suppression of impulse noise in mr images using artificial intelligent based neuro-fuzzy adaptive...

Digital Signal Processing 18 (2008) 391–405

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Suppression of impulse noise in MR images using artificialintelligent based neuro-fuzzy adaptive median filter

Abdullah Toprak a, Mehmet Siraç Özerdem b, Inan Güler c,∗

a Dicle University, Meslek Yüksek Okulu, Elektrik-Elektronik Bölümü, 21280 Diyarbakır, Turkeyb Dicle University, Mühendislik Mimarlık Fakültesi Elektronik Bölümü, 21280 Diyarbakır, Turkey

c Gazi University, Teknik Egitim Fakültesi, Elektronik-Bilgisayar Bölümü, 0650 Teknikokullar, Ankara, Turkey

Available online 1 May 2007

Abstract

This paper presents a new artificial intelligent based neuro-fuzzy rule base adaptive median filter for removing highly impulsenoise. Since the filter is rule base, it is called neuro-fuzzy rule base adaptive median (NFRBAM) filter. The NFRBAM filter is animproved version of switch mode fuzzy adaptive median filter (SMFAMF) and is presented for the purpose of noise reduction ofimages corrupted with additive impulse noise. The NFRBAM filter consists of a decision unit and three different types of filters.In the decision unit, the noisy input image is directed to the proper filter with respect to the noise density. Neuro-fuzzy rule basedapproach is used in both decision and filtering parts. In artificial neural network, multi layer perceptron (MLP) architecture withbackpropagation (BP) algorithm is used for noise detection and removing highly impulse noise corrupted MR images. In fuzzylogic, bell-shaped membership function is employed in order to obtain better results. Experimental results indicate that the proposedfilter is improvable with the increased fuzzy rules to reduce more noise corrupted images and preserve image details more thanSMFAMF.© 2007 Elsevier Inc. All rights reserved.

Keywords: Fuzzy adaptive median filter; Noise reduction; Neuro-fuzzy rule base adaptive median filter

1. Introduction

Image processing with fuzzy logic, starts with the fuzzy digital topology and continues with the studies on sup-pression of noise or improvement of image by using fuzzy logic rules [1,2]. While some researchers have realizedhistogram based noise suppression by using fuzzy logic [3], some others have deteriorated the impulse noise in theimages by using adaptive filter techniques [4]. The studies on purification of the noise from an image by using fuzzyrule base have been performed by [5,6]. The studies on purifying high amounts of impulse noise from images usingfuzzy logic has been performed by [7]. Median filter (MF) applications for image improvement studies have beenperformed by [8].

Median filter is an image filter that is more effective in situations where the impulse noise is less than 0.2. AlthoughMF is a method used in suppressing noises with low density (0.2), it may even be insufficient in destroying the noises.

* Corresponding author.E-mail address: [email protected] (I. Güler).

1051-2004/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2007.04.008

392 A. Toprak et al. / Digital Signal Processing 18 (2008) 391–405

While the MF suppresses the noise, some of the details of the image could be lost and the details of image could notbe seen in the screen. This situation is especially a major drawback in medical imaging and diagnosis. When the MFscreening window is downsized although it suppresses more noise, the most of the details in the picture are lost whenthe window is enlarged. When MF is used, due to the fact that even the pixel value is switched with the median value,nearly all pixels shall be switched with an erroneous value. If the ratio exceeds 0.2, adaptive median filter (AMF) isused. As is the case in other filters, an Sxy window is selected for the AMF. However a feature that differentiates theAMF from the other filters is the fact that the size of this window can be changed. Unfortunately, while AMF removesthe impulse noise, it also deteriorates the details in the image it accompanies.

In order to prevent the details of an image, the authors have been described a new adaptive median filter in theirprevious paper [9]. In [9], they have been described a switch mode fuzzy adaptive median filter (SMFAMF) for sup-pressing highly corrupted impulse noises. Although the SMFAMF provides better results in suppressing the impulsenoises but the edges of the image were still deteriorated.

In this study, in order to keep the details while removing the impulse noise, the neuro-fuzzy rule base adaptivemedian filter (NFRBAM) filter is described. Since the pixel values of image are determined by fuzzy logic rules, thefilter can be implemented as artificial intelligent base. NFRBAM filter is made up of three basic sections. These aremedian filter, fuzzy logic rules, and the decision unit that decides whether there is noise or not. The main purpose ofusing this filter is to avoid losing details in the image while destroying the noise in the image.

2. Materials and method

If X[(i,j)] is defined as image matrix p(k, l) to assume as pixel values, then a matrix of 3 × 3 will be in the formof W [(k, l)] ∈ X[(i,j)]. This window matrix will scan the whole X[(i,j)] matrix from top to bottom and left to right.In every 3 × 3 scan it will classify 9 pixels according to gray intensity. P(k, l) ∈ Ximp, if p(k, l) = min{W [k, l]} ormax{W [k, l]}.

The window W [(k, l)], which scans over the entire image to clarify the noise from is a 3×3 matrix is shown below.Ximp matrix is a noise matrix that is mixed up with X[(i,j)] image matrix as

p(k − 1, l − 1) p(k − 1, l) p(k − 1, l + 1)

p(k, l − 1) p(k, l) (k, l + 1)

p(k + 1, l − 1) p(k + 1, l − 1) p(k + 1, l + 1)

, (1)

X =

⎡⎢⎢⎣

x11 x12 . . . xij . . . x1W

x21 x22 . . . x2j . . . x2W

......

......

xH1 xH2 . . . xHj . . . xHW

⎤⎥⎥⎦ = [xij ]HxW . (2)

In Eq. (2), H and W are height and width and xij ∈ {0,1,2, . . . ,255} shows the gray intensity of pixel in i,jcoordinate of X matrix.

x1 = p(k − 1, l − 1), x2 = p(k − 1, l), x3 = p(k − 1, l + 1), x4 = p(k, l − 1), x5 = p(k, l),

x6 = (k, l + 1), x7 = p(k + 1, l − 1), x8 = p(k + 1, l − 1), x9 = p(k + 1, l + 1).

In this case,

W[(k, l)

] = [p(k − 1, l − 1),p(k − 1, l),p(k − 1, l + 1),p(k, l − 1),p(k, l), (k, l + 1),p(k + 1, l − 1),

p(k + 1, l − 1),p(k + 1, l + 1)].

Each component is defined as a fuzzy variable and the membership function is the intensity value of each inputpixel.

MF makes possible for the elimination of a divergent value by changing the divergent value in a finite series withthe medium value in the same series [10]. When it is two dimensions, the MF for images is as

m(k) = MEDw(k) = MED{x−n(k), . . . , x−1(k), x0(k), x1(k), . . . , xn(k)

}. (3)

For the adaptive median filter an Sxy window were chosen [9]. The flow diagram of adaptive median filter can beanalyzed in two levels. Let us label these levels as A and B .

A. Toprak et al. / Digital Signal Processing 18 (2008) 391–405 393

Level A: A1 = Zmed − ZminA2 = Zmed − ZmaxIf A1 > 0 and A2 < 0 then go to level B

If it is not, then increase the size of the windowIf window size is �Smax repeat level A

If not output Zxy

Level B: A1 = Zxy − ZminA2 = Zxy − ZmaxIf B1 > 0 and B2 < 0, output Zxy

If not, output Zmed,

where

Zmin = Sxy is the lowest gray level value inside,

Zmax = Sxy is the highest gray level value inside,

Zxy = (x, y) gray level at the subject matter coordinates,

Zmed = is the maximum possible Sxy window size.

Because of the window size can be adjusted depend on impulse noise ratio and as it can be understood from theflow diagram of AMF given above, it is an effective filter why is used in eliminating the impulse noise. Let us assumethat the median value at level A was equal to the noise, in a case as such the size of the window to be examined willbe changed and another median value will be calculated. This process will be continued until the median value comesout as different from the minimum or the maximum value. But it can never be guaranteed that the value obtained isnot the noise [12]. However depending on the size of the window the probability of obtaining a noise value will bereduced. While the enlargement of the window suppresses the noise to a great extent, at the same time, in proportionto its size, the details on the image will be harmed [11].

3. Neuro-fuzzy rule based adaptive median filter

The proposed filter system is shown in Fig. 1. As it is seen in Fig. 1, NFRBAM filter is the first stage of theflowchart of the elimination of noises. NFRBAM filter is used for obtaining the image with details, then the decision

Fig. 1. The flow chart of NFRBAM filter model.


units is used for another process called last stage. Last stage has three filters in which the image is directed to theproper filter with respect to the remaining noise of density.

In this section, proposed NFRBAM filter is described. Fuzzy ruled based and artificial neural network approachesconstitute NFRBAM filter. First, neural network is used for noise detection and removing highly impulse noise cor-rupted MR images. Second, fuzzy rule base is used for obtaining the image with details having more quality. Thedetails of each approach and their applications are described in Sections 3.1 and 3.2.

3.1. Fuzzy rule based approach

This approach needs an image that is mixed with impulse noise, so that elimination of noise can be performed.For this purpose, let us assume that the image has 128 × 128 pixels and that it has gray levels between 0 and 255. Itcan be defined that the noise amount and uncertainty are included in the image with the fuzzy logic variables. If it isphrased NFRBAM filter is a system that can be formed by defining nine fuzzy logic membership functions and ninevariables for each one of these, and the fuzzy variables can be named as mf 1,mf 2, . . . ,mf 9. As such we will acceptthe intensity of input pixel p(x, y) as a fuzzy variable and define the membership levels of the fuzzy sets sequentiallyas follows; mf 1 (blackest), mf 2 (less black), . . . , mf 8 (very white), and mf 9 (whitest). However in practice we arenot required to limit the membership functions with nine, we can increase the number of membership function. Inorder to conduct fuzzy image processing, first we have to give fuzziness to each pixel input intensity value and thennormalize them to 0 � p(x, y) � 1 range. In this study, the bell shaped membership function was used for NFRBAMfilter as the following. We use a 3 × 3 image matrix to scan the entire image. Here the filtered output will be changedwith the central pixel of this matrix, and the noise in the image will be suppressed by using the fuzzy filter.

If we define Nimp matrix as the pixel values in noise form, then an 3 × 3 matrix will be in the form of W [(k, l)].This window matrix will scan the entire shape from right to left and top to bottom. In each 3 × 3 scan with 9 pixelswill be classified according to gray intensity. P(k, l) ∈ Nimp, if p(k, l) = min{W [k, l]} or max{W [k, l]}. In a case assuch

x1 = p(k − 1, l − 1), x2 = p(k − 1, l), x3 = p(k − 1, l + 1), x4 = p(k, l − 1), x5 = p(k, l),

x6 = (k, l + 1), x7 = p(k + 1, l − 1), x8 = p(k + 1, l − 1), x9 = p(k + 1, l + 1).

Each element will be defined as a fuzzy variable and the membership function is the intensity value of each inputpixel. The bell shaped membership function was used for NFRBAM filter as the following:

pj (xi) = 1

1 + ((xi − cj )/aj )2bj

, i = 1,2, . . . ,9, j = 1,2,3. (4)

In Eq. (4) aj , bj , cj are the parameters that can be adjusted according to requirements. Mamdani method was usedfor fuzzy rules deduction in NRBFMF. The basic rules are as the following:

If p(k, l) is close to the blackest Y = is the blackest, else,

If p(k, l) is close to gray Y = is gray, else,

......

If p(k, l) close to the blackest Y = whitest.

By adjusting the parameters in bell-shaped membership function and by applying Eq. (4) to the given sub imagewith nine pixels, the fuzziness was obtained. Choosing the value of b much higher than a and c makes possible for themembership function of impulse noise to be filtered properly. In the 2nd step, normalized values for each pixel valueare calculated using Eq. (5) given as

wij = mj(xi)∑9i=1 mj(xi)

, i = 1,2, . . . ,9, j = 1,2,3. (5)

The following rules are used to determine whether the medium pixel values of NFRBAM filter or noise.


Rule 1. If x1 = p(k − 1, l − 1) ∈ mf (1), x2 = p(k − 1, l) ∈ mf (1), x3 = p(k − 1, l + 1) ∈ mf (1)4, x4 = p(k, l − 1) ∈mf (1), x5 = p(k, l) ∈ mf (1), x6 = (k, l + 1) ∈ mf (1), x7 = p(k + 1, l − 1) ∈ mf (1), x8 = p(k + 1, l − 1) ∈mf (1), x9 = p(k + 1, l + 1) ∈ mf (1), then x5 = p(k, l) /∈ Ximp.

Rule 2. If x1 = p(k − 1, l − 1) ∈ mf (2), x2 = p(k − 1, l) ∈ mf (2), x3 = p(k − 1, l + 1) ∈ mf (2), x4 = p(k, l − 1) ∈mf (2), x5 = p(k, l) ∈ mf (2), x6 = (k, l + 1) ∈ mf (2), x7 = p(k + 1, l − 1) ∈ mf (2), x8 = p(k + 1, l − 1) ∈mf (2), x9 = p(k + 1, l + 1) ∈ mf (2), then x5 = p(k, l) /∈ Ximp.





Rule 7. If x1 = p(k − 1, l − 1) ∈ mf (7), x2 = p(k − 1, l) ∈ mf (7), x3 = p(k − 1, l + 1) ∈ mf (7), x4 = p(k, l − 1) ∈mf (7), x5 = p(k, l) ∈ mf (7), x6 = (k, l + 1) ∈ mf (7), x7 = p(k + 1, l − 1) ∈ mf (7), x8 = p(k + 1, l − 1) ∈mf (7), x9 = p(k + 1, l + 1) ∈ mf (7), then x5 = p(k, l) /∈ Ximp

Rule 8. If x1 = p(k − 1, l − 1) ∈ mf (8), x2 = p(k − 1, l) ∈ mf (8), x3 = p(k − 1, l + 1) ∈ mf (8), x4 = p(k, l − 1) ∈mf (8), x5 = p(k, l) ∈ mf (8), x6 = (k, l + 1) ∈ mf (8), x7 = p(k + 1, l − 1) ∈ mf (8), x8 = p(k + 1, l − 1) ∈mf (8), x9 = p(k + 1, l + 1) ∈ mf (8), then x5 = p(k, l) /∈ Ximp


The following methodology was followed for fuzzy logic filter. First, an image of 128 × 128 in size and 8 bit indepth (256 gray levels) impulse was added at the rate of 70%. The pixel values for the image with noise, in the size of128 × 128, were written in sequential lines following each other, and as such a column matrix was formed in the sizeof 1×16,384. Second, the n−1, n and n+1 series were obtained and the data in one column matrix form is convertedto input set of three columns. The purpose here is to be able to see the previous and next values of each pixel. In thenext stage, logical relationships among these three column inputs sets were established. This logical relationship wasdefined using the mathematical expression given in Eq. (6) as [9].

Xi =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Xmin < Xn−1 < Xmak, Xn−1 ifnotXmin < Xn < Xmak and Xmin < Xn+1 < Xmak, (Xn + Xn+1)/2 ifnotXmin < Xn < Xmak, Xn ifnotXmin < Xn+1 < Xmak, Xn+1 ifnot0,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (6)

One column data matrix that was obtained by using Eq. (6) was taken as the only input of the fuzzy system. Assuch the input file of the fuzzy system was prepared. Output file was prepared in the form of a one column matrixsimilar to the 128×128 sized matrix of the original image. The membership functions of both the input and the outputwere chosen as 36. The number of rules was determined as 36 being equal to the number of membership functions.The mathematical expression of the membership functions are as the following:

UA−(x) = UA−

(x;a, b, c) =⎡⎣ (x − a)/(b − a), if a � x < b,

(c − x)/(c − b), if b � x < c,

0, if x > c or x < a.

(7)

The weighting coefficients of the rules were determined to be 1. Centroid method was used as a clarificationmethod.


Fig. 2. (a) Noisy input training image for NN, (b) the structure of three layered NN in present study.

Fig. 3. General architecture of the learning and identification system.

A part of the noise in the image was first suppressed with AMF, and then the rest of the noise (especially the noisethat AMF has failed to suppress) was eliminated by using the fuzzy model.

3.2. Artificial neural network (ANN) approach

An ANN is a mathematical model consisting of a number of highly interconnected processing elements organizedinto layers, the geometry and functionality of which have been likened to that of the human brain. ANNs learn byexperience, generalize from previous experiences to new ones, and can make decisions [13]. Neural elements of a


Table 1Results obtained using MLP with BP learning algorithm

Model No. Structure Training accuracy (%) Testing accuracy (%)

1 9-10-1 76.73 63.122 9-20-1 72.19 65.473 9-30-1 95.85 98.024 9-40-1 80.21 72.185 9-50-1 89.56 84.886 9-60-1 63.14 89.197 9-70-1 70.38 74.618 9-80-1 71.84 65.109 9-90-1 78.62 69.7710 9-100-1 81.45 77.51

Fig. 4. Performance changing of neural network in training stage.

human brain have a computing speed of a few milliseconds, whereas the computing speed of electronic circuits ison the order of microseconds. The ANNs are parallel processing elements and have the following characteristics:(a) ANN is a mathematical model of a biological neuron, (b) ANN has processing elements which are related withanother, (c) ANN keeps knowledge with connection weights.

Back-propagation learning algorithm is mainly used for classification problems and so it can be use for detectionnoisy pixels. The BP learning algorithm is simple and effective, and has found home in a wide assortment of machinelearning applications, such as detection of noise for image processing. The advantage of the generalized back prop-agation is that it adapts the internal parameters of the hidden neurons too. So the center selection problem is partlysolved by the back propagation training.

The network has one input layer, one hidden layer and one output layer as shown in Fig. 2. The input layer consistsof all the input factors. Information from the input layer is then processed in the course of one hidden layer, followingoutput vector is computed in the output layer. A schematic description of the layers is given in Fig. 2b. In developingan ANN model, the available data set was divided into two sets, one to be used for training of the network, and theremaining was used to verify the generalization capability of the network. Input-output pairs are presented to thenetwork and weights are adjusted to minimize the error between the network output and actual value [14].


Table 2MLP architecture and training parameters

The number of layers 3The number of neuron on the layers Input: 9, hidden: 30, output: 1The initial weights and biases Randomly between −1 and 1Activation functions for hidden and output layers Log-sigmoidTraining parameters learning rule Back-propagationAdaptive learning rate for hidden layer 0.9Adaptive learning rate for output layer 0.7Number of iteration 39Momentum constant 0.9Duration of learning time 36 sAcceptable mean-squared error 0.005

(a) (b) (c)

(d) (e)

Fig. 5. Noise reduction with NFRBAM: (a) original MR-1 image, (b) γ = 0.6 image with noise, (c) noise suppression with MF, (d) noise suppressionwith SMFAMF, (e) noise suppression with NFRBAM.

Among the various kinds of ANN approaches that exit, the multilayer perceptron (MLP) architecture with back-propagation (BP) learning algorithm, which has become the most popular in engineering and biological applications,was used in this study. BP algorithm, which is common in literature, has been used to update the forward pathparameters in ANN. This method is based on minimization of the quadratic cost function by tuning the networkparameters [15]. The mean square error is considered as a measurement criterion for a training set. Parameters whichminimize this cost function are determined. The averaged square error is given by

ej (n) = dj (n) − yj (n), (8)

ε(n) = 1

2P

∑j∈C

P∑n=1

e2j (n). (9)


(a)

(b)

(c)

Fig. 6. Comparison of NFRBAM: (a) MSE, (b) RMSE, (c) PSNR values prepared for MR-1 image.

In these equations e, n, d , y, P , and C indicate error signal at the output, iteration number, desired output, generatedoutput by network, total number of patterns contained in the training set and number of neurons at output layer,respectively [15]. The adjustment of synaptic weights between hidden layer and output layer is given by the equations:

�wji(n) = ηδj (n)yi(n), (10)

δj (n) = ej (n)ϕ′(

m∑i=0

wji(n)yi(n)

). (11)

The adjustment of synaptic weight coefficients between input layer and hidden layer are given by the equation:

δj (n) = ϕ′(

m∑wji(n)yi(n)

)∑δk(n)wkj (n). (12)

i=0 k


(a) (b) (c)

(d) (e)


In Eq. (10), η indicates learning-rate parameters and has different values in different problems. In case of thenetwork is not converge, the formula which is used to determine weight coefficients including α momentum parameterhas been generalized by the equation [15]:

�wji(n) = α�wji(n − 1) + ηδj (n)yi(n). (13)

The general architecture of learning and identification system is given in Fig. 3. As given in Fig. 3, learningalgorithm stage is presented symbolically as a block having delta rule.

4. Results and discussions

The aim of using the ANN model considered as a practical approach is to test the ability to identify the restoredoutput image. The network has 9 input parameters: 3 × 3 pixel filtering window and one output parameter, restoredoutput image.

In order to obtain the architecture having the best identification accuracy we developed a number of feed-forwardneural network using exactly the same training and testing data sets. We used a number of different structures and theBP learning algorithm for training the neural network models. The results are summarized in Table 1. For each NNidentifier, Table 1 depicts the structure of the network (for example, the sequence 9-10-1 that corresponds to the firstidentifier means that it consists of 9 input neurons, 10 hidden neurons, and 1 output neuron), the best identificationaccuracy based on 9-30-1 structure.

The experimental data set includes 15,876 patterns (each pattern contains of 3 × 3 pixels data), of which 12,876patterns were used for training the network and 3000 patterns were selected randomly to test the performance of thetrained network. All the input and output values were normalized between 0.1 and 0.9 using linear scaling (Fig. 2a).During the training period, the averaged square error decreased with increasing number of iteration. The performancechanging of ANN in training stage is given in Fig. 4. The MLP architecture and training parameters used in learningstage for ANN structure are presented in Table 2.

Impulse noise is added to the (MR-1, 2, 3) image in different ratios and MR images with noise were obtained.Then, with the usage of different filtration techniques, the results obtained with the suppression of impulse noise inthe images were compared with the resulting values of NFRBAM filter. These images are given in Figs. 5, 7, and 9.


(a)

(b)

(c)


In Figs. 5, 7, and 9 (a) the original MR image, (b) 60% impulse noise added to image, (c) MR image that its noiseis suppressed with MF, (d) MR image that its noise is suppressed with FAMF, and (e) MR image that its noise issuppressed with NFRBAM. As is it clearly seen in Figs. 5, 7, 9, NFRBAM protects the details the best comparedwith other filters, while suppressing the impulse noise. The reliability of the images which were used for this purposecan be evaluated by mean square error (MSE), root mean square error (RMSE), signal-to-noise ratio (SNR), and peaksignal-to-noise ratio (PSNR) criteria. The MSE, RMSE, and PSNR are given in Eqs. (14), (15), and (16), respectively.MSE, RMSE, and PSNR values in the output images of the filters are shown in Tables 3, 4, and 5. As it is understoodfrom these tables, while MSE value was quiet high comparing with NFRBAM. Nevertheless, it is clearly seen thatwhile SNR ratio has low values in other filters, but it is higher in NFRBAM. These values can be observed moreclearly and graphically in Figs. 6, 8, and 10, respectively.


(a) (b) (c)

(d) (e)


Table 3Comparisons of MSE, RMSE, NMSE, and PSNR for NFRBAM, median (3 × 3), FAMF, SMFAMF using MR-1 images (128 × 128)

Image Filter Impulse-to-noise ratio

20% 30% 40% 50% 60%

MSE NFRBAM 28 85 223 433 786No. filter 1325 2456 4582 7913 10,457Median (3 × 3) 458 495 1142 2356 3956FAMF 701 688 928 1493 2309SMFAMF 214 392 569 785 989

RMSE NFRBAM 5.3 9.2 14.9 20.8 28.0No. filter 36.4 49.6 67.7 89.0 102.3Median (3 × 3) 21.4 22.2 33.8 48.5 62.9FAMF 26.5 26.2 30.5 38.6 48.1SMFAMF 14.6 19.8 23.9 28.0 31.4

PSNR NFRBAM 35.50 31.70 28.40 24.60 22.00No. filter 19.80 17.20 14.30 12.20 11.00Median (3 × 3) 24.60 23.50 20.80 17.50 15.00FAMF 22.70 22.80 21.50 19.40 17.00SMFAMF 30.43 27.30 26.34 23.21 21.23

NMSE NFRBAM 5.000E–07 0.000E+00 0.000E+00 0.000E+00 0.000E+00No. filter 2.010E–05 0.000E+00 0.000E+00 1.000E–01 0.000E+00Median (3 × 3) 5.600E–06 0.000E+00 0.000E+00 0.000E+00 0.000E+00FAMF 9.400E–06 0.000E+00 0.000E+00 0.000E+00 0.000E+00SMFAMF 6.500E–06 0.000E+00 0.000E+00 0.000E+00 0.000E+00

MSE =∑N

i=1∑M

j=1 ‖F(i, j) − G(i, j)‖2

, (14)
NM


Table 4Comparisons of MSE, RMSE, NMSE and PSNR for NFRBAM, median (3 × 3), FAMF, SMFAMF using MR-2 images (128 × 128)


20% 30% 40% 50% 60%

MSE NFRBAM 97.88 179.43 907.73 2283.70 4092.20No. filter 939.82 2702.60 11,089.00 24,045.00 44,217.00Median (3 × 3) 7356.40 7399.30 7359.80 7391.90 11,949.00FAMF 22,848.68 22,949.00 25,094.68 27,995.35 32,361.68SMFAMF 220.68 350.00 1977.00 6880.46 8970.35



NMSE NFRBAM 0.00 0.01 0.02 0.06 0.11No. filter 0.03 0.07 0.30 0.64 1.18Median (3 × 3) 1.96 1.97 1.96 1.97 0.32FAMF 0.61 0.61 0.67 0.75 0.86SMFAMF 0.83 0.82 0.92 1.01 1.15

Table 5Comparisons of MSE, RMSE, NMSE and PSNR for NFRBAM, median (3 × 3), FAMF, SMFAMF using MR-3 images (128 × 128)


20% 30% 40% 50% 60%

MSE NFRBAM 96.32 177.87 906.17 2282.14 4090.64No. filter 938.26 2701.04 11,087.44 24,043.44 44,215.44Median (3 × 3) 7354.84 7397.74 7358.24 7390.34 11,947.44FAMF 22,847.44 22,947.44 25,093.44 27,993.44 32,360.44SMFAMF 747.44 1557.44 4546.44 5656.44 6766.44



NMSE NFRBAM 0.0001 0.0002 0.0010 0.0026 0.0046No. filter 0.0711 0.0703 0.0813 0.0927 0.0950Median (3 × 3) 0.0475 0.0569 0.0834 0.0837 0.1019FAMF 0.0258 0.0260 0.0284 0.0317 0.0366SMFAMF 0.0351 0.0349 0.0390 0.0429 0.0486

RMSE = √MSE, (15)

PSNR = 20 log

{255

RMSE

}. (16)


(a)

(b)

(c)


5. Conclusion

MR images having different ratio of noises are tested with proposed filter and better results are observed. The mainobservation is that the details of image are preserved by the proposed filter. As a result of the study, we conductedit was seen that the noise suppression process which was performed by using NFRBAM filter is more successful inpreserving the details on the MR image. This situation can be seen clearly in Figs. 5, 7, and 9. The basic purpose ofusing this filter is not only to eliminate the noise of the image but also to prevent the disappeared details of the image.It is for this reason that NFRBAM filter is an effective filter in destroying the impulse noise. If it was possible todefine 9 input pixels and 9 membership functions we were required to define 99 rules. Since this process will be verydifficult to perform with the computers, we defined only 758 rules. With the increase in number of rules the image


gets better but the processing of the image on the computer becomes much slower. In our future studies, we plan touse the reduced number of fuzzy logic rules.

Acknowledgment

This work has been supported by Gazi University Scientific and Research Project Fond (Project No: 07/2006/-05).

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