INTRODUCTION
In the field of signal processing, the median filter is a wellknown filter
which demonstrates effectively in suppressing impulsive noises. However, the
median filter also removes fine components of signals, causing distortion of
signal waveform. Therefore, modifications to the median filter are needed and
there are many studies about modification of the median filter. For example,
the Weighted Median Filter (WMF) was given in Yin et
al. (1996) and Lukac (2004) where the modification
is to take the median value in a signal group in which some input signals are
multiplied. However, setting the weights is difficult for actual signal processing.
In Ko and Lee (1991) and Lin (2007),
the Center Weighted Median Filter (CWMF) was introduced. In fact, it is a special
case of the WMF that gives weight to only the central pixel in the filter window.
Moreover, the Switching Median Filters (SMF) have been studied by Sun
and Neuvo (1994) and Wang et al. (2010).
These filters mainly use a detection process for separating the uncorrupted
pixels from the corrupted. Therefore, pixels are left unchanged if they are
judged as noisefree pixels. In addition, an extension of the Vector Median
Filter (VMF) has been presented by Smolka and Chydzinski
(2005) and Lukac et al. (2006) where there
is an adaptively switching filtering design. The switching concept depends on
a threshold and the detection process is based on the peer group concept and
the statistical measure of the vector’s deviation. These switching median
filters yield satisfactory results when the parameters concerned are properly
set. However, it is not easy to exploit fixed decisionmaking parameters, since
the parameters are obtained at a preassumed noise density level.
Since possibilistic linear regression analysis was first introduced the literature
dealing with possibilistic regression analysis and Possibilistic Linear Model
(PLM) has grown rapidly. Ge et al. (2008) investigated
the Regularized Possibilistic Linear Model (RPLM) which is a regularized version
of PLM and can enhance the generalization capability of PLM. In this study,
we propose a novel Adaptive RPLMs Based Median Filter (ARBMF) to improve the
medianbased filters, especially for suppressing fixed and randomvalued noises
while preserving image details. In this filter, the judgment of the existence
of impulsive noises is expressed by RPLMs and the filter parameter is controlled
by the models. Examples of processing actual images with impulsive noises are
shown to verify the high performance of this filter. Moreover, the new filter
also provides excellent robustness with respect to various percentages of impulse
noise in our testing examples.
THE DESIGN OF THE FILTER ARBMF
The principle of the filter ARBMF: Suppose that impulsive noises are added to a twodimensional image X with its height r and width c. Let x (i, j)ε{0, 1, 2,..., 255} be the pixel value at position (i, j) in image X. Let the observed sample matrix W (i, j) represents a filter window at location (i, j) with its size (2t+1)^{2}, where t+1≤i≤rt, t+1≤j≤ct and the central pixel value of the window W (i, j) be x (i, j).
The output value y (i, j) of the filter ARBMF at the processed pixel x (i, j) is obtained as follows:
Here, β (i, j) denotes the weight indicating to what extent an impulsive noise is considered to be located at the pixel x (i, j). If β (i, j), an impulsive noise is considered to be located at the pixel x (i, j) and the output of ARBMF is equal to the median value of the input pixel values in filter window. If β (i, j) = 0, an impulsive noise is not located at x (i, j) and the output is equal to the input as it is.
To judge whether an impulsive noise exists or not, β (i, j) should take a continuous value from 0 to 1 to cope with ambiguous case. Therefore, the major concern of filter ARBMF is how to decide the value of β (i, j) at the pixel x (i, j).
The weight β (i, j) can be set by the local characteristics of the input
signals. In general, the amplitudes of most impulsive noises are larger than
the fine changes of signals. Hence, we can define the variable u (i, j) as follows
(Arakawa, 1996):
where, u (i, j) denotes the absolute difference between x (i, j) and the median value m (i, j).
Obviously, if u (i, j) is large then an impulsive noise is assumed present,
else {u (i, j)≈0} no impulsive noise is assumed present. The variable
u (i, j) is a measure for detecting the possibility whether the input x (i,
j) is contaminated or not (Wang, 1997). However, it is
difficult to separate the impulsive noises sufficiently only by the value of
u (i, j). For example, suppose that an image contains very fine components such
as line components, the width of which is just one pixel and x (i, j) is located
on the line with no impulsive noise. The value u (i, j) is large since m (i,
j) must not be close to x (i, j) but to the background of this line and accordingly,
an impulsive noise is assumed to be located at the pixel x (i, j), although
no noise is there. To avoid the wrong judgment, it is necessary to add variable
v (i, j) to improve the filter’s performance. The variable v (i, j) can
be defined as follows (Arakawa, 1996):
where, x_{1} (i, j) and x_{2} (i, j) are selected to be the pixel values closest to that of x (i, j) in its adjacent pixels in the filter window. If v (i, j) is large then an impulsive noise is assumed present, else {v (i, j))≈0} no impulsive noise is assumed present.
The variable v (i, j) takes the isolation of impulsive noises into consideration so as to separate the impulsive noises from the fine components of signals. When a line component appears in the filter window, v (i, j) must be small since the two input signals selected in formula (3), that is, x_{1} (i, j) and x_{2} (i, j) must be located on the line, Thus, we can judge that no noise is located at the pixel x (i, j).
The partitioning of the observation vector space: In present study, according to the variables u (i, j) and v (i, j), the observation vectors are given by:
A partition is defined that the observation vector space Ω subset of R^{2} is classified into a set of N mutually exclusive blocks, defined as Ω_{1}, Ω_{2},..., Ω_{N} given by:
where, the classifier f (.) is defined as a function of the observation vector
O (i, j). It determines the output from a partition of the vector space Ω
into N nonoverlapping blocks according to the value of O (i, j), Thus, each
input data x (i, j) corresponding to its O (i, j) is only classified into one
of N blocks. In general, the classifier f (.) can be obtained by different methods
to determine to which block the vector O (i, j) belongs. Owing to simple computation
and efficiency, the Scalar Quantization (SQ) (Lin and Yu,
2004; Chen and Wu, 2001) is considered to be the
classifier f (.) on the design of filter ARBMF.
In order to diminish the complexity, all the block boundaries on the partition
Ω are restricted to be parallel to the coordinate axes and their projections
on the coordinate axes are mutually exclusive or identical, Based on the special
case, each block Ω_{k} can be represented as a Cartesian product
of two interval blocks, s_{1} and s_{2}; that is, Ω_{k}
= s_{1}xs_{2}. Then each scalar component O_{a} (i,
j)ε{u (i, j), v (i, j)}, a = 1, 2 of O (i, j) can be classified independently
by using SQ (Lin and Yu, 2004) which is a very simple
process whose quantizer consists of an encoder mapping process and a decoder
mapping process (Sayood, 2000). The encoder mapping process
includes receiving the input value O_{a} (i, j) and providing an output
codeword which depends on the interval in which the valued falls and the decoder
mapping process provides the codeword to a representative value d. In this study,
encoder mapping process divides the range [0,255] into five intervals such that
each scalar component O_{a} (i, j) belongs to one of the five intervals
as shown in Fig. 1.
The following is the algorithm for the partitioning of the observation vector space.
• 
Input x (i, j) and the filter window w (i, j) centers around
x (i, j) 
• 
Compute u (i, j) and v (i, j) by using Eq. 2 and 3,
respectively 
• 
Obtain O (i, j) by using Eq. 4 
• 
Decide which interval u (i, j) belongs to and provide a representative
value d_{1} 
• 
Decide which interval v (i, j) belongs to and provide a representative
value d_{2} 
• 
Evaluate O (i, j) that belongs to block k of the partition such that O
(i, j)εΩ_{k}, kε {1, 2,..., N} by using the equation
k = (d_{1}1)xB+d_{2}, B, denotes the number of intervals
of u (i, j) and v (i, j) 

Fig. 1(ab): 
The quantizer inputoutput map for scalar observation vectors 
Figure 1 shows that the range [0,255] is divided into five intervals. Of course other possibilities may also exit. In present study, the quantization interval values are obtained empirically through extensive experiments and remain fixed in filtering process throughout all the experiments. That is, they are part of the filter ARBMF and can be applied to all situations. Despite its simplicity and low computational complexity using the SQ classifier, the filter ARBMF has shown desirable robustness in dealing with a variety of images corrupted by different impulsive noises.
The creation of regularized possibilistic linear models: In general, a possibilistic linear model can be written as:
where,
denotes a symmetric triangular fuzzy number, i.e., ,
whose membership function is defined as:
where, ξ_{i} is a center and η_{i} is a radius.
By fuzzy number arithmetic, we have
Thus, let:
Then:
Now, we can derive the acceptable fuzzy membership function μ_{Y} (y) of output y as:
We can easily establish a regularized possibilistic linear model for each block,
according to the partitioning of the observation vector space.

Fig. 2: 
The original reference image ‘parlor’ 
The input/output data set for the creation of regularized possibilistic linear
models can be obtained from a reference image. Figure 2 shows
the original reference image ‘parlor’ used in out experiments. In
our experiments, the reference image corrupted by 20% randomvalued impulsive
noise. According to the original image, the reference image and the formula
(1), we can obtain the input/output data set. In this data set, the input variable
is the observation vector and the output variable is the desired weight for
the pixel. The data set D can be described as:
Another representation is also possible. If we take the observation vector u and v as the input variables, written as x_{1} and x_{2}, respectively and take weight β as the output variables, written as y, The data set D can also be defined as:
where,
According to the partitioning of the observation vector space, the data set D can be divided into N data subsets which are corresponding to those observation vector subspaces, defined as D_{1}, D_{2},..., D_{N} given by:
Assume we have obtained the data subset D_{k} (k = 1, 2,..., N), then we can establish the RPLM for the block Ω_{k} (k = 1, 2,..., N), defined as M_{k} given by:
where,
and
denote the estimates of
respectively.
In RPLM, we need to choose an appropriate free parameter λ (i.e., the
threshold value used to measure degree of fit), such that
i.e. (Ge et al., 2008):
where, η^{T} x_{i} denotes a radius of
and ξ^{T} x_{i} denotes the center of
. That is, the degree of fitting the estimated regularized possibilistic linear
regression model
to the given output data is determined by the corresponding λlevel set.
Then, the regularized possibilistic linear regression analysis in Eq.
13 will become the following optimization problem (Ge
et al., 2008):
where, c is a predefined constant, Δ^{―} and Δ^{+} denote the latent variables of the upper/lower bounds of the output, respectively.
The RPLMs can be established after solving the optimization problems mentioned
above. In our experiments, we let c = 80 and λ = 0.5. Although, we use
reference image ‘parlor’ to establish RPLMs, our experimental results
show that the performance of filter ARBMF is not dependent on the reference
image. For example, the filtering results by the ‘parlor’ reference
image are very close to the filtering results by ‘Lena’ or other reference
images. For, in general speaking, even if we use other benchmark images as the
reference images, the models established from our methods are steady.
It should be emphasized that after the RPLMs have been established, these models will be used in all our experiments and will never be changed again. In fact, these models have become the fixed parts of the filter ARBMF.
The operating procedure of filter ARBMF: Assume the filter window of ARBMF is sliding on the image X from left to right, top to bottom in a raster scan fashion. Now, we can express the operating procedure of the filter ARBMF as follows: the conventional median filter m (i, j) and observation vector O (i, j) are first computed and the k th block is detected for each input data x (i, j) by using the function f (O (i, j) and the value of β (i, j) associated with its block Ω_{k} is obtained according to the regularized possibilistic linear model M_{k}. Of course, the corresponding output of M_{k} is a fuzzy number, we choose the center of the fuzzy number as the value of β (i, j). Finally, the output of filter ARBMF can be obtained by using Eq. 1.
The operating procedure can also be described in detail as follows:
EXPERIMENTAL RESULTS
Several experiments for benchmark images are organized to demonstrate how well the proposed filter ARBMF can suppress impulsive noises and enhance the image restoration performance for signal processing. In order to evaluate and compare the performance of the proposed filter ARBMF with a number of existing impulse removal techniques which are variances of the standard median filter in the literature, we adopt the peak signaltonoise ratio PSNR criterion to measure the image restoration performance and the noise suppression capability.
In addition, 3x3 filter windows were used in all the experiments. In particular, the quantization interval values employed in the partitioning processed were obtained experimentally and the satisfactory interval values shown in Fig. 2 are used throughout the experiments. The five intervals can be [0,5), [5,20), [20,35), [35,60), [60,255) and [0,2), [2,5), [5,15), [15,35), [35,255) for the variables u (i, j) and v (i, j), respectively. In the experiments, the image ‘parlor’ corrupted by 20% impulsive noise as shown in Fig. 2 was taken as the reference image. The RPLMs for the corresponding blocks can stay constant during the filtering stage throughout the experiments.
The first experiment is to compare the filter ARBMF with the standard median
filter MEDF, the Center Weighted Median Filter (CWMF), the Switching Median
Filter (SMF) (Sun and Neuvo, 1994) and the Vector Sigma
Median Filter (VSMF) (Lukac et al., 2006) which
was proposed in terms of noise removal capability. Table 1
serves to compare the PSNR results of removing both the fixed and randomvalued
impulsive noise with p = 20% and it reveals that the filter ARBMF achieves significant
improvement on the other filters for suppressing both types of impulsive noises.
In order to show the excellent capability for preserving image details while effectively suppressing impulse noise, Fig. 3, 4 are given here. These figures are the comparative restoration results of the several filters mentioned above for the benchmark images corrupted by randomvalued impulsive noise at 20%. In our experiments, the filter VSMF seems to always outperform MEDF, CWMF and SMF, so the filter ARBMF is just compared with VSMF. Apparently, the filter ARBMF produced a better subjective visual quality restored image with more noise suppression and detail preservation. Especially, in Fig. 4, we can see that the ARBMF has preserved the top thin line more completely than the VSMF.
The second experiment is to demonstrate the robustness of the weights obtained
from RPLMs with different percentages of impulsive noises, regardless of what
is used as the reference image. In this experiment, the image ‘parlor’
corrupted by 20% impulsive noise is also taken as the reference image, independent
of the actual corruption percentage.

Fig. 3(ad): 
Restoration performance comparison. (a) original image, (b)
the image degraded by 20% randomvalued impulsive noise, (c) the restoration
result of VSMF and (d) the restoration result of ARBMF 

Fig. 4(ad): 
Restoration performance comparison. (a) original image, (b)
the image degraded by 20% randomvalued impulsive noise, (c) the restoration
result of VSMF and (d) the restoration result of ARBMF 
Table 1: 
Comparative restoration results in PSRN (dB) for 20% impulsive
noise 

(a): Fixedvalued and (b): Randomvalued impulsive noise 
Figure 5 shows the comparative PSNR results of the restored
image ‘boats’ when corrupted by the fixed and random valued impulsive
noise of 530%, respectively. From Fig. 5, the filter ARBMF
has exhibited a satisfactory performance in robustness, regardless of the reference
image used in experiments. In addition, Fig. 6 shows the restoration
performance comparisons of different methods in filtering the image corrupted
by randomvalued impulsive noises at various noise ratios. From these figures,
we can also see that the filter ARBMF produced better subjective visual quality
restored images.

Fig. 5(ab): 
Restoration performance comparison of different methods in
filtering the ‘Boat’ image corrupted by impulsive noises at various
noise ratios. (a) image corrupted by fixedvalued impulsive noises, (b)
image corrupted by randomvalued impulsive noises 

Fig. 6(ah): 
Restoration performance comparison of different methods in
filtering the image corrupted by randomvalued impulsive noises at various
noise ratios. (ab) original image, (c) noised image (p = 15%), (d) noised
image (p = 25%), (ef) the restoration results of VSMF and (gh) the restoration
results of ARBMF 
CONCLUSIONS
In present study, a novel adaptive RPLMs based mediantype filter ARBMF has been proposed to preserve image details while effectively suppressing impulsive noises. The proposed filter achieves its effect through a summation of the input signal and the output of median filter. With the filtering framework, the efficient SQ method is used to partition the observation vector space and the observation vector is classified as one of N mutually exclusive blocks, then the weight associated with the corresponding block is obtained according to the RPLM. Some results of image denoising show the high performance of this filter and the filter ARBMF is not only capable of showing desirable robustness in suppressing noise but also able to gain appreciated image quality.
ACKNOWLEDGMENTS
This study is supported by National Natural Science Foundation of China (Grant No.60975027), New century Outstanding Young Scholar Grant of Ministry of Education of China.