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Impulse noise reduction in medical images with the use of switch mode fuzzy adaptive median filter
Digital Signal Processing 17 (2007) 711–723 www.elsevier.com/locate/dsp
Impulse noise reduction in medical images with the use of switch mode fuzzy adaptive median filter Abdullah Toprak a , ˙Inan Güler b,∗ a Dicle University, Meslek Yüksek Okulu, Elektrik-Elektronik Bölümü, 21280 Diyarbakır, Turkey b Gazi University, Teknik E˘gitim Fakültesi, Elektronik-Bilgisayar Bölümü, 06500 Teknikokullar, Ankara, Turkey
Available online 15 December 2006
Abstract In this paper, a novel fuzzy adaptive median filter is presented for the noise reduction in MR images corrupted with heavy impulse (salt&pepper) noise. We propose a switch mode fuzzy adaptive median filter (SMFAMF) for removing highly corrupted salt&pepper noise without destroying edges and details in the image. The SMFAMF filter is an improved version of adaptive median filter (AMF) in order to reduce additive impulse noise in the images. The proposed filter can preserve details in the images better than AMF while suppressing additive salt&pepper or impulse type noises. In this paper, we placed our preference on bellshaped membership function with adaptive parameters instead of triangular membership function without variable coefficients in order to observe better results. Experiments with the magnetic resonance (MR) image from healthy subject, an MR image having the opaque material, and an MR image having disease demonstrate the mean square error (MSE), root mean square error (RMSE), signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR) of the proposed method. The results show that the proposed method can be useful for MR images with impulse type noises. © 2006 Elsevier Inc. All rights reserved. Keywords: Adaptive median filter; Fuzzy adaptive median filter; Impulse noise; Noise reduction
1. Introduction Medical images are often deteriorated by noise due to various sources of interference and other phenomena that affect the measurement processes in imaging and data acquisition systems. Median filtering is a common nonlinear method for noise suppression that has unique characteristics. It does not use convolution to process the image with a kernel of coefficients. Rather, in each position of the kernel frame, a pixel of the input image contained in the frame is selected to become the output pixel located at the coordinates of the kernel center. The kernel frame is centered on each pixel (m, n) of the original image, and the median value of the pixels within the kernel frame is computed. The pixel at the coordinates (m, n) of the output image is set to this median value [1]. Median filter (MF) is a 2D image filter that is more effective in situations where the impulse noise is less than 0.2 [2]. If this ratio exceeds 0.2, adaptive median filter (AMF) is used. As it is the case in the other filters, an Sxy window is selected for the AMF [3]. However, a feature that differentiates the AMF from the other filters is the fact * Corresponding author.
E-mail addresses: [email protected] (A. Toprak), [email protected] (˙I. Güler). 1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2006.11.008
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that the size of this window can be changed. Unfortunately, while AMF removes the impulse noise, it also deteriorates the details in the image it accompanies. In this study, in order to prevent the deterioration, the pixel values of the image are determined by the fuzzy adaptive median filter (FAMF) [4–6], and such FAMF with adaptive membership parameters is defined. This is called switch mode fuzzy adaptive median filter (SMFAMF). SMFAMF is made up of three basic sections. These are decision unit which is MF, fuzzy logic rules, and noise. The basic purpose of using this filter is to suppress the noise in the image, while it keeps the details in the image without deterioration. It is for this reason that SMFAMF is an effective filter for removing the impulse noise in MR images [7–10]. 2. Method and model 2.1. General definitions If we define X[(i,j )] image matrix p(k, l) to be made up of pixel values, a matrix of 3 × 3 will be in the form of 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 grey intensity. p(k, l) ∈ Ximp , if p(k, l) = min{W [k, l]} or max{W [k, l]}. The window W [(k, l)], that we will scan over the entire image, to clarify the noise from image, is a 3 × 3 matrix and this matrix is given as follows [11]: p(k − 1, l − 1) p(k − 1, 1) p(k − 1, l + 1) W (k, l) = . (1) p(k, l − 1) p(k, l) p(k, l + 1) p(k + 1, l − 1) p(k + 1, l) p(k + 1, l + 1) On the other hand, Ximp In a case as such ⎡ x11 x12 ⎢ x21 x22 X=⎣ ... ... xH1 xH2
matrix is a noise matrix that is mixed up with X[(i,j )] image matrix. ... ... ...
xij x2j ... xHj
... ... ...
⎤ x1W x2W ⎥ ⎦ = [xij ]H×W . ... xHW
(2)
Here H and W are height and width, respectively, and xij ∈ {0, 1, 2, . . . , 255} shows the grey intensity of pixel in i, j coordinate 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 input pixel. 2.2. Median filter This is a filter that makes possible for the elimination of a divergent value by changing the divergent value in a finite series with the medium value in the same series [9]. When it is of two dimensions, the MF for images would be developed as follows:
(3) m(k) = med w(k) = med x−n (k), . . . , x−1 (k), x0 (k), x1 (k), . . . , xn (k) . 2.3. Adaptive median filter MF is an image filter that can be more effective under conditions where the noise rate is less than 0.2. However, under conditions when the rate of noise exceeds 0.2 the AMF must be considered. Another advantage of AMF is making sure that the details on the screen are not lost while the noise is suppressed. Like the other filters, an Sxy
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window will be chosen for AMF. However, in AMF the size of this window can be changed even if the size of the window is changed the result of the operation conducted will be one and single, and this value can be changed with the central pixel value of the window. We take the notations below into consideration: Zmin = Sxy is the lowest grey level value inside. Zmax = Sxy is the highest grey level value inside. Zxy = (x, y) grey level at the subject matter coordinates. Zmed = is the maximum possible Sxy window size. We should analyze the flow diagram of AMF at two levels. Let us name them as levels A and B. Level A: A1 = Zmed − Zmin A2 = Zmed − Zmax If A1 > 0 and A2 < 0 then go to level B. If it is not the case, then increase the size of the window. If window size Smax , repeat level A. Level B: A1 = Zxy − Zmin A2 = Zxy − Zmax If B1 > 0 and B2 < 0, output Zxy . If not, output Zmed . From the flow diagram of AMF, it can be understood that it is an effective filter which is used in eliminating the impulse noise. The purpose of level A here is to determine whether the output value of the filter Zmed is in the noise or not. If Zmin < Zmed < Zmax then Zmed value is not a noise value and it must be transmitted to the exit. In a case like this, we have to proceed to level B. In level B, it is determined whether Zxy itself is a noise level and a new value will be determined according to this. Let us assume that the median value at level A equals to the noise, in a case like this the size of the window to be examined will be changed and another median value will be calculated. This process will be continued until the median value comes out different from the minimum or the maximum value. Nevertheless, it can never be guaranteed that the value obtained is not the noise. However, depending on the size of the window the probability of obtaining a noise value will be reduced. While the enlargement of the window suppresses the noise to a great extent, at the same time, in proportion to its size, the details on the image will be harmed [11]. 2.4. Switch mode fuzzy adaptive median filter with adaptive membership parameters First of all we need an image that is mixed with impulse noise, so that elimination of noise can be performed. For this, let us assume that the image has 128 × 128 pixels and that it has grey levels between 0 and 255. We can define the noise amount and uncertainty that is included in the image with the fuzzy logic variables. If we phrase it differently, SMFAMF is a system that can be formed by defining nine fuzzy logic membership functions and nine variables for each one and the fuzzy variables can be named as mf1, mf2, . . . , mf9. As such we will accept the intensity of input pixel p(x, y) as a fuzzy variable and express the membership levels of the fuzzy sets sequentially as follows: mf1 (blackest), mf2 (less black), . . . , mf8 (very white), and mf9 (whitest). However, in practice we are not required to limit the membership functions with nine, we can increase this number. For us to be able to conduct fuzzy image processing, first we have to give fuzziness to each pixel input intensity value and then normalize them to 0 p(x, y) 1 range. We will use a 3 × 3 image matrix for scanning the entire image. Here the filtered output will be changed with the central pixel of this matrix, and the noise in the image will be suppressed by using the fuzzy filter [12]. If we define Nimp matrix as the pixel values in noise form, then a 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, 9 pixels will be classified according to grey intensity. p(k, l) ∈ Nimp , if p(k, l) = min{W [k, l]} or max{W [k, l]}. In a case like
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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 input pixel. In Eq. (4) the bell-shaped membership function used for SMFAMF is given: pj (xi ) =
1 , 1 + ((xi − cj )/aj )2bj
i = 1, 2, . . . , 9, j = 1, 2, . . . , 9.
(4)
Here aj , xj , cj are the parameters that can be adjusted according to desired values [12]. For fuzzy meaning deduction rules that will be used in adaptive fuzzy median filter (AFMF), fundamental Mamdani method was utilized. The following are the basic rules. By adjusting the parameters in bell-shaped membership function and by applying Eq. (1) to the given sub image with nine pixels, the fuzziness was obtained. Choosing the value of xi is much higher than ai and ci makes possible for the membership function of impulse noise to be filtered properly. In the second step, normalized values are calculated for each pixel value by using the equation mj (xi ) wij = 9 , i=1 mj (xi )
i = 1, 2, . . . , 9, j = 1, 2, 3, . . . , 9.
(5)
The following rules are used to determine the noise in image having sampling window 3 × 3 in a pixel that is used in SMFAMF. 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) x4 = p(k, 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, / XVblack . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 2. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XBblack . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 3. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XLblack . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 4. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xgray . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 5. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xgray . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 6. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xgray . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 7. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XLwhite . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 8. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xwhite . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 9. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XVwhite . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈
1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈
As mentioned, the bell-shape membership function parameters can be adjusted according to the rules above [13]. If x5 = p(k, l) ∈ / XVblack then a1 = 0.1, c1 = 0.2, b = 15. / Xblack then a1 = 0.2, c1 = 0.3, b = 15. If x5 = p(k, l) ∈
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If x5 = p(k, l) ∈ / XLblack then a1 = 0.2, c1 = 0.4, b = 15. / Xgray then a1 = 0.4, c1 = 0.5, b = 15. If x5 = p(k, l) ∈ / XLwhite then a1 = 0.2, c1 = 0.6, b = 15. If x5 = p(k, l) ∈ / Xwhite then a1 = 0.2, c1 = 0.7, b = 15. If x5 = p(k, l) ∈ / XVwhite then a1 = 0.1, c1 = 0.8, b = 15. If x5 = p(k, l) ∈ For fuzzy logic filter, the following methodology was followed. First, impulse noise was added at the rate of 60% to the image of 128 × 128 in size and 8 bit resolution (256 grey levels). The pixel values for the image with noise, in the size of 128 × 128, were written in sequential lines following each other, and such a column matrix was formed in the size of 1 × 16384. Second, the n − 1, n, and n + 1 series were obtained and the data in one column matrix form is converted into an input set of three columns. The purpose here is to be able to see the previous and next values of each pixel together. In the next stage logical relationships among these three column sets were established. This logical relationship was defined by using the mathematical expression given as ⎫ ⎧ Xmin < Xn−1 < Xmax , Xn−1 if not ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ Xmin < Xn < Xmax and Xmin < Xn+1 < Xmax , (Xn + Xn+1 )/2 ⎪ Xi = Xmin < Xn < Xmax , Xn if not . (6) ⎪ ⎪ ⎪ ⎪ < X < X , X if not X ⎪ ⎪ min n+1 max n+1 ⎪ ⎪ ⎭ ⎩ 0 One column data matrix that was obtained by using Eq. (5) was taken as the only input of the fuzzy system. As such the input file of the fuzzy system was prepared. On the other hand, output file was prepared in the form of a one column matrix similar to the 128 × 128 sized matrix of the original image. The membership functions of both the input and the output were 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 was obtained as the equation ⎡ (x − a)/(b − a), a x < b, UA (x) = UA (x; a, b, c) = ⎣ (c − x)/(c − b), b x < c, (7) − − 0, x > c or x < a. The weighting coefficients of the rules were determined to be 1. As a clarification method, centroid method was used. A part of the noise in the image was first suppressed with AMF, and then the rest of the noise (especially the noise that AMF has failed to suppress) was eliminated by using the fuzzy model. The flow chart of the model was designed as stated in Fig. 1. In this study, we placed our preference on bell-shaped membership function instead of triangular membership function in order to observe better results. We adjusted a, b, c parameters to get better result for removal noise in image. Membership function is illustrated in Fig. 2.
Fig. 1. The flow chart of SMFAMF model.
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Fig. 2. The bell-shaped membership function.
3. Results and discussion The reliability of the images which were used for this purpose can be evaluated by mean square error (MSE), root mean square error (RMSE), signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR) criteria. The MSE, NMSE, SNR, and PSNR are given in Eqs. (8)–(11), respectively. While SNR and PSNR values give the information on the quality of the image, MSE and NMSE values demonstrate the error values of same the images compared with the original one. The gray levels of the pixels in the image are taken as data. When the signal ratio of an image is high and the noise ratio is low, this image is considered to be good quality. Therefore, in the analysis of an image, SNR and PSNR values should absolutely be examined. Since the magnitude of this ratio is proportional to the quality of the image, the SNR of the original image and the image containing noise is determined. N M 2 i=1 j =1 F (i, j ) − G(i, j ) , (8) MSE = NM √ RMSE = MSE, (9) 255 , (10) PSNR = 20 log RMSE N M 2 i=1 j =1 F (i, j ) − G(i, j ) NMSE = . (11) N M 2 i=1 j =1 F (i, j ) In the calculation of normalized MSE value, values obtained using Eq. (8) were compared with other techniques. In this study, SMFAMF’s effects on MR images are examined using 3 types of MR images. These images are the most used three MR images of various types, especially the MR image of a healthy brain, MR image of a brain with tumor and MR image obtained by using an opaque substance. With the examination of the findings, SMFAMF’s superiority related to the other MR images according to few other filtration techniques is determined. They are evaluated comparison with MSE, RMSE, SNR and NMSE values of output images. 3.1. Experiment 1 Impulse noise is added to the healthy MR (MR-1) image in different ratios and healthy MR image with 60% noise (γ = 0.6) is obtained. Then, with the use of MF, FAMF, and SMFAMF techniques, the results of impulse noise suppression in the images are compared with each other as shown in Fig. 3. Figure 3a shows the original MR image, Fig. 3b shows 60% impulse noise added to the image, Fig. 3c shows the noise suppression with MF, Fig. 3d shows the noise suppression with FAMF, and Fig. 3e shows the noise suppression with SMFAMF. As is it clearly seen in Fig. 3, SMFAMF protects the details best compared with the other filters while suppressing the impulse noise. MF contains
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(a)
(b)
(d)
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(c)
(e)
Fig. 3. Noise reduction with SMFAMF: (a) original MR-1 image, (b) γ = 0.6 image with noise, (c) noise suppression with median filter, (d) noise suppression with FAMF, (e) noise suppression with SMFAMF. Table 1 Comparison of MSE, RMSE, NMSE, and PSNR for SMFAMF, median (3 × 3), median (5 × 5) using MR-1 images (128 × 128) Filter
Noise ratio 20%
30%
40%
50%
60%
MSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
36.9 1645.28 1361.4 453.4 284
87.3 3824.54 2460.9 580.8 356
188.2 8246.25 4772.5 1077.5 659
454.8 11,258.31 7828.9 2309.7 883
821.0 14,324.36 10,648.0 4153.0 1298
RMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
6.1 40.56 36.9 21.3 16.9
9.3 61.84 49.6 24.1 18.9
13.7 90.80 69.1 32.8 25.7
21.3 106.10 88.5 48.1 29.7
29 119.68 103.0 64.0 36.0
PSNR
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
33.5 32.5 16.8 21.6 27.43
31.5 28.70 14.2 20.5 24.30
35.20 25.4 11.3 17.8 23.34
40.5 21.6 9.2 14.5 20.21
50.0 19.0 8.0 12.0 18.23
NMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
17.6 3.E−07 1.1E−5 3.6E−6 5.4E−6
18.2 0.0 0.0 0.0 0.0
17.2 0.0 0.0 0.0 0.0
16.0 0.0 0.1 0.0 0.0
14.0 0.0 0.0 0.0 0.0
more noises and, therefore, prevents some details in the image. On the other hand, FAMF deteriorates the edges of the images so causes the absence of some details concerning with the edges. Comparing with MF and FAMF, the SMFAMF provides better image in the case of noise reduction and edge detection. MSE, RMSE, PSNR, and NMSE values of the filters are shown in Table 1. Median filter values were obtained for two different sampling in the pixels
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(a)
(b)
(c) Fig. 4. Comparison of SMFAMF: (a) MSE, (b) NMSE, (c) PSNR values prepared for MR-1 image.
such as 3 × 3 and 5 × 5 for comparison. As it is clearly seen from Table 1, MSE value is high in MF and FAMF, and is quite low in SMFAMF. On the other hand, it is clearly seen that PSNR ratio has low values in MF and FAMF filters, while it was found to be high in SMFAMF. Figure 4 shows graphics of the MSE, NMSE and PSNR values of no filter (NF), MF(3 × 3), MF(5 × 5), FAMF, and SMFAMF. It is obviously seen that SMFAMF filter has superior performance in the case of all statistical cases while the noise increases to 60%. 3.2. Experiment 2 In this experiment, noisy MR image having opaque material (MR-2) is obtained from original image as shown in Fig. 5. Same procedure as in experiment 1 was applied to this experiment. Table 2 shows the MSE, RMSE, PSNR, and NMSE values of the filters. Figure 6 shows the graphical forms of values given in Table 2. The results show that when the noise is increased PSNR values of the filters are increased. SMFAMF values, in all statistical cases, are in acceptable limits for noise reduction when the noise increased to 60% values.
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(a)
(b)
(d)
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(c)
(e)
Fig. 5. Noise reduction with SMFAMF: (a) original MR-2 image, (b) γ = 0.6 image with noise, (c) noise suppression with median filter, (d) noise suppression with FAMF, (e) noise suppression with SMFAMF.
Table 2 Comparison of MSE, RMSE, NMSE, and PSNR for SMFAMF, median (3 × 3), median (5 × 5) using MR-2 images (128 × 128) Filter
Noise ratio 20%
30%
40%
50%
60%
MSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
112.88 954.82 7371.4 22,864 764
194.43 2717.6 7414.3 22,964.0 1574
922.73 11,104.0 7374.8 25,110.0 4563
2298.7 24,060.0 7406.9 28,010.0 5673
4107.2 44,232.0 11,964.0 32,377.0 6783
RMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
10.624 30.9 85.857 151.21 27.64
13.944 52.131 86.106 151.54 39.67
30.376 105.37 85.877 158.46 67.55
47.945 155.11 86.064 167.36 75.32
64.087 210.31 109.38 179.94 82.36
PSNR
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
27.605 18.332 9.4553 4.5393 22.35
25.243 13.789 9.4301 4.5203 19.78
18.48 7.6762 9.4533 4.1324 13.95
14.516 4.3178 9.4344 3.6577 5.72
11.995 1.6734 7.3522 3.0284 5.67
NMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
0.108 0.9155 7.0678 21.922 29.784
0.18642 2.6057 7.1089 22.018 29.628
0.88473 10.646 7.0711 24.075 33.104
2.2041 23.069 7.1019 26.856 36.355
3.938 42.41 11.471 31.044 41.237
3.3. Experiment 3 Impulse noise is added to the MR image having brain tumor (MR-3) in different ratios so that the diseased MR image with 60% noise is obtained. Then, with the usage of same filtration techniques as in experiment 1, the results obtained with the suppression of impulse noise in the image are compared with the resulting values of SMFAMF filter.
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(b)
(c) Fig. 6. Comparison of SMFAMF: (a) MSE, (b) NMSE, (c) PSNR values prepared for MR-1 image.
These images are given in Fig. 7. Figure 7a shows original MR image, Fig. 7b shows original MR image having 60% impulse noise, Fig. 7c shows MF image output, Fig. 7d shows FAMF output, and Fig. 7e shows SMFAMF output. As is it clearly seen in Fig. 7, SMFAMF protects the details the best compared with other filters, while suppressing the impulse noise. MSE, RMSE, PSNR, and NMSE values are shown in Table 3. As it is understood from the table, while MSE value was quiet high with other filters, it is seen to be lower in SMFAMF. Nevertheless, it is clearly seen that while SNR ratio has low values in other filters, it was found be higher with SMFAMF. Besides, these values can be observed more clearly and graphically in Fig. 8. 4. Conclusion As a result of the study, we conducted that the noise suppression process which was performed using SMFAMF is more successful in preserving the details on the MR image while it suppresses the highly corrupted impulse noise. This can obviously be seen from Figs. 3, 5, and 7. This situation is especially important in medical imaging techniques so that we prefer to use MR images for processing. However, the image window we dealt with was of 3 × 3 dimensions
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(b)
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(c)
(e)
Fig. 7. Noise reduction with SMFAMF: (a) original MR-3 image, (b) γ = 0.6 image with noise, (c) noise suppression with median filter, (d) noise suppression with FAMF, (e) noise suppression with SMFAMF.
Table 3 Comparison of MSE, RMSE, NMSE, and PSNR for SMFAMF, median (3 × 3), median (5 × 5) using MR-3 images (128 × 128) Filter
Noise ratio 20%
30%
40%
50%
60%
MSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
1171.2 2005 2791.4 30,874 236.45
1510.5 3122.1 3162.4 31,364 365.32
2181 10,111 3599.3 32,675 1992.40
4494.8 25,373.0 6355.3 39,517.0 6895.76
6995.9 43,012.0 9422.4 43,389.0 8985.98
RMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
34.223 44.778 52.834 175.71 15.37
38.866 55.876 56.235 177.1 19.11
46.701 100.55 59.994 180.76 34.53
67.043 159.29 79.72 198.79 83.04
83.629 207.39 97.069 208.3 94.79
PSNR
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
17.444 15.11 13.673 3.2349 1.8558
16.339 13.186 13.131 3.1665 1.7438
14.744 8.0828 12.569 2.9887 1.6615
11.604 4.0871 10.099 2.163 0.7682
9.6836 1.7949 8.3892 1.757 0.67384
NMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
1.1323 1.9384 2.6986 29.847 41.003
1.4603 3.0183 3.0572 30.321 42.074
2.1084 9.7748 3.4796 31.588 42.878
4.3453 24.529 6.1439 38.203 52.671
6.7613 41.582 9.1091 41.946 53.828
and there were 9 input pixels and to be able to get a better image we have to define more rules. When we define 9 input pixels and 9 membership functions we are required to define 99 rules. Since this process is very difficult to perform and also very slow using with the computers, we defined only 758 rules. If it is possible to increase the number of rules and number of membership functions, then the image gets better but the processing of the image on
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(a)
(b)
(c) Fig. 8. Comparison of SMFAMF: (a) MSE, (b) NMSE, (c) PSNR values prepared for MR-1 image.
the computer becomes much slower. In our future studies, we will use the fuzzy logic rules in reduced number but will employ different membership functions in a dedicated fuzzification. Acknowledgment This work has been supported by Gazi University Scientific and Research Project Fund (Project No. 07/2006/-05). References [1] I.N. Bankman, Handbook of Medical Imaging, Academic Press, San Diego, CA, 2000. [2] E. Abreu, M. Lightstone, S.K. Mitra, K. Arakawa, A new efficient approach for the removal of impulse noise from highly corrupted images, IEEE Trans. Image Process. 5 (1992) 1012–1025. [3] J.H. Wang, L.D. Lin, M.D. Yu, Histogram-based adaptive neuro-fuzzy filter for image restoration, Proc. Natl. Sci. Counc. ROC (A) 21 (6) (1997) 556–572.
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[4] A. Rosenfeld, Fuzzy digital topology, Inform. Control 40 (1979) 76–87. [5] H. Kong, L. Guan, A noise-exclusive adaptive filtering framework for removing impulse noise in digital image, IEEE Trans. Circuits Syst. 45 (3) (1998) 422–428. [6] Y. Li, G.R. Arce, J. Bacca, Weighted median filters for multichannel signals, IEEE Trans. Signal Process. 51 (11) (2006) 4271–4281. [7] S. Peng, L. Lucke, Fuzzy filtering for mixed noise removal during image processing, in: Third IEEE Int. Conf. Fuzzy System, Orlando, FL, June 26–29, 1994, pp. 89–93. [8] K. Arakawa, Median filter based on fuzzy rules and its application to image restoration, Fuzzy Sets Syst. 77 (1996) 3–13. [9] C.S. Lee, Y.H. Kuo, P.T. Yu, Weighted fuzzy mean filters for image processing, Fuzzy Sets Syst. 89 (1997) 157–180. [10] S. Zhang, M.A. Karim, A new impulse detector for switching median filters, IEEE Signal Process. Lett. 9 (2002) 360–363. [11] R.C. Gonzalez, R.E. Woods, Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1992, pp. 241–243. [12] A. Toprak, ˙I. Güler, Suppression of impulse noise in medical images with the use of fuzzy adaptive median filter, J. Med. Syst. (2006), in press. [13] J.H. Wang, W.J. Liu, L.D. Lin, Histogram-based fuzzy filter for image restoration, IEEE Trans. Syst. Man Cybernet. B Cybernet. 32 (2) (2002) 230–238.
Abdullah Toprak was born in Diyarbakır, Turkey in 1972. He received the diploma in electronic engineering from Erciyes University, Kayseri, Turkey in 1993, the M.S. degree from Dicle University, Diyarbakır, Turkey in 2002, and the Ph.D. degree from Gazi University in 2006, both in electronic engineering. Since 1995, he has been a Lecturer at Dicle University and Chairman of electronic section. His researches have been focused on the modeling of fuzziness and uncertainty. His current research interests include fuzzy and fuzzy relations, fuzzy topology and fuzzy image processing, digital signal processing, content-based image/video indexing and retrieval, pattern recognition, clustering, biomedical signals and images processing.
˙ Inan Güler was born in Düzce, Turkey in 1956. He graduated from Erciyes University in 1981. He took the M.S. degree from Middle East Technical University in 1985 and the Ph.D. degree from ˙Istanbul Technical University in 1990, both in electronic engineering. He is a Professor at Gazi University where he is a Head of Department. His interest areas include biomedical instrumentation, biomedical signal processing, electronic instrumentation, neural networks, and artificial intelligence. He has written more than 150 articles related with his interest areas.
Impulse noise reduction in medical images with the use of switch mode fuzzy adaptive median filter Abdullah Toprak a , ˙Inan Güler b,∗ a Dicle University, Meslek Yüksek Okulu, Elektrik-Elektronik Bölümü, 21280 Diyarbakır, Turkey b Gazi University, Teknik E˘gitim Fakültesi, Elektronik-Bilgisayar Bölümü, 06500 Teknikokullar, Ankara, Turkey
Available online 15 December 2006
Abstract In this paper, a novel fuzzy adaptive median filter is presented for the noise reduction in MR images corrupted with heavy impulse (salt&pepper) noise. We propose a switch mode fuzzy adaptive median filter (SMFAMF) for removing highly corrupted salt&pepper noise without destroying edges and details in the image. The SMFAMF filter is an improved version of adaptive median filter (AMF) in order to reduce additive impulse noise in the images. The proposed filter can preserve details in the images better than AMF while suppressing additive salt&pepper or impulse type noises. In this paper, we placed our preference on bellshaped membership function with adaptive parameters instead of triangular membership function without variable coefficients in order to observe better results. Experiments with the magnetic resonance (MR) image from healthy subject, an MR image having the opaque material, and an MR image having disease demonstrate the mean square error (MSE), root mean square error (RMSE), signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR) of the proposed method. The results show that the proposed method can be useful for MR images with impulse type noises. © 2006 Elsevier Inc. All rights reserved. Keywords: Adaptive median filter; Fuzzy adaptive median filter; Impulse noise; Noise reduction
1. Introduction Medical images are often deteriorated by noise due to various sources of interference and other phenomena that affect the measurement processes in imaging and data acquisition systems. Median filtering is a common nonlinear method for noise suppression that has unique characteristics. It does not use convolution to process the image with a kernel of coefficients. Rather, in each position of the kernel frame, a pixel of the input image contained in the frame is selected to become the output pixel located at the coordinates of the kernel center. The kernel frame is centered on each pixel (m, n) of the original image, and the median value of the pixels within the kernel frame is computed. The pixel at the coordinates (m, n) of the output image is set to this median value [1]. Median filter (MF) is a 2D image filter that is more effective in situations where the impulse noise is less than 0.2 [2]. If this ratio exceeds 0.2, adaptive median filter (AMF) is used. As it is the case in the other filters, an Sxy window is selected for the AMF [3]. However, a feature that differentiates the AMF from the other filters is the fact * Corresponding author.
E-mail addresses: [email protected] (A. Toprak), [email protected] (˙I. Güler). 1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2006.11.008
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that the size of this window can be changed. Unfortunately, while AMF removes the impulse noise, it also deteriorates the details in the image it accompanies. In this study, in order to prevent the deterioration, the pixel values of the image are determined by the fuzzy adaptive median filter (FAMF) [4–6], and such FAMF with adaptive membership parameters is defined. This is called switch mode fuzzy adaptive median filter (SMFAMF). SMFAMF is made up of three basic sections. These are decision unit which is MF, fuzzy logic rules, and noise. The basic purpose of using this filter is to suppress the noise in the image, while it keeps the details in the image without deterioration. It is for this reason that SMFAMF is an effective filter for removing the impulse noise in MR images [7–10]. 2. Method and model 2.1. General definitions If we define X[(i,j )] image matrix p(k, l) to be made up of pixel values, a matrix of 3 × 3 will be in the form of 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 grey intensity. p(k, l) ∈ Ximp , if p(k, l) = min{W [k, l]} or max{W [k, l]}. The window W [(k, l)], that we will scan over the entire image, to clarify the noise from image, is a 3 × 3 matrix and this matrix is given as follows [11]: p(k − 1, l − 1) p(k − 1, 1) p(k − 1, l + 1) W (k, l) = . (1) p(k, l − 1) p(k, l) p(k, l + 1) p(k + 1, l − 1) p(k + 1, l) p(k + 1, l + 1) On the other hand, Ximp In a case as such ⎡ x11 x12 ⎢ x21 x22 X=⎣ ... ... xH1 xH2
matrix is a noise matrix that is mixed up with X[(i,j )] image matrix. ... ... ...
xij x2j ... xHj
... ... ...
⎤ x1W x2W ⎥ ⎦ = [xij ]H×W . ... xHW
(2)
Here H and W are height and width, respectively, and xij ∈ {0, 1, 2, . . . , 255} shows the grey intensity of pixel in i, j coordinate 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 input pixel. 2.2. Median filter This is a filter that makes possible for the elimination of a divergent value by changing the divergent value in a finite series with the medium value in the same series [9]. When it is of two dimensions, the MF for images would be developed as follows:
(3) m(k) = med w(k) = med x−n (k), . . . , x−1 (k), x0 (k), x1 (k), . . . , xn (k) . 2.3. Adaptive median filter MF is an image filter that can be more effective under conditions where the noise rate is less than 0.2. However, under conditions when the rate of noise exceeds 0.2 the AMF must be considered. Another advantage of AMF is making sure that the details on the screen are not lost while the noise is suppressed. Like the other filters, an Sxy
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window will be chosen for AMF. However, in AMF the size of this window can be changed even if the size of the window is changed the result of the operation conducted will be one and single, and this value can be changed with the central pixel value of the window. We take the notations below into consideration: Zmin = Sxy is the lowest grey level value inside. Zmax = Sxy is the highest grey level value inside. Zxy = (x, y) grey level at the subject matter coordinates. Zmed = is the maximum possible Sxy window size. We should analyze the flow diagram of AMF at two levels. Let us name them as levels A and B. Level A: A1 = Zmed − Zmin A2 = Zmed − Zmax If A1 > 0 and A2 < 0 then go to level B. If it is not the case, then increase the size of the window. If window size Smax , repeat level A. Level B: A1 = Zxy − Zmin A2 = Zxy − Zmax If B1 > 0 and B2 < 0, output Zxy . If not, output Zmed . From the flow diagram of AMF, it can be understood that it is an effective filter which is used in eliminating the impulse noise. The purpose of level A here is to determine whether the output value of the filter Zmed is in the noise or not. If Zmin < Zmed < Zmax then Zmed value is not a noise value and it must be transmitted to the exit. In a case like this, we have to proceed to level B. In level B, it is determined whether Zxy itself is a noise level and a new value will be determined according to this. Let us assume that the median value at level A equals to the noise, in a case like this the size of the window to be examined will be changed and another median value will be calculated. This process will be continued until the median value comes out different from the minimum or the maximum value. Nevertheless, it can never be guaranteed that the value obtained is not the noise. However, depending on the size of the window the probability of obtaining a noise value will be reduced. While the enlargement of the window suppresses the noise to a great extent, at the same time, in proportion to its size, the details on the image will be harmed [11]. 2.4. Switch mode fuzzy adaptive median filter with adaptive membership parameters First of all we need an image that is mixed with impulse noise, so that elimination of noise can be performed. For this, let us assume that the image has 128 × 128 pixels and that it has grey levels between 0 and 255. We can define the noise amount and uncertainty that is included in the image with the fuzzy logic variables. If we phrase it differently, SMFAMF is a system that can be formed by defining nine fuzzy logic membership functions and nine variables for each one and the fuzzy variables can be named as mf1, mf2, . . . , mf9. As such we will accept the intensity of input pixel p(x, y) as a fuzzy variable and express the membership levels of the fuzzy sets sequentially as follows: mf1 (blackest), mf2 (less black), . . . , mf8 (very white), and mf9 (whitest). However, in practice we are not required to limit the membership functions with nine, we can increase this number. For us to be able to conduct fuzzy image processing, first we have to give fuzziness to each pixel input intensity value and then normalize them to 0 p(x, y) 1 range. We will use a 3 × 3 image matrix for scanning the entire image. Here the filtered output will be changed with the central pixel of this matrix, and the noise in the image will be suppressed by using the fuzzy filter [12]. If we define Nimp matrix as the pixel values in noise form, then a 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, 9 pixels will be classified according to grey intensity. p(k, l) ∈ Nimp , if p(k, l) = min{W [k, l]} or max{W [k, l]}. In a case like
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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 input pixel. In Eq. (4) the bell-shaped membership function used for SMFAMF is given: pj (xi ) =
1 , 1 + ((xi − cj )/aj )2bj
i = 1, 2, . . . , 9, j = 1, 2, . . . , 9.
(4)
Here aj , xj , cj are the parameters that can be adjusted according to desired values [12]. For fuzzy meaning deduction rules that will be used in adaptive fuzzy median filter (AFMF), fundamental Mamdani method was utilized. The following are the basic rules. By adjusting the parameters in bell-shaped membership function and by applying Eq. (1) to the given sub image with nine pixels, the fuzziness was obtained. Choosing the value of xi is much higher than ai and ci makes possible for the membership function of impulse noise to be filtered properly. In the second step, normalized values are calculated for each pixel value by using the equation mj (xi ) wij = 9 , i=1 mj (xi )
i = 1, 2, . . . , 9, j = 1, 2, 3, . . . , 9.
(5)
The following rules are used to determine the noise in image having sampling window 3 × 3 in a pixel that is used in SMFAMF. 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) x4 = p(k, 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, / XVblack . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 2. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XBblack . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 3. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XLblack . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 4. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xgray . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 5. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xgray . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 6. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xgray . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 7. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XLwhite . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 8. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / Xwhite . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈ Rule 9. If x1 = p(k − 1, l − 1) ∈ mf(1) x2 = p(k − 1, l) ∈ mf(1) x3 = p(k − 1, l + 1) ∈ mf(1) x4 = p(k, 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, / XVwhite . mf(1) x9 = p(k + 1, l + 1) ∈ mf(1) then x5 = p(k, l) ∈
1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈ 1 − 1) ∈ l − 1) ∈
As mentioned, the bell-shape membership function parameters can be adjusted according to the rules above [13]. If x5 = p(k, l) ∈ / XVblack then a1 = 0.1, c1 = 0.2, b = 15. / Xblack then a1 = 0.2, c1 = 0.3, b = 15. If x5 = p(k, l) ∈
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If x5 = p(k, l) ∈ / XLblack then a1 = 0.2, c1 = 0.4, b = 15. / Xgray then a1 = 0.4, c1 = 0.5, b = 15. If x5 = p(k, l) ∈ / XLwhite then a1 = 0.2, c1 = 0.6, b = 15. If x5 = p(k, l) ∈ / Xwhite then a1 = 0.2, c1 = 0.7, b = 15. If x5 = p(k, l) ∈ / XVwhite then a1 = 0.1, c1 = 0.8, b = 15. If x5 = p(k, l) ∈ For fuzzy logic filter, the following methodology was followed. First, impulse noise was added at the rate of 60% to the image of 128 × 128 in size and 8 bit resolution (256 grey levels). The pixel values for the image with noise, in the size of 128 × 128, were written in sequential lines following each other, and such a column matrix was formed in the size of 1 × 16384. Second, the n − 1, n, and n + 1 series were obtained and the data in one column matrix form is converted into an input set of three columns. The purpose here is to be able to see the previous and next values of each pixel together. In the next stage logical relationships among these three column sets were established. This logical relationship was defined by using the mathematical expression given as ⎫ ⎧ Xmin < Xn−1 < Xmax , Xn−1 if not ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ Xmin < Xn < Xmax and Xmin < Xn+1 < Xmax , (Xn + Xn+1 )/2 ⎪ Xi = Xmin < Xn < Xmax , Xn if not . (6) ⎪ ⎪ ⎪ ⎪ < X < X , X if not X ⎪ ⎪ min n+1 max n+1 ⎪ ⎪ ⎭ ⎩ 0 One column data matrix that was obtained by using Eq. (5) was taken as the only input of the fuzzy system. As such the input file of the fuzzy system was prepared. On the other hand, output file was prepared in the form of a one column matrix similar to the 128 × 128 sized matrix of the original image. The membership functions of both the input and the output were 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 was obtained as the equation ⎡ (x − a)/(b − a), a x < b, UA (x) = UA (x; a, b, c) = ⎣ (c − x)/(c − b), b x < c, (7) − − 0, x > c or x < a. The weighting coefficients of the rules were determined to be 1. As a clarification method, centroid method was used. A part of the noise in the image was first suppressed with AMF, and then the rest of the noise (especially the noise that AMF has failed to suppress) was eliminated by using the fuzzy model. The flow chart of the model was designed as stated in Fig. 1. In this study, we placed our preference on bell-shaped membership function instead of triangular membership function in order to observe better results. We adjusted a, b, c parameters to get better result for removal noise in image. Membership function is illustrated in Fig. 2.
Fig. 1. The flow chart of SMFAMF model.
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Fig. 2. The bell-shaped membership function.
3. Results and discussion The reliability of the images which were used for this purpose can be evaluated by mean square error (MSE), root mean square error (RMSE), signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR) criteria. The MSE, NMSE, SNR, and PSNR are given in Eqs. (8)–(11), respectively. While SNR and PSNR values give the information on the quality of the image, MSE and NMSE values demonstrate the error values of same the images compared with the original one. The gray levels of the pixels in the image are taken as data. When the signal ratio of an image is high and the noise ratio is low, this image is considered to be good quality. Therefore, in the analysis of an image, SNR and PSNR values should absolutely be examined. Since the magnitude of this ratio is proportional to the quality of the image, the SNR of the original image and the image containing noise is determined. N M 2 i=1 j =1 F (i, j ) − G(i, j ) , (8) MSE = NM √ RMSE = MSE, (9) 255 , (10) PSNR = 20 log RMSE N M 2 i=1 j =1 F (i, j ) − G(i, j ) NMSE = . (11) N M 2 i=1 j =1 F (i, j ) In the calculation of normalized MSE value, values obtained using Eq. (8) were compared with other techniques. In this study, SMFAMF’s effects on MR images are examined using 3 types of MR images. These images are the most used three MR images of various types, especially the MR image of a healthy brain, MR image of a brain with tumor and MR image obtained by using an opaque substance. With the examination of the findings, SMFAMF’s superiority related to the other MR images according to few other filtration techniques is determined. They are evaluated comparison with MSE, RMSE, SNR and NMSE values of output images. 3.1. Experiment 1 Impulse noise is added to the healthy MR (MR-1) image in different ratios and healthy MR image with 60% noise (γ = 0.6) is obtained. Then, with the use of MF, FAMF, and SMFAMF techniques, the results of impulse noise suppression in the images are compared with each other as shown in Fig. 3. Figure 3a shows the original MR image, Fig. 3b shows 60% impulse noise added to the image, Fig. 3c shows the noise suppression with MF, Fig. 3d shows the noise suppression with FAMF, and Fig. 3e shows the noise suppression with SMFAMF. As is it clearly seen in Fig. 3, SMFAMF protects the details best compared with the other filters while suppressing the impulse noise. MF contains
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Fig. 3. Noise reduction with SMFAMF: (a) original MR-1 image, (b) γ = 0.6 image with noise, (c) noise suppression with median filter, (d) noise suppression with FAMF, (e) noise suppression with SMFAMF. Table 1 Comparison of MSE, RMSE, NMSE, and PSNR for SMFAMF, median (3 × 3), median (5 × 5) using MR-1 images (128 × 128) Filter
Noise ratio 20%
30%
40%
50%
60%
MSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
36.9 1645.28 1361.4 453.4 284
87.3 3824.54 2460.9 580.8 356
188.2 8246.25 4772.5 1077.5 659
454.8 11,258.31 7828.9 2309.7 883
821.0 14,324.36 10,648.0 4153.0 1298
RMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
6.1 40.56 36.9 21.3 16.9
9.3 61.84 49.6 24.1 18.9
13.7 90.80 69.1 32.8 25.7
21.3 106.10 88.5 48.1 29.7
29 119.68 103.0 64.0 36.0
PSNR
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
33.5 32.5 16.8 21.6 27.43
31.5 28.70 14.2 20.5 24.30
35.20 25.4 11.3 17.8 23.34
40.5 21.6 9.2 14.5 20.21
50.0 19.0 8.0 12.0 18.23
NMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
17.6 3.E−07 1.1E−5 3.6E−6 5.4E−6
18.2 0.0 0.0 0.0 0.0
17.2 0.0 0.0 0.0 0.0
16.0 0.0 0.1 0.0 0.0
14.0 0.0 0.0 0.0 0.0
more noises and, therefore, prevents some details in the image. On the other hand, FAMF deteriorates the edges of the images so causes the absence of some details concerning with the edges. Comparing with MF and FAMF, the SMFAMF provides better image in the case of noise reduction and edge detection. MSE, RMSE, PSNR, and NMSE values of the filters are shown in Table 1. Median filter values were obtained for two different sampling in the pixels
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(a)
(b)
(c) Fig. 4. Comparison of SMFAMF: (a) MSE, (b) NMSE, (c) PSNR values prepared for MR-1 image.
such as 3 × 3 and 5 × 5 for comparison. As it is clearly seen from Table 1, MSE value is high in MF and FAMF, and is quite low in SMFAMF. On the other hand, it is clearly seen that PSNR ratio has low values in MF and FAMF filters, while it was found to be high in SMFAMF. Figure 4 shows graphics of the MSE, NMSE and PSNR values of no filter (NF), MF(3 × 3), MF(5 × 5), FAMF, and SMFAMF. It is obviously seen that SMFAMF filter has superior performance in the case of all statistical cases while the noise increases to 60%. 3.2. Experiment 2 In this experiment, noisy MR image having opaque material (MR-2) is obtained from original image as shown in Fig. 5. Same procedure as in experiment 1 was applied to this experiment. Table 2 shows the MSE, RMSE, PSNR, and NMSE values of the filters. Figure 6 shows the graphical forms of values given in Table 2. The results show that when the noise is increased PSNR values of the filters are increased. SMFAMF values, in all statistical cases, are in acceptable limits for noise reduction when the noise increased to 60% values.
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(b)
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Fig. 5. Noise reduction with SMFAMF: (a) original MR-2 image, (b) γ = 0.6 image with noise, (c) noise suppression with median filter, (d) noise suppression with FAMF, (e) noise suppression with SMFAMF.
Table 2 Comparison of MSE, RMSE, NMSE, and PSNR for SMFAMF, median (3 × 3), median (5 × 5) using MR-2 images (128 × 128) Filter
Noise ratio 20%
30%
40%
50%
60%
MSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
112.88 954.82 7371.4 22,864 764
194.43 2717.6 7414.3 22,964.0 1574
922.73 11,104.0 7374.8 25,110.0 4563
2298.7 24,060.0 7406.9 28,010.0 5673
4107.2 44,232.0 11,964.0 32,377.0 6783
RMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
10.624 30.9 85.857 151.21 27.64
13.944 52.131 86.106 151.54 39.67
30.376 105.37 85.877 158.46 67.55
47.945 155.11 86.064 167.36 75.32
64.087 210.31 109.38 179.94 82.36
PSNR
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
27.605 18.332 9.4553 4.5393 22.35
25.243 13.789 9.4301 4.5203 19.78
18.48 7.6762 9.4533 4.1324 13.95
14.516 4.3178 9.4344 3.6577 5.72
11.995 1.6734 7.3522 3.0284 5.67
NMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
0.108 0.9155 7.0678 21.922 29.784
0.18642 2.6057 7.1089 22.018 29.628
0.88473 10.646 7.0711 24.075 33.104
2.2041 23.069 7.1019 26.856 36.355
3.938 42.41 11.471 31.044 41.237
3.3. Experiment 3 Impulse noise is added to the MR image having brain tumor (MR-3) in different ratios so that the diseased MR image with 60% noise is obtained. Then, with the usage of same filtration techniques as in experiment 1, the results obtained with the suppression of impulse noise in the image are compared with the resulting values of SMFAMF filter.
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(a)
(b)
(c) Fig. 6. Comparison of SMFAMF: (a) MSE, (b) NMSE, (c) PSNR values prepared for MR-1 image.
These images are given in Fig. 7. Figure 7a shows original MR image, Fig. 7b shows original MR image having 60% impulse noise, Fig. 7c shows MF image output, Fig. 7d shows FAMF output, and Fig. 7e shows SMFAMF output. As is it clearly seen in Fig. 7, SMFAMF protects the details the best compared with other filters, while suppressing the impulse noise. MSE, RMSE, PSNR, and NMSE values are shown in Table 3. As it is understood from the table, while MSE value was quiet high with other filters, it is seen to be lower in SMFAMF. Nevertheless, it is clearly seen that while SNR ratio has low values in other filters, it was found be higher with SMFAMF. Besides, these values can be observed more clearly and graphically in Fig. 8. 4. Conclusion As a result of the study, we conducted that the noise suppression process which was performed using SMFAMF is more successful in preserving the details on the MR image while it suppresses the highly corrupted impulse noise. This can obviously be seen from Figs. 3, 5, and 7. This situation is especially important in medical imaging techniques so that we prefer to use MR images for processing. However, the image window we dealt with was of 3 × 3 dimensions
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Fig. 7. Noise reduction with SMFAMF: (a) original MR-3 image, (b) γ = 0.6 image with noise, (c) noise suppression with median filter, (d) noise suppression with FAMF, (e) noise suppression with SMFAMF.
Table 3 Comparison of MSE, RMSE, NMSE, and PSNR for SMFAMF, median (3 × 3), median (5 × 5) using MR-3 images (128 × 128) Filter
Noise ratio 20%
30%
40%
50%
60%
MSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
1171.2 2005 2791.4 30,874 236.45
1510.5 3122.1 3162.4 31,364 365.32
2181 10,111 3599.3 32,675 1992.40
4494.8 25,373.0 6355.3 39,517.0 6895.76
6995.9 43,012.0 9422.4 43,389.0 8985.98
RMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
34.223 44.778 52.834 175.71 15.37
38.866 55.876 56.235 177.1 19.11
46.701 100.55 59.994 180.76 34.53
67.043 159.29 79.72 198.79 83.04
83.629 207.39 97.069 208.3 94.79
PSNR
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
17.444 15.11 13.673 3.2349 1.8558
16.339 13.186 13.131 3.1665 1.7438
14.744 8.0828 12.569 2.9887 1.6615
11.604 4.0871 10.099 2.163 0.7682
9.6836 1.7949 8.3892 1.757 0.67384
NMSE
SMFAMF No filter Median (3 × 3) Median (5 × 5) FMF
1.1323 1.9384 2.6986 29.847 41.003
1.4603 3.0183 3.0572 30.321 42.074
2.1084 9.7748 3.4796 31.588 42.878
4.3453 24.529 6.1439 38.203 52.671
6.7613 41.582 9.1091 41.946 53.828
and there were 9 input pixels and to be able to get a better image we have to define more rules. When we define 9 input pixels and 9 membership functions we are required to define 99 rules. Since this process is very difficult to perform and also very slow using with the computers, we defined only 758 rules. If it is possible to increase the number of rules and number of membership functions, then the image gets better but the processing of the image on
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(c) Fig. 8. Comparison of SMFAMF: (a) MSE, (b) NMSE, (c) PSNR values prepared for MR-1 image.
the computer becomes much slower. In our future studies, we will use the fuzzy logic rules in reduced number but will employ different membership functions in a dedicated fuzzification. Acknowledgment This work has been supported by Gazi University Scientific and Research Project Fund (Project No. 07/2006/-05). References [1] I.N. Bankman, Handbook of Medical Imaging, Academic Press, San Diego, CA, 2000. [2] E. Abreu, M. Lightstone, S.K. Mitra, K. Arakawa, A new efficient approach for the removal of impulse noise from highly corrupted images, IEEE Trans. Image Process. 5 (1992) 1012–1025. [3] J.H. Wang, L.D. Lin, M.D. Yu, Histogram-based adaptive neuro-fuzzy filter for image restoration, Proc. Natl. Sci. Counc. ROC (A) 21 (6) (1997) 556–572.
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[4] A. Rosenfeld, Fuzzy digital topology, Inform. Control 40 (1979) 76–87. [5] H. Kong, L. Guan, A noise-exclusive adaptive filtering framework for removing impulse noise in digital image, IEEE Trans. Circuits Syst. 45 (3) (1998) 422–428. [6] Y. Li, G.R. Arce, J. Bacca, Weighted median filters for multichannel signals, IEEE Trans. Signal Process. 51 (11) (2006) 4271–4281. [7] S. Peng, L. Lucke, Fuzzy filtering for mixed noise removal during image processing, in: Third IEEE Int. Conf. Fuzzy System, Orlando, FL, June 26–29, 1994, pp. 89–93. [8] K. Arakawa, Median filter based on fuzzy rules and its application to image restoration, Fuzzy Sets Syst. 77 (1996) 3–13. [9] C.S. Lee, Y.H. Kuo, P.T. Yu, Weighted fuzzy mean filters for image processing, Fuzzy Sets Syst. 89 (1997) 157–180. [10] S. Zhang, M.A. Karim, A new impulse detector for switching median filters, IEEE Signal Process. Lett. 9 (2002) 360–363. [11] R.C. Gonzalez, R.E. Woods, Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1992, pp. 241–243. [12] A. Toprak, ˙I. Güler, Suppression of impulse noise in medical images with the use of fuzzy adaptive median filter, J. Med. Syst. (2006), in press. [13] J.H. Wang, W.J. Liu, L.D. Lin, Histogram-based fuzzy filter for image restoration, IEEE Trans. Syst. Man Cybernet. B Cybernet. 32 (2) (2002) 230–238.
Abdullah Toprak was born in Diyarbakır, Turkey in 1972. He received the diploma in electronic engineering from Erciyes University, Kayseri, Turkey in 1993, the M.S. degree from Dicle University, Diyarbakır, Turkey in 2002, and the Ph.D. degree from Gazi University in 2006, both in electronic engineering. Since 1995, he has been a Lecturer at Dicle University and Chairman of electronic section. His researches have been focused on the modeling of fuzziness and uncertainty. His current research interests include fuzzy and fuzzy relations, fuzzy topology and fuzzy image processing, digital signal processing, content-based image/video indexing and retrieval, pattern recognition, clustering, biomedical signals and images processing.
˙ Inan Güler was born in Düzce, Turkey in 1956. He graduated from Erciyes University in 1981. He took the M.S. degree from Middle East Technical University in 1985 and the Ph.D. degree from ˙Istanbul Technical University in 1990, both in electronic engineering. He is a Professor at Gazi University where he is a Head of Department. His interest areas include biomedical instrumentation, biomedical signal processing, electronic instrumentation, neural networks, and artificial intelligence. He has written more than 150 articles related with his interest areas.