Automatic wavelet base selection and its application to contrast enhancement

ARTICLE IN PRESS Signal Processing 90 (2010) 1279–1289

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Automatic wavelet base selection and its application to contrast enhancement H.D. Cheng , Rui Min, Ming Zhang Department of Computer Science, Utah State University, Logan, UT 84322-4205, USA

a r t i c l e in fo

abstract

Article history: Received 25 November 2008 Received in revised form 16 July 2009 Accepted 19 October 2009 Available online 1 November 2009

In this paper, we propose a novel approach to automatic selecting wavelet bases and parameters which is an important and essential issue for implementing wavelet algorithms. The proposed approach is applied to contrast enhancement which is one of the fundamental topics in image processing, pattern recognition and computer vision. Our method utilizes wavelet transform to decompose the image, and then modifies the coefficients by employing the proposed method to perform contrast enhancement. Experimental results demonstrate that the proposed method is very efficient in contrast enhancement without under-enhancement and over-enhancement, and it is superior to some other existing methods. & 2009 Elsevier B.V. All rights reserved.

Keywords: Wavelet transform Wavelet basis Image enhancement Over-enhancement Under-enhancement

1. Introduction Contrast enhancement is one of the most important tasks in image processing, pattern recognition and computer vision, which can sharpen the edges of objects, and attain more information from the enhanced image. Contrast enhancement methods can be categorized as either indirect or direct approaches [1]. The indirect approaches change intensity values of image. Histogram equalization and histogram specification are two commonly used indirect algorithms [2]. They re-distribute the intensities to enhance the image, and only use the global information. If there is a lot of texture information, indirect approaches cannot work well. Local histogram processing was proposed [2,3], and the main idea is to modify histogram distribution over a local window. There is a major drawback of local histogram processing since local regions may be so outstanding that the entire image loses its uniform hierarchy, whereas in direct

 Corresponding author.

E-mail address: [email protected] (H.D. Cheng). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.10.013

methods, a function of the contrast is defined, and the contrast values of objects are computed. These values are modified, the intensities of the objects are updated, and the contrast enhancement completes. Some definitions of contrast are summarized in [4]. Recently, fuzzy technologies, such as fuzzy neural networks [5], and fuzzy edge detection approach [6,7] were applied to contrast enhancement. In general, direct methods have better performance than that of indirect methods. However, some of the direct methods still suffer noise amplification, under-enhancement and over-enhancement which should not be generated by an ideal contrast enhancement algorithm. Wavelet transformation has been widely applied in image processing and pattern recognition [8,9]. Some contrast enhancement methods based on wavelet transform were developed for medical image analysis [10,11], fingerprint analysis [12], etc. In this paper, we propose a novel approach to automatically select wavelet bases which is an essential and critical issue for wavelet algorithm implementation. Then, we apply the proposed approach for contrast enhancement to demonstrate its effectiveness and usefulness.

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2. Discrete wavelet transforms After transforming signal to frequency domain using Fourier transform, the space/time information is totally lost. Wavelets are localized in both time and frequency, whereas the Fourier transform is only localized in frequency [13]. Wavelet transforms are classified into continuous wavelet transforms (CWTs) [14] and discrete wavelet transforms (DWTs) [15]. In this paper, the proposed algorithm is based on discrete wavelet transform. 2.1. Fast wavelet transform (FWT) FWT is a computationally efficient implementation of DWT [16]. If scaling vector, h, and wavelet vector, g, are known, scaling and wavelet functions can be derived [16,17], X pffiffiffi fð2j x  kÞ ¼ hðm  2kÞ 2fð2j þ 1 x  mÞ ð1aÞ m

X pffiffiffi cð2 x  kÞ ¼ gðm  2kÞ 2fð2j þ 1 x  mÞ j

ð1bÞ

m

from scale j+1 to scale j. Here c is the ‘‘mother wavelet’’ R satisfying cðtÞ dt ¼ 0, and f is the scaling function of the chosen wavelet transform. Scaling and wavelet coefficients can be computed as below, X hðm  2kÞWf ðj þ1; mÞ ð2aÞ Wf ðj; kÞ ¼ m

Wc ðj; kÞ ¼

X

gðm  2kÞWc ðj þ 1; mÞ

ð2bÞ

m

After wavelet transform is applied to an image, the V H D above four functions fðx; yÞ, c ðx; yÞ, c ðx; yÞ, and c ðx; yÞ will be generated. They are denoted by functions A, V, H, and D, respectively. Function A is the trend of the image, and functions V, H, and D measure the fluctuations along the horizontal, vertical and diagonal directions, respectively. Two-dimensional fast wavelet transform (FWT2) [16,18] is computationally efficient. FWT2 decomposing an image (m  n) from level j+ 1 to level j is shown in Fig. 1 [16,17]. Two-dimensional inverse fast wavelet transform (IFWT2) is an inverse process of FWT2 which uses the scaling and wavelet vectors that are exactly the same as those of FWT2. Fig. 2 illustrates the process of reconstructing an image using IFWT2 [16,17]. 2.3. Orthogonality and compact support For image processing, we should consider two kinds of DWTs: the orthogonal wavelet and the compact supported wavelet. The orthogonality can provide the convenience in computations. A wavelet filter with compact support is non-zero only in a finite interval and the compact support decides the filter width. A wavelet vector fgl : l ¼ 0; . . . ; L  1g with length L should have three basic properties [19], L1 X

L1 X

Scaling and wavelet coefficients can be written [16,17], Wf ðj; kÞ ¼ hðnÞ  Wf ðj þ 1; nÞjn ¼ 2k;kZ0

ð3aÞ

Wc ðj; kÞ ¼ gðnÞ  Wc ðj þ1; nÞjn ¼ 2k;kZ0

ð3bÞ

where h(n) is the time-reversed vector of h(n), and g(n) is the time-reversed vector of g(n). Eqs. (3a) and (3b) indicate that the wavelet coefficients at scale j can be computed by the convolving of the wavelet coefficients at scale j+1 and down-sampling by 2 of the indices with the time reversed wavelet vector.

gl ¼ 0

ð6aÞ

gl2 ¼ 1

ð6bÞ

l¼0

l¼0 L1 X l¼0

gl gl þ 2n ¼

1 X

gl gl þ 2n ¼ 0

ð6cÞ

l ¼ 1

For all nonzero integer n, g0 a0 and gL1 a0. Fig. 3 shows an image and its Haar wavelet transform which is a kind of wavelet with both orthogonality and compact support. 2.4. Other considerations for implementing DWT

2.2. DWT for image processing Multi-resolution analysis is one of the main features in DWT, i.e., signal features are presented at different scales or resolutions. In other words, it is easy to characterize the gross features in a large ‘‘window’’ and the local features in a small ‘‘window.’’ Images are two-dimensional, and wavelet transform should be extended to two-dimensions. At a certain level of a wavelet transform, a two-dimensional basis consists of a two-dimensional scaling function [16,17],

fðx; yÞ ¼ fðxÞfðyÞ

ð4Þ

and three two-dimensional wavelet functions,

cV ðx; yÞ ¼ fðxÞcðyÞ cH ðx; yÞ ¼ cðxÞfðyÞ cD ðx; yÞ ¼ cðxÞcðyÞ

ð5Þ

We should consider three major issues for implementing DWT. (1) How to select suitable wavelet: The choice of wavelet should depend on the image nature. [19] discussed two considerations: (i) if the width of the selected wavelet is too short, some undesirable artifacts, such as unrealistic block effects, may be introduced; and (ii) if the width of the selected wavelet is too large, more wavelet coefficients are affected by the boundaries, features in local area are blurred, and longer computational time is needed. In our experiment, we need keep the local area distinguish and we choose the wavelet with the smallest length among that of all applicable wavelets. (2) How to handle boundaries: Generally, the sampled data are treated as circular. However, if DWT uses circularity, heavy boundaries may be generated in H, V,

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h (-m) h (-n)

2 Columns (along n)

g (-m)

Aj+1 (m,n) h (-m) g (-n)

2 Rows (along m) 2 Rows 2

Aj (m,n)

Vj (m,n)

Hj (m,n)

Rows

2 Columns

1281

g (-m)

2

Dj (m,n)

Rows Fig. 1. Decomposition of the two-dimensional fast wavelet transform. 2k means down sampling.

Aj (m,n)

2

h (m) Rows (along m)

Vj (m,n)

2

g (m)

2

h (n)

Columns (along n) Aj+1 (m,n)

Rows Hj (m,n)

2

h (m) Rows

Dj (m,n)

2

g (m)

2

g (n)

Columns

Rows Fig. 2. Reconstruction of the two-dimensional inverse fast wavelet transform. 2m means up sampling.

and D. How to handle the coefficients affected by the boundaries will be discussed in Section 3.4. (3) How to handle image sizes not fitting a specific wavelet: If an n-level wavelet is employed, the image width and height should be a multiple of 2n; otherwise, the image should be filled by certain values. In our experiments, the rightmost and bottommost lines are duplicated to meet such requirement. 3. Proposed method 3.1. The algorithm There are three major steps in the proposed image enhancement algorithm, and they use both global and local information [20]: 1. choose a specific wavelet transform to convert the image and obtain wavelet coefficients; 2. use a filter to adjust the wavelet coefficients; and 3. apply the inverse wavelet transform to map the result to the space domain. The flowchart of the algorithm is shown in Fig. 4.

Wavelet transform utilizes local information to decompose an image, and the maximum absolute value of each part of the coefficients is used for adjusting the coefficients, i.e., the global information is also employed.

3.2. Selection of wavelets Several wavelet transforms have been developed, and the selection of suitable DWT is very critical for implementing wavelet algorithms. All available wavelets in our experiments were able to perform contrast enhancement at a certain degree; however, the results varied. In addition, the proposed algorithm can enhance images with different resolutions. The shift variance, regularity and number of vanishing moments will affect the wavelet selection [21]. Table 1 lists commonly used compact support DWTs with orthogonal bases: Haar, Daubechies, Coiflets, Symlets and discrete Meyer wavelets. Totally, we considered 66 wavelets. In the experiments, a wavelet database of the commonly used wavelets was developed, as presented in Table 1. Wavelets in the database were applied to the P images one by one, the coefficients of Hð x2H jxjÞ and

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Vð

H.D. Cheng et al. / Signal Processing 90 (2010) 1279–1289

P

x2V jxjÞ

P x¼

were computed, and x is defined as

x2H jxjþ

P

x2V jxj

3.3. Design of filter After the ‘‘optimal’’ wavelet is selected, four parts A, H, V, and D at each level will be extracted. As usual, we leave trend coefficients (A) unchanged and only change fluctuation coefficients (H, V, and D). For each part (H, V, and D) in the decomposition at level j, we do the following: Step 1. Normalize the coefficients.

ð7Þ

2mn

Here, x are the wavelet coefficients, and m  n is the size of H and V. According to Eqs. (4) and (5), the coefficients of H and V are the measurements of the horizontal and vertical edges. If wavelet window size in horizontal direction is exactly matched with the section of objects P in this direction, x2H jxj is 0 according to Eq. (6a), conclusion similarly holds in the vertical direction, P x2V jxj ¼ 0. Therefore, the wavelet with the smallest x is the optimal wavelet which best matches the objects.

xjnor ¼ xjold =MM ¼ maxfjxjold jg

ð8Þ

where xold is original value at level j. Step 2. The following function is employed for adjusting the normalized coefficients: ( j ðjxnor jP  T 1P Þ signðxjnor Þ 0rjxjnor jrT j xadj ¼ j j ð1  ð1  jxnor jÞP ð1  TÞ1P Þ signðxnor Þ T ojxjnor jr1 ð9Þ where P 2 ð1; 1Þ and T 2 ½0; 1, are the parameters of the proposed algorithm, and 8 > < 1 x 40 0 x¼0 signðxÞ ¼ > : 1 x o 0 Step 3. Compute enhanced coefficients using the following formula: xjenh ¼ xjadj  M

ð10Þ

The curve of the function in step 2 is shown in Fig. 5, where T=0.18 and P= 2.6. The regions whose wavelet coefficients are less than threshold T are de-enhanced, while other regions are enhanced. A larger P will make the image sharper; however, if P is too large, the image may be over-enhanced. In Section 3.3.2, we will discuss how to select P automatically. 3.3.1. Selection of T The selection of T depends on the wavelet function and the noise level in the original image. As mentioned above, after a wavelet transform is applied to an image, noise is more distinguishable in wavelet coefficient D than other subbands. Since the threshold value T is mainly for

Table 1 Wavelet database.

Fig. 3. (a) The original image. (b) Four parts after wavelet transform. (c) The decomposition of Haar wavelet.

Original Image

Wavelet Transform

Wavelet Coefficients

Filter

Wavelet type

Parameter

Number of wavelets

Haar Daubechies Coiflets Symlets discrete Meyer

n/a N, N = 1, 2,y, 30 N, N = 1, 2, 3, 4, 5 N, N = 2, 3, y, 30 n/a

1 30 5 29 1

Adjusted Coefficients

Fig. 4. Scheme of the proposed algorithm.

Wavelet Transform

Enhanced Image

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Enhanced Values Original Values

-1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Adjusted Coefficients

Coefficient Adjustment Curve (T = 0.18, P = 2.6) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1

Normilized Coefficients

Fig. 6. Regression surface of P on ln x and ln T based on 30 observations. Each open dot represents one observation. The shaded surface represents the regression surface.

Fig. 5. Coefficient adjustment curve.

from boundaries should not be utilized for computing the maximum value and the average value in Eqs. (7) and (11), respectively. In the experiments, wrap-around approach is applied.

Table 2 The parameters for the images. Parameter estimates Variable

DF

Estimate

Error

t value

Pr 4|t|

4. Experimental results and discussions

Intercept ln x ln T

1 1 1

9.97450 0.35483 2.18888

0.39973 0.08148 0.09715

24.95 4.36 22.53

o .0001 0.0002 o .0001

4.1. Wavelet bases and parameters

eliminating noise, T could be computed as follows: P x2D jxj T¼ m  n  maxx2D ðjxjÞ

ð11Þ

where m  n is the size of D. 3.3.2. Selection of P Parameter P decides the enhancement strength. Because P is an exponential value, a standard linear model with two parameters is developed for estimating the value of P. P ¼ b0 þ b1 ln x þ b2 ln T

ð12Þ

where b0, b1, and b2 are constants. We can easily observe that P should be 41 for enhancement from Eq. (9). Therefore, if the estimate of P is r1, the image is deenhanced or not enhanced. We estimate the coefficients of Eq. (12) using 30 images. The regression results from SAS (statistics software) are presented in Table 2. All estimates are significant at 1% significance level. The regression surface is plotted in Fig. 6. 3.4. Handling boundaries Because wavelet boundary may affect experimental results, boundaries should be handled carefully. If zeropadding or wrap-around is employed, the coefficients

The wavelet database consists of 66 wavelets, listed in Table 1. For a testing image, x is computed for each wavelet in the database, and the wavelet with the lowest x is chosen. This process is quite fast, which only takes 4.9 s for an image with size of 256  384. The FWT [16,18] is implemented by Visual C++ running on Windows XP with Pentium IV 2.4 GHz. According to the experiments, b0 =9.97450, b1 = 0.35483, and b2 = 2.18888 for Eq. (9), 95.71% of the testing data can be estimated by the model. Actually, the algorithm is very robust since small variations of threshold T and value P will not affect the enhanced results much. Fig. 7(a) is the image using the estimated P with the largest difference from the best desired P. The value P of Fig. 7(b) is decided by using Eq. (12) of the model while that of Fig. 7(c) is the desired (best) value which is manually decided. As can be observed, Figs. 7(b) and (c) are quite close. 4.2. Experimental results We apply the proposed method to Figs. 7–11 and list the parameters in Table 3. Fig. 7(a) is a low contrast and vague image. In Fig. 7(b), the texture of the roads and the windows on the buildings become more distinct. Especially in Fig. 7(a), one can hardly recognize the objects which are on the right side of the buildings and the objects in the shadows. However, the objects are quite clear in Fig. 7(b). Fig. 8(a) is relatively blurry and dark. The detailed woodwork was sharpened

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Fig. 7. (a) The original image with size 256  256 (city). (b) The image enhanced by the proposed algorithm with Symlets 24 wavelet, P = 4.938 and T= 0.077. (c) The image enhanced by the proposed algorithm with Symlets 24 wavelet, P =4.100 and T= 0.077.

Fig. 8. (a) The original image with size 256  256 (house). (b) The image enhanced by the proposed algorithm with Symlets 4 wavelet, P= 4.695 and T= 0.081.

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Fig. 9. (a) The original image with size 256  256 (bridge). (b) The image enhanced by the proposed algorithm with Symlets 12 wavelet, P= 5.988 and T= 0.111.

Fig. 10. (a) The original image with size 256  256 (panda). (b) The image enhanced by HE method. (c) The image enhanced by contourlet method. (d) The image enhanced by the proposed algorithm with Symlets 20 wavelet, P= 4.637 and T= 0.061.

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Fig. 11. (a) The original image with size 256  256 (eagle). (b) The image enhanced by HE method. (c) The image enhanced by contourlet method (d) The image enhanced by the proposed algorithm with Symlets 9 wavelet, P =4.757 and T= 0.054. Table 3 The parameters determined by the automated process. Image

Size

Wavelet

Fig. Fig. Fig. Fig. Fig.

256  256 256  256 256  256 256  256 256  256

Symlets Symlets Symlets Symlets Symlets

7 (city) 8 (house) 9 (bridge) 10 (panda) 11 (eagle)

24 4 12 20 9

P

T

4.938 4.695 5.988 4.637 4.757

0.077 0.081 0.111 0.061 0.054

considerably in Fig. 8(b). Even the shrubs in front of the house become much clearer in Fig. 8(b). In Fig. 9(b), the pavement of the bridge and the chain on the right side of the bridge are much clear. The ripples of water, the leaves of the background trees, and the branches of the bushes in the front are much more distinct in Fig. 9(b). 4.3. Comparison with other enhancement methods The proposed method is first compared with histogram equalization (HE) [15] and contourlet enhancement [22]

in Figs. 10 and 11. Then it is compared with three unsharp masking (UM) methods: cubic UM [23,24], statistics UM [25], and rational UM [26,27] in Figs. 12 and 13. Histogram equalization is a commonly used contrast enhancement method. It maps the distribution of image intensities to the uniform distribution which spreads the grayscale values of given image to a larger range of the grayscale values [15]. Contourlet forms a multiresolution directional tight frame to efficiently approximate images and makes smooth regions separated by smooth boundaries [22]. The contourlet transform is based on a Laplacian pyramid decomposition followed by directional filterbanks applied to each bandpass subband [28]. It can be applied to denoising, enhancement, and contour detection [29]. The unsharp masking (UM) technology is a widely used method of contrast enhancement [15]. The fundamental idea is subtract from the input signal to a lowpass filtered version of the signal and then to add the input signal with a processed version of the signal in which highfrequency components are enhanced [30]. Because the UM method has two principle shortcomings (sensitive to

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Fig. 12. (a) The original image with size 256  256 (boats). (b) The image enhanced by the rational UM method. (c) The image enhanced by the proposed algorithm.

noise and over-enhancing high-contrast areas), many algorithms have been created for overcoming the drawbacks [26,27]. Two examples are the cubic UM [23,24] and the order statistics UM [25]. Authors proposed an adaptive UM method in [27], and authors improved the method in [26,27], called ‘‘rational unsharp masking’’ technique. Fig. 10(a) is a panda image. Edges of bamboo are vague. Fig. 10(b) is the result after applying histogram equalization. The face of panda and leaves of bamboo are overenhanced. Fig. 10(c) is the result of using contourlet enhancement method [29]. The entire image looks unnatural, the face of panda gets dark which is obviously over-enhanced. Fig. 10(d) is the result using the proposed method, the edges of bamboo are distinct and hair of the panda is clear. Fig. 11(a) is the image of two eagles. Histogram equalization in Fig. 11(b) makes the wings over-enhanced. Contourlet method in Fig. 11(c) enhances the background and causes the background nonuniform. The proposed

method in Fig. 11(d) makes wings of the eagles much clearer than that in Fig. 11(a). Figs. 12(a) and (b) are the original image of ‘‘boats’’ and the enhanced image in [26] by using rational UM, respectively. Fig. 12(c) is the result using the proposed algorithm. Comparing Figs. 12(a) and (b), it is easy to see that the algorithm in [26] over-enhanced the original image. There are many white lines around the backstays, especially around the backstays against clouds. This kind of problems exists as long as a black edge of an object is adjacent to a lighter background. Further, the body of the tower is not uniform, especially on the sides. The waves of the water are also over-enhanced. A relatively large white patch on the life preserver can be noticed. In the foreground, white blocks appear in Fig. 12(b), and these problems do not exist in Fig. 12(c). Figs. 13(a)–(e) are the original image of ‘‘lena,’’ the image enhanced by the conventional UM, the cubic UM [26], the order statistics UM method [26], and the rational UM algorithm [26], respectively. Fig. 13(e) has the best

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Fig. 13. (a) The original image with size of 256  256 (lena). (b) The image enhanced by the conventional UM method. (c) The image enhanced by the cubic UM method. (d) The image enhanced by the order statistics UM method. (e) The image enhanced by the rational UM method. (f) The image enhanced by the proposed algorithm.

result among the five enhanced images by applying the rational UM algorithm with different parameters in [26]; however, it was also obviously over-enhanced. It has the

problems similar to those in Fig. 12(b). There are a lot of white spots and lines that do not exist in the original image. The worse problem is that the noise is amplified, so

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that the large areas of the face are not uniform. The images enhanced by the conventional UM in Fig. 13(b), the cubic UM method in Fig. 13(c), and the order statistics method in Fig. 13(d), also have the problems as discussed in [26]. In fact, local regions containing noise are very distinct. These UM methods use only local information, therefore, some local regions may be over-enhanced while other regions may be under-enhanced. By examining Figs. 12(c) and 13(e), we can conclude that the proposed algorithm makes the features of the images clearer, without over-enhancement and underenhancement. 4.4. Multiple-level wavelets To achieve multiresolution analysis, multiple-level wavelet transform may be employed. We applied multiple-level wavelets to the test images, and enhanced them using the proposed algorithm; however, the experiments show that the improvement is not significant. Furthermore, it is much more time-consuming to use high level wavelets. Therefore, only one-level wavelet is employed in the experiments. 5. Conclusions Wavelet transform gives attention to the features both in the space domain and frequency domain. Selecting best bases of wavelets is critical and important for implementing wavelet algorithms. In this paper, we propose a novel approach for automatic selection of wavelet bases and parameters. The method employs both global and local information. For validating the proposed approach, we apply it to contrast enhancement, and test it on a variety of images. The experimental results demonstrate that the proposed method is superior to other methods. It has no over-enhancement or under-enhancement. The proposed approach may be very useful for many applications in image processing, pattern recognition and computer vision. References [1] L. Dash, B.N. Chatterji, Adaptive contrast enhancement and deenhancement, Pattern Recognition 24 (1991) 289–302. ¨ [2] B. Jahne, Digital Image Processing, sixth rev. and extended ed., Springer, Berlin, New York, 2005. [3] J.A. Stark, Adaptive image contrast enhancement using generalizations of histogram equalization, IEEE Transactions on Image Processing 9 (2000) 889–896. [4] E.L. Hall, Computer Image Processing and Recognition, Academic Press, New York, 1979. [5] F. Russo, An image enhancement technique combining sharpening and noise reduction, IEEE Transactions on Instrumentation and Measurement 51 (2002) 824–828.

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