Weighted Median filter is obtained. Oneneeds only to change the first Reference signal calcu-lation to be done by the Recursive weighted medianfilter with weights ai.

This gives more freedom for the designer. Furthermore, one can completely reject potential outliers byletting the weights be zero when the difference be-tween the filtered signal and the original signal ex-ceeds a certain level.Steps involved in the Median controlled algorithm areas follows1. Get the median filtered image using the windowsliding W, store the result in REFERENCE image.2. Calculate the weight as

Weight(i,j)= exp{-α|original(i,j) - Reference(i,j)|}3. Using the above weights, perform the Recursiveweighted median operation and store the output asreference image.4. The process is done iteratively, so that outputimage is produced with least mean square error.

3. STRUCTURE OF THE FILTER

The general structure of the recursive weightedmedian filter [4] is given as equation 6,

Y (n) = MED(|A1| ¦ sgn(A1)Y (n− 1)N |+ Bk| ¦ sgn(X(n + k)M2)) (6)

Let us consider the algorithm as stages as shown inthe block diagram (Fig 1).

Fig.1: Block diagram of Median controlled adaptiveRWM filter

Stage 1: Determination of the window size:Zmin = minimum intensity value in Sxy

Zmax = maximum intensity value in Sxy

ZRWM = RWM intensity value in Sxy

Zxy = intensity value at coordinates Sxy

The adaptive Recursive weighted median filteringalgorithm works in two levels

Level A : If Zmin < ZRWM < Zmax, go to level BElse increase the window sizeIf window size ≤ Smax, repeat level AElse output ZRWM

Level B : If Zmin < Zxy < Zmax , output Zxy

Else output ZRWM

Stage 2: Filtering operation:The Recursive weighted median filtering operation

is carried out based on the adaptive window size de-termined. The algorithm for the recursive weightedmedian filter is given as:

Input s/ Outputs : M × N imageMoving window W, |W|=N=2k+1Weight vector a=(a1, a2. . .aN)Let Half Sum=

∑Ni = 1 ai /2

for i=1to Number of Rowsfor j=1 to Number of Columns

place the window W at (i,j)store the image values inside W and thecorrespondingweights in x= ((X1,a1),(X2,a2),. . .(XN,aN))sort x with respect to Xis, store the result iny = ((X1,a1),(X2,a2),. . .(XN,aN))let Sum = 0, m =1repeatlet Sum = Sum+a(m)let m= m+1until Sum ≥ Half Sumlet Output(i,j) = y(m-1)Recmed(X1,X2,. . .XN) =MED(Y1,Y2,. . .Yk,Xk+1,. . .XN)

endend


3.1 RESULTS

In MATLAB7.1 the proposed ARWM filter usingmedian controlled algorithm is tested using the Grayscale images (Elaine, Lena and Pepper) and colour(Zelda) image. Fig2 show the results Elaine imagecorrupted by 20% of noise density. Fig 2 (a, b, c,d, e, f, g) are the original image, corrupted image,standard median filter (SMF) output, Weighted me-dian filter (WMF) output, Recursive Weighted me-dian filter (RWMF) output, Median controlled usingLin’s algorithm (MC Lin’s) output and the proposedARWMF using median controlled algorithm. Fig3show the results of Lena image corrupted by 60%of noise density and results of the different filters.Fig 4 (a, b, c, d, e, f, g, h) are the original pepperimage, corrupted image at 90% noise density, stan-dard median filter (SMF) output, Center Weightedmedian filter (CWMF) output, Weighted median fil-ter (WMF) output, Recursive Weighted median filter(RWMF) output, Median controlled using Lin’s algo-rithm (MC Lin’s) output and the proposed methodoutput. Fig.5 shows the Zelda original color imagecorrupted by 60% noise density and restored resultsof various filters. For colour image, the proposed al-gorithm is applied to each plane (R, G, and B) sepa-rately and later the outputs are combined. The per-formance metrics such as Peak signal to noise ratio(PSNR), Mean square error (MSE) and Mean abso-lute error (MAE) are evaluated using the followingformulas

PSNR = 10 log10

(2252

MSE

)(7)

MSE =1

MN

∑

ij

(yij − xij)2 (8)

MAE =1

MN

∑

ij

|yij − xij | (9)

For color image the equation of MSE and MAE ismultiplied by a factor 1/3. From the results, at thehigher noise densities the proposed adaptive recursiveweighted median filter produces better results. Theedges and fine details are preserved without blurringclearly shown in the visual results.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig.2: (a) Original 512x512 Elaine image (b) Noisyimage (Noise Density =20%) Outputs of(c)SMF (d)CWMF (e) WMF (f) MC with Lin’s AW (g) RWMF(h) Proposed Method

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig.3: (a) Original 512X512 lena image (b) Noisyimage (Noise Density =60%). Outputs of(c)SMF (d)CWMF (e) WMF (f) MC with Lin’s AW (g) RWMF(h) Proposed Method


(a) (b) (c) (d)

(e) (f) (g) (h)

Fig.4: (a) Original 256X256 Pepper Image (b) Noisy Image (Noise Density=90%). Outputs of (c) SMF (d)CWMF (e) WMF (f) MC with Lin’s AW (g) RWMF (h) Proposed Method

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig.5: (a) Original 256X256 Zelda Image (b) Noisy Image (Noise Density=60%). Outputs of (c) SMF (d)CWMF (e) WMF (f) MC Lin’s (g) RWMF (h) Proposed Method


Table 1: Comparison table of PSNR of different filters for Lena.png (Gray scale Image)Noise SMF CWMF WMF MC RWM Proposed

Density Lin’s Algorithm10 33.72 33.67 34.22 34.48 33.28 29.2920 29.62 25.81 27.08 31.70 32.22 28.5230 24.03 20.04 21.66 27.53 31.08 28.0940 19.03 16.19 17.57 22.30 29.14 27.1950 15.45 13.12 14.22 17.40 25.96 26.7160 12.44 10.59 11.64 13.72 21.88 26.2270 10.09 9.12 9.49 10.70 17.56 25.6380 8.19 7.64 7.90 8.36 14.14 24.2290 6.69 6.46 6.58 7.91 11.91 22.97

Table 2: Comparison table of MSE of different filters for Lena.png (Gray scale Image)Noise SMF CWMF WMF MC RWMF Proposed


Table 3: Comparison table of MAE of different filters for Lena.png (Gray scale Image)Noise SMF CWMF WMF MC RWM Proposed



Fig.6: Comparison graph of PSNR at different noise density of Zelda image

Fig.7: Comparison graph of MSE at different noise density of Zelda image

Fig.8: Comparison graph of MAE at different noise density of Zelda image

4. CONCLUSION

Generally, the RWM filters are designed only forthe fixed window length. This causes blurring in theoutput samples. Because in the fixed window length,the noise may absent in some windows in that con-dition, the filtering operation is done for the originalsamples which causes the blurring in the output. Toovercome the problem, this filter is designed where

the window length is determined by the width of theimpulsive noise presented in the input sample. Hence,there is no chance of filtering the uncorrupted pixeland it reduces the blurring in the output sample. Andthe weights calculated by using the median controlledalgorithm is producing very effective result and causesless blurring and the MSE and MAE are also veryless when compared to other median type algorithms.But the proposed algorithm has three different stages


for compute the result and hence the processing timeis more compare to the existing methods.

References

[1] Ho-Ming Lin and Alan, “Median filters withAdaptive Length,” IEEE transactions of the cir-cuits and systems, vol.35, no.6, June 1988.

[2] S.Manikandan, O.Uma Maheswari, D.Ebenezer,“Adaptive length Recursive weighted medianfilter with improved performance in impul-sive noisy environnent,” WSEAS transaction onElectronics, issue 3, Vol.1,july 2004

[3] G. Arce, “A General Weighted Median FilterStructure Admitting Negative Weights,” IEEETr. On Signal Proc., vol.46, Dec. 1998

[4] G. Arce and J. Paredes, “Recursive WeightedMedian Filters Admitting Negative Weights andTheir Optimization,” IEEE Tr.on Signal Proc.,vol.48, nr. 3, March 2000.

[5] O. Yli-Harja, J. Astola and Y. Neuvo, “Anal-ysis of the Properties of Median and WeightedMedian Filters Using Threshold Logic and StackDecomposition,” IEEE Tr. Signal Proc., vol.39,no. 2, pp. 395-410, Feb. 1991.

[6] O.Yli-Harja, Heikki Huttunen, Antti and Karen,“Design of Recursive weighted median filterswith negative weights,” signal processing lab,Tampere university of tech, Finland

[7] I..Pitas and A.N.Venetsanopoulos, Nonlineardigital filters Principles and applications, 1990

[8] Gonzalez and Woods, “Digital Image processingusing matlab,” Pearson Education , pp no.178-179, 2004

[9] S. Zhang and M. A. Karim, “A new impulse de-tector for switching median filters,” IEEE SignalProcessing Letters,9, pp. 360-363, 2002.

[10] Z. Wang and D. Zhang, “Progressive switchingmedian filter for the removal of impulse noisefrom highly corrupted images,” IEEE Transac-tions on Circuits and Systems II, 46, pp. 78-80,1999.

[11] M. Nikolova, “A variational approach to removeoutliers and impulse noise”, Journal of Mathe-matical Imaging and Vision, vol.20, pp. 99-120,2004.

[12] H. Hwang and R. A. Haddad, “Adaptive me-dian filters: new algorithms and results,” IEEETransactions on Image Processing, 4, pp. 499-502, 1995.

[13] J. Astola and P. Kuosmanen, Fundamentals ofNonlinear Digital Filtering. Boca Raton, CRC,1997.

[14] 14 Raymond H. Chan, Chung-Wa Ho, and MilaNikolova , “Salt-and- Pepper Noise Removalby Median-type Noise Detectors and Detail-preserving Regularization,” July, 2004.

V.R.Vijaykumar is currently work-ing as a lecturer in the departmentof Electrionics and Communication En-gineering, PSG College of Technology,Coimbatore. He received his bachelordegree from Government of college ofTechnology, Vellore and Master degreefrom Thiagarajar college of Engineering,Madurai. He is currently doing his Ph.Dunder Anna University, Tamilnadu. Hisresearch interest is digital image pro-

cessing, nonlinear filters, and digital signal processing.

S.Manikandan received his bachelordegree from Madras University andMaster degree from Anna University.His area of interest includes nonlinearsignal processing, VLSI design and im-age processing. He is currently workingas lecturer in the department of Elec-tronics and Communication Engineer-ing, Srikrishna college of Engineeringand Technology, Coimbatore.

P.T. Vanathi is working as AssistantProfessor in the department of Elec-tronics and Communication Engineer-ing, PSG College of Technology, Coim-batore. Her area of interest includesSpeech Signal Processing, Non linearsignal processing, Digital communica-tion and VLSI Design. She has pub-lished many papers in international jour-nals and international conferences. Sheis also member of review committees for

many national and international conferences.

P.Kanagasabapathy is working asDean, Madras Institute of Technology,Anna University. His area of interest in-cludes Speech Signal Processing, Digi-tal Image Processing, and Process Con-trol Instrumentation. He has publishedmany papers in international journalsand international conferences. He is alsomember of review committees for manynational and international Journals.

D. Ebenezer is currently working as aprofessor in the department of Electron-ics and Communication Engineering,Srikrishna college of Engineering andTechnology, Coimbatore. He workedas a Assistant .Professor in College ofEngg. Guindy, Anna University. Hisarea of interest includes Digital signalprocessing, Non-linear signal processing,Digital communication. He has pub-lished many papers on Non Linear signal

processing. And he is also member of review committees formany national and international conferences.

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