A New Adaptive Median Filter Based on Possibilistic Linear Models for Image Restoration

Hongwei Ge , Shitong Wang School of information, Southern Yangtze University

Wuxi, Jiangshu , People’s Republic of China ghw8601@163.com

Abstract

A novel partition center weighted median-type(PCWM) filter is proposed for improving theperformance of median-based filters. The proposedfilter achieves its effect through the linear combinations of the weighted output of the medianfilter and the related weighted input signal, and the weights are set based on possibilistic linear modelsconcerning the states of the input signal sequence. Theextensive experimental results included in this paperhave demonstrated that the proposed filter is superior to a number of well-accepted median-based filters in the literature.

1. Introduction

Median filter is well known for removing impulsivenoises but this filter distorts the fine structure ofsignals as well. Therefore, modifications to the medianfilter are needed and the literature about modificationof the median filter has grown rapidly. For example,the weighted median (WM) filter was given in [1], thecenter weighted median (CWM) filter was introducedIn [2,5], and the switching median filters have beenstudied in [3,4]. Recently, an extension of the vector sigma median (VSM) filter has been presented in [6].

In this paper, we propose a new adaptive medianfilter based on possibilistic linear models, i.e. a novel partition center weighted median-type (PCWM) filter for suppressing fixed- and random-valued noises whilepreserving image details. In this filter, the judgment of the existence of impulsive noises is expressed bypossibilistic linear models, and the filter parameter is controlled by the models. Examples of processingactual images with impulsive noises are shown toverify the high performance of this filter. Moreover,the new filter also provides excellent robustness withrespect to various percentages of impulse noise in ourtesting examples.

2. The design of the PCWM filter

2.1. The principle of The PCWM filter

Suppose that impulsive noises are added to a two-dimensional image. The central pixel value of the filter window at location is . The output value of the PCWM filter at theprocessed pixel is obtained as follows:

),( ji }255,2,1,0{),( jix),( jiy),( jix

),(),(1),(),(),( jixjijimjijiy . (1) Here, ),( ji denotes the weight indicating to whatextent an impulsive noise is considered to be located atthe pixel . If ),( jix 1),( ji , an impulsive noise isconsidered to be located at the pixel , and theoutput of PCWM filter is equal to the median value ofthe input pixel values in filter window. If

),( jix

0),( ji ,an impulsive noise is not located at the pixel ,and the output is equal to the input as it is.

),( jix

To judge whether an impulsive noise exists or not,),( ji should take a continuous value from 0 to 1 to

cope with ambiguous case. Therefore, the majorconcern of PCWM filter is how to decide the value of

),( ji . The weight ),( ji can be set by the localcharacteristics of the input signals. The amplitudes ofmost impulsive noises are larger than the fine changesof signals. Hence, we can define as follows [7]:),( jiu

),(),(),( jimjixjiu , (2) where denotes the absolute difference between the input and the median value .

),( jiu),( jix ),( jim

Obviously, if is large then an impulsive noiseis assumed present, else no impulsivenoise is assumed present. The variable is ameasure for detecting the possibility whether the input

is contaminated or not [7]. However, it isdifficult to separate the impulsive noises sufficientlyonly by the value of . For example, suppose that

),( jiu}0),({ jiu

),( jiu

),( jix

),( jiu

0-7695-2882-1/07 $25.00 ©2007 IEEE

an image contains very fine components such as linecomponents, the width of which is just one pixel, and

is located on the line with no impulsive noise.The value is large since must not be close to but to the background of this line, andaccordingly, an impulsive noise is assumed to be located at the pixel , although no impulsive noiseis there. To avoid the wrong judgment, it is necessaryto add variable to improve the filter’s performance. The variable can be defined as follows [7]:

),( jix),( jiu ),( jim

),( jix

),( jix

),( jiv),( jiv

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

),( 21 jixjixjixjixjiv , (3)

where , are selected to be the pixelvalues closest to that of in its adjacent pixels in the filter window . If is large then an impulsivenoise is assumed present, else no impulsivenoise is assumed present.

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

),( jiv}0),({ jiv

The variable takes the isolation of impulsivenoises into consideration so as to separate theimpulsive noises from the fine components of signals.When a line component appears in the filter window,

must be small since the two input signalsselected in formula (3) , that is, and

must be located on the line, Thus, we can judgethat no impulsive noise is located at the pixel .

),( jiv

),( jiv),(1 jix

),(2 jix),( jix

2.2. The partitioning of the observation vectorspace

According to the variables and , the observation vectors are given by

),( jiu ),( jiv

),(),,(),( jivjiujiO . (4) A partition is defined that the observation vector space

subset of 2R is classified into a set of N mutuallyexclusive blocks, defined as given by

N21,

,,

..,,2,1},),(:),({

1

lkforand

tsNkkjiOfjiO

lk

N

kk

k (5)

where the classifier is defined as a function of theobservation vector . It determines the outputfrom a partition of the vector space

)(f),( jiO

into Nnonoverlapping blocks according to the value of

, Thus, each input data corresponding toits is only classified into one of

),( jiO ),( jix),( jiO N blocks.

In general, the classifier can be obtained bydifferent methods to determine to which block thevector belongs. Owing to simple computationand efficiency, the scalar quantization (SQ) [8] isconsidered to be the classifier .

)(f

),( jiO

)(fIn order to diminish the complexity, all the block

boundaries on the partition are restricted to beparallel to the coordinate axes, and their projections onthe coordinate axes are mutually exclusive or identical, Based on the special case, each block

k can be

represented as a Cartesian product of two intervalblocks, and ; that is , . Then each scalar component

1s 2s 21 ssk

2,1)},,(),,({),( ajivjiujiOaof

can be classified independently by using SQ, which is a very simple process whose quantizerconsists of an encoder mapping process and a decodermapping process. The encoder mapping processincludes receiving the input value and providing an output codeword which depends on theinterval in which the valued falls, and the decodermapping process provides the codeword to arepresentative value d . In this work, the encodermapping process divides the range [0,255] into five

intervals such that each scalar componentbelongs to one of the five intervals as shown in Fig.1.

),( jiO

),( jiOa

),( jiOa

Figure 1. The quantizer input-output map for scalar observation vectors

The following is the algorithm for the partitioning ofthe observation vector space. (1) Input and the filter window centers around .

),( jix ),( jiw),( jix

(2) Calculate ),(),,(),( jivjiujiO . (3) Decide which interval belongs to and provide a representative value .

),( jiu

1d(4) Decide which interval belongs to and provide a representative value .

),( jiv

2d(5) Evaluate that belongs to block k of thepartition such that by usingthe equation .

),( jiO},,2,1{,),( NkjiO k

21 5)1( ddkFig.1 shows that the range [0,255] is divided into

five intervals. The quantization interval values areobtained empirically through extensive experimentsand remain fixed in filtering process throughout all the experiments. That is, they are part of the PCWM filterand can be applied to all situations. Despite its simplicity and low computational complexity using theSQ classifier, the PCWM filter has shown desirablerobustness in dealing with a variety of imagescorrupted by different impulsive noises.

2.3. The creation of possibilistic linear models and the operation procedure of PCWM filter

According to the partitioning of the observationvector space , we establish a possibilistic linear modelfor each block. The possibilistic linear model for the block can be denoted as .Nkk ,,2,1, kM

A possibilistic linear model can be written asnn xAxAxAAfY 22110),( Ax , (6)

where denotes a symmetric triangularfuzzy number, i.e.

)0( niAi

),( iiiA , where i

is a center and

iis a radius. By fuzzy number arithmetic, we

have |)|,( iiiiii xxxA , ),( jijiji AA .Thus, let

Tnxxx ),,,,1( 21x , ),,,,( 210 n

T ,T

nxxx |)|,|,||,|,1(|| 21x , , then .

),,,,( 210 nT

|)|,( xx TTYThe input/output dataset for the creation of the

possibilistic linear model can be obtained from a reference image. In this data set, the input variable isthe observation vector, and the output variable is thedesired weight for the pixel. Fig.2 shows the originalreference image ‘parlor‘ used in out experiments. In our experiments, the reference image corrupted by20% impulsive noise.

Assume we obtain the dataset D for block ,which can be defined as

k

)},(,),,(),,{( 2211 LL yyyD xxx , (7) Then, we can establish the corresponding modelaccording to the dataset . Here, we try to find out the

appropriate triangular fuzzy numbers such that, where , ,

,

kMD

iT

iY xA Tiii xx ),,1( 21x TAAA ),,( 210A

TAAA ),,( 210A Li ,,2,1 , and denote theestimates of .

A,iY

A,iYIn this possibilistic linear regression model, we have

need to choose an appropriate free parameter (i.e.threshold ) such that )( iY

yi

, i.e.

||||1)(

iT

iT

iiY

yy

i xx , (8)

where denotes a radius of , denotes a

center of . If we take as the objective

index, then possibilistic linear regression analysis willbecome the following LP problem:

|| iT x iY i

T x

iYL

ii

T

1

|| x

Minimize,i

iTJ ||)( x , (9)

subject to

.,,2,1,0

,||)1(

,||)1(

Liallfor

y

y

iiT

iT

iiT

iT

xxxx

Now, we can express the operating procedure of thePCWM filter as follows: the conventional median filter

and observation vector are firstcomputed, and the k th block is detected for each input data by using the function , and thevalue of

),( jim ),( jiO

),( jix ),( jiOf),( ji associated with its block

k is

obtained according to the possibilistic linear model. Of course, the corresponding output of is a

fuzzy number, we choose the center of the fuzzynumber as the value of

kM kM

),( ji . Finally, the output of PCWM filter can be obtained by using Eq. (1).

3. Experimental results

Several experiments have been conducted on avariety of benchmark images to evaluate and comparethe performance of the PCWM filter with a number ofexisting impulse removal techniques which arevariances of the standard median filter in the literature.Here, we adopt the peak signal-to-noise ratio PSNRcriterion to measure the image restoration performanceand the denoising capability.

In addition, 33 filter windows were used in all theexperiments, and the image ‘parlor’ corrupted by 20% impulsive noise as shown in Fig.2 was taken as thereference image. The possibilistic linear models for thecorresponding blocks can stay constant during thefiltering stage throughout the experiments.

The first experiment is to compare the PCWM filter with the standard median (MED) filter, the centerweighted median (CWM) filter and the vector sigmamedian (VSM) filter. Table 1 serves to compare thePSNR results of removing both the fixed- and random-valued impulsive noise with and it revealsthat the PCWM filter achieves significantimprovement on the other filters.

%20p

The second experiment is to demonstrate therobustness of the weight obtained from possibilistic linear model with different percentages of impulsivenoises. In this experiment, the image ‘parlor’ corrupted by 20% impulsive noise is also taken as the referenceimage, independent of the actual corruption percentage.Table 2 shows the comparative PSNR results of therestored image ‘boats’ when corrupted by the fixed-and random- valued impulsive noise of 10-30%. FromTable 2, the PCWM filter has exhibited a satisfactoryperformance in robustness.

4. Conclusions

In this work, the PCWM filter has been proposed to preserve image details while effectively suppressingimpulsive noises. The filter achieves its effect througha summation of the input signal and the output of median filter. With the filtering framework, the SQ method is used to partition the observation vectorspace, and the observation vector is classified as one ofN mutually exclusive blocks, then the weightassociated with the corresponding block is obtainedaccording to the possibilistic linear model. Someresults of image denoising show the high performanceof this filter.

5. References

[1] L. Yin, R. Yang, M. Gabbouj, Y. Neuvo, “Weighted median filters: a tutorial”, IEEE Transactions on Circuits and Systems, vol.43, pp. 157-192,1996 [2] S.J. Ko, Y.H. Lee, “Center weighted median filters and their applications to image enhancement”, IEEE Trans.Circuits Systems, vol.38, pp. 984-993, 1991 [3] T. Sun, Y. Neuvo, “Detail-preserving median-based filters in image processing”, Pattern Recognition Letters.vol.15, pp.341-347,1994 [4] T. Chen, K.K. Ma, L.H. Chen, “Tri-state median filter forimage denoising”, IEEE Trans. Image Process, vol.8, pp.1834-1838,1999[5] T.Chen, H.R. Wu, “Adaptive impulse detection using center-weighted median filters”, IEEE Signal Processing Letters, vol.8, pp.1-3. 2001 [6] R.Lukac, et al, “Vector sigma filters for noise detectionand removal in images”, Journal of Visual Communication and Image Representation, no.1, pp.1-26, 2006

[7] Li-Xin Wang, “A Course in Fuzzy Systems and Control”, Prentice-Hall PTR, Upper Saddle River. NJ 07458, 1997. [8] T. Chen, H.R. Wu, “Application of partition-based median type filters for suppressing noise in images”, IEEETrans. Image Process, vol.6, pp.829-836, 2001

Figure 2. The original reference image

Table I. Comparative restoration results in PSRN (dB) for 20% impulsive noise (a)fixed-

valued and (b) random-valued impulsive noise Filters Images

Lena Goldhill Boats Bridge Lake(a)

MED 30.2 28.8 29.2 24.9 27.2CWM 30.4 29.9 29.8 25.7 28.1VSM 31.3 30.5 31,1 26.9 28.5

PCWM 35.1 33.9 33.8 29.2 31.1(b)

MED 31.7 29.7 30.1 25.4 27.8CWM 32.4 30.8 31.0 26.4 28.9VSM 32.9 31.2 31.3 26.3 29.2

PCWM 34.1 32.6 32.8 27.7 30.1

Table 2. Comparative restoration results in PSRN (dB) for filtering the ‘boats’ image

corrupted by (a) fixed-valued and (b) random-valued impulsive noise, with different

percentagesFilters Percentage of impulse noise

10% 15% 20% 25% 30%(a)MED 31.7 30.6 29.2 27.4 25.4CWM 32.5 31.3 29.8 28.3 26.2VSM 34.3 33.0 31.1 29.9 27.1PCWM 36.9 35.5 33.8 32.8 31.2(b)MED 31.8 31.0 30.1 29.1 28.1CWM 32.5 31.6 31.0 29.8 28.9VSM 33.2 32.1 31.3 30.3 29.5PCWM 35.8 34.2 32.8 31.7 30.0