I Introduction
Images are often corrupted by impulse noise during acquisition and transmission. Therefore, an efficient noise suppression method is required before subsequent image processing operations [1]. Many recent methods [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
first detect the corrupted pixels and then restore them without affecting the uncorrupted pixels. Various solutions for estimating the intensity of noisy pixels can be divided into four categories of medianbased filters, fuzzybased algorithms, adhoc ideas and weightedaverage filters.
Most of impulse noise removal algorithms are variations of median filtering. Best examples are DecisionBased Algorithm (DBA) [3], Medianbased Switching filter (MS) [9], and Modified Decision Based Unsymmetric Trimmed Median Filter (MDBUTMF) [10]. Also, due to the nature of impulse noise, some methods are proposed based on fuzzy logics, such as DetailPreserving Filter (DPF) [6], Noise Adaptive Fuzzy Switching Median (NAFSM) filter [8]
, and Turbulent Particle swarm optimization based Fuzzy Filtering (TPFF)
[13].Other methods employed different ideas. In [2], Specialized Regularization (SR) method is proposed to restore noisy pixels. OpeningClosing Sequence (OCS) filter is presented in [4] based on mathematical morphology. In [5], EdgePreserving Algorithm (EPA) is proposed which adopts a directional correlationdependent filtering technique. In [11], Robust Outlyingness Ratio is combined with the NonLocal Means (RORNLM) to detect and filter the noisy pixels. In [14], a method is presented which employs an iterative impulse detector and an Adaptive Iterative Mean (AIM) filter to remove the general fixedvalued impulse noise.
Another wellknown approach is weightedaverage filtering, which exploits the correlation among neighboring pixels to restore the corrupted pixels. The Switchingbased Adaptive Weighted Mean (SAWM) filter [7] and the Cloud Model (CM) filter [12] employ this approach for impulse suppression. Both filters adaptively determine the filtering window and use complex weighting rules. In SAWM method, the weights are specified based on the degree of compatibility between pixels, and the CM filter uses the certainty degrees of uncorrupted pixels as their weights. These filters are timevarying; that is they have to perform pixelbypixel restoration, rather than processing the image as a whole. This constraint opposes efficient implementation.
In this paper, we propose a twostep method for realtime impulse noise suppression. First, we employ an impulse detector to identify the corrupted pixels. It examines the spatial correlation of suspicious image pixels to decrease the false detection of uncorrupted pixels as corrupted. Second, we restore the image using a weightedaverage filter. The filter operates on the nearest neighboring interpolated image and can be implemented using matrixbased operations.
The rest of this paper is organized as follows. Section II defines the impulse noise model. The method is presented in section III. The experimental results and comparisons are provided in section IV and section V concludes the paper.
Ii Impulse Noise Model
FixedValued Impulse Noise (FVIN), also known as SaltandPepper Noise (SPN), is commonly modeled by:
(1) 
where , and are the original and corrupted images and noise density, respectively, and is the image coordinate. This model implies that the pixels are randomly corrupted by two fixed extreme greyvalues, and , with the same probability.
For impulse noise suppression, we first specify the impulse values and locate the corrupted pixels, and then estimate their original values using the information provided by the uncorrupted pixels.
Iii The Proposed Method [15]
The proposed noise suppression method consists of two steps: Impulse Detection and Image Restoration. Each step is further discussed in the following.
Iiia Impulse Detector
To identify the corrupted pixels, we first determine the impulse values, and . However, marking all pixels with an extreme greyvalue as noisy pixels results in a false detection of some uncorrupted pixels as corrupted. Therefore at the next step, we should locate the noisy pixels by discriminating the uncorrupted pixels which have an impulse value. For this, we use the fact that SPN alters the pixel values to one of the two extreme greyvalues with equal probabilities. Thus, a strong inclination toward one of the impulse values in a neighborhood indicates that there are some uncorrupted pixels with an impulse value. We examine the inclination for each neighborhood and the correlation of each suspicious pixel with its neighbors to distinguish between the corrupted pixels and the uncorrupted ones which have one of the impulse values.
Computing the window size: Given the estimated noise probability is , for a pixel the probability of being uncorrupted is approximately
. Thus, using binomial distribution, the expected number of uncorrupted pixels in a window of size
is. If we set this value equal to five, the window size is obtained as the smallest odd integer greater than
. The window size is specified such that the expected number of uncorrupted pixels in the pixels’ neighborhood becomes five, and, as a result, there would be enough uncorrupted pixels to examine the center pixel.We describe the details of the impulse detector in the following:

Specify the impulse values, and , by finding the two extreme greyvalues of the corrupted image, and then construct the set as:
(2) where the symbol denotes the logical OR. The set includes the indices of suspicious pixels, i.e. pixels with one of the impulse values.

Compute the estimated noise probability as the rate of the suspicious pixels and set equal to the smallest odd integer greater than .

Compute and as the number of pixels, in the neighborhood of the pixel at coordinate , with greyvalues equal to and , respectively.

Construct the sets and as follows:
(3) (4) where the symbol denotes logical AND.

Compute the set of corrupted pixels as:
(5) where and denotes the intersection and union operations, respectively, and stands for the complement set of .
Equations (35) imply that a pixel with an impulse value is considered to be uncorrupted if:

All of its neighbors have values equal to the impulse values,

The majority of its neighbors is inclined to one of the impulse values,

It contributes in this inclination.
We define the mask matrix, corresponding to the set of corrupted pixels , as:
(6) 
After impulse detection, we rename the uncorrupted and corrupted pixels as known and noisy pixels, respectively.
IiiB Image Restoration
For image restoration, we propose an Efficient WeightedAverage (EWA) filter. In the proposed method, first we construct an initial image using the Nearest Neighboring Interpolation (NNI). In this image, each noisy pixel takes the greyvalue of its nearest known pixel. We then improve the initial image by employing a weightedaverage filter, which applies different procedures for weighting the known and noisy pixels.
Note that in this subsection, all multiplications and divisions are pixelwise.
IiiB1 Weight Assignment Procedures
The weight assignment procedures are described in this following.
Weights of Known Pixels
The weight assignment to known pixels is based on the fact that due to the spatial correlation of image pixels, the information of adjacent pixels overlaps. In other words, two separate pixels have more information than two adjacent pixels. This confirms the observations that, with a fixed noise density, random losses can be restored better than block/burst losses [16].
Therefore, to quantify the value of unique information in each known pixel, we should determine the solitariness of that pixel. For this, we define the Information Matrix (IM) as follows:
(7) 
where * denotes the convolution operation, is the convolution kernel, which is an all one matrix of size , and Mask is obtained from (6).
The denominator of (7) represents a matrix, which elements specify the number of known pixels in the neighborhood of the corresponding image pixel. The set of neighboring pixels is determined by the kernel . The center pixel is also included in the set of neighboring pixels to avoid infinite information value for a solitary known pixel.
Finally, the weights of known pixels are computed as:
(8) 
The weights are normalized such that the lowest weight for known pixels is one. Therefore, the range of is between 1 and 9. Note that these values are not valid in the positions of noisy pixels.
Weights of Noisy Pixels
The weight assignment to noisy pixels is based on this image property that farther image pixels have lesser correlation with each other. Thus in the initial image, noisy pixels which are farther from their nearest known pixel take less accurate value.
We compute the Distance Transform of the image to obtain the Distance Matrix (DM) and the ClosestPixel Map (CPM). Each element of DM and CPM contain the Euclidean distance of the corresponding image pixel with the nearest known pixel and the index of that pixel, respectively.
The weights of noisy pixels are computed as:
(9) 
Equation (9) implies that is inversely proportional to the DM and is set to be in the range of .
IiiB2 Implementation
One key factor for fast restoration is matrix implementation. The proposed filter has the advantage that all steps are implemented using matrixbased operations. The implementation details of the restoration stage are described in the following:

Compute the matrices DM and CPM from the corrupted image . For computing the Euclidean distance transform, we used the fast algorithm described in [17].

Using the CPM, employ NNI on the image to obtain the initial image .

Construct the overall Weight Matrix (WM) as:
(10) 
Restore the image as:
(11)
Figure 1 depicts the proposed filter of the restoration stage.
IiiC Computational Complexity
In this subsection, we compute the complexity of restoring an image from r% SPN, using the EWA filter. The major complexity of the proposed filter is for processing NNI, three convolutions, three pixelwise multiplications and three pixelwise divisions. The complexity of NNI is [17]. Also, we implement the convolution by twodimensional FFT (FFT2) using the following relation:
(12) 
The complexity of FFT2 is . As a result, the complexity of computing the twodimensional convolution of an image with a kernel matrix is approximately . Finally, division and multiplication have the complexity of . Therefore, regardless of the noise density, the overall complexity is .
Iv Simulation Results
The proposed algorithm is compared with the best existing methods for SPN removal. Comparisons include the quantitative evaluation, the visual quality and the time complexity. Simulations are run on four greyscale test images Lena, Peppers, Boat and Bridge. The Peak SignaltoNoise Ratio (PSNR) is employed for objective performance assessment. To make a reliable comparison, each method is run 20 times with different impulse noise patterns and the result is obtained by averaging over all experiments.
In section I, we divided the SPN removal methods into four categories. For the sake of brevity, in simulations, we compare the proposed method with the best method of each category. For medianbased filters, fuzzybased algorithms, adhoc ideas and weightedaverage filters, DBA [3], TPFF [13], EPA [5] and SAWM [7] have the best results, respectively.
Figure 6 depicts the restoration results of different methods for different images corrupted by SPN with various noise densities. The results demonstrate that the EWA filter performs better than other methods. Figures 1116 exhibit the restored images of the three best filters for images Lena and Bridge corrupted by 80% and 90% SPN, respectively. Clearly, the proposed method outperforms other methods in preserving the image details. Also, Table I lists the running time of the most competitive filters in the MATLAB environment. The simulation results verify that the proposed filter is very efficient for high density impulse noise removal.
50%  60%  70%  80%  90%  
DBA [3]  1.2  1.4  1.6  1.8  2.1 
EPA [5]  0.54  0.62  0.70  0.74  0.72 
SAWM [7]  10  11  13  15  16 
CM [12]  15  22  25  29  39 
AIM [14]  0.14  0.16  0.19  0.25  0.31 
EWA [Proposed]  0.075  0.075  0.075  0.075  0.075 
V Conclusion
In this paper, we proposed a method for fast impulse noise removal from images. The proposed filter first constructs an initial image using the nearest neighboring interpolation and then improves it by employing a weightedaverage filter, which applies different procedures for weighting the known and noisy pixels. Experimental results verify that the proposed method outperforms the best existing methods in both qualitative and quantitative measures and is quite suitable for realtime applications.
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