- Email: [email protected]
WITHDRAWN: A texture-preserved image denoising algorithm based on local adaptive regularization
Accepted Manuscript Title: A texture-preserved image denoising algorithm based on local adaptive regularization Author: Li Guo Weilong Chen Yu Liao Honghua Liao Jun Li PII: DOI: Reference:
S0030-4026(15)00988-2 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.221 IJLEO 56145
To appear in: Received date: Accepted date:
12-9-2014 29-8-2015
Please cite this article as: L. Guo, W. Chen, Y. Liao, H. Liao, J. Li, A texture-preserved image denoising algorithm based on local adaptive regularization, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.221 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A texture-preserved image denoising algorithm based on local adaptive regularization Li Guo1, Weilong Chen2,Yu Liao1*, Honghua Liao1,and Jun Li1 Information Engineering Department, Hubei Minzu University, Enshi, China 2
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1
Digital Media College, Sichuan Normal University, Chengdu, China
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Abstract Denoising methods based on gradient dependent energy functional, such as Perona-Malik and total variation denoising, modify images towards piecewise constant function. In these methods, some important information such as edge sharpness and location is
us
well preserved, but encoded in image features like texture or certain detail is often compromised in the process of denoising. For this reason, we propose an image denoising method based on local adaptive regularization in this paper, which can adaptively adjust denoising degree of noisy image by adding spatial variable fidelity term, so as to better preserve fine scale features of image.
signal-noise-ratio (SNR) is also objectively improved by 0.1-0.3dB.
an
Experimental results show that the proposed denoising method can achieve state-of-the-art subjective visual effect, and the
Keywords: adaptive regularization term; texture preserved; image denoising; total-variation function
M
Ⅰ. INTRODUCTION
In the process of image acquisition and transmission, image is inevitably polluted by noise due to many factors. Noise deteriorates the quality of image and causes difficulty in image observation, feature extraction and image
d
analysis. In order to effectively remove noise, some filters such as mean filter and Gaussian filter are applied on the noisy image. It is limited that filters can lose large edge and texture information of image in denoising process. In order
te
to avoid this problem, many researchers proposed various methods. Weickert proposed a partial differential equation (PDE) method to remove noise while keep the image edge information [1, 2]. Tony proposed a total variation method
Ac ce p
based on L1 norm [3]. May proposed a spatial-adaptive image restoration method [4], this method consider the noise as image edge and produce stair effect in smoothing area. Mithun proposed a generalized total variation denoising model, which can remove false edge, but it is sensitive to the selection of P factor [5]. Jiabing Y. proposed a denoising method based on local structure and symmetric four-order partial differential equation respectively [6, 7]. These methods can improve the image denoising quality in some extent, but destroy high-frequency information of image inevitably. In recent years, Liu proposed a map image adaptive regularization denoising [8]. Xu proposed a image denoising method based non-local regularization [9, 10]. Chen proposed an adaptive image denoising model based on regularization and fidelity total variation [11]. Yan proposed an imaging denoising by generalized total variation regularization and least squares fidelity [12]. Such image denoising method described above can achieve acceptable denoising effect, but it is a problem that how to select an appropriate regularized factor and set reasonable iteration. For these analyses, this paper proposed an image denoising method based on local adaptive regularization, which can adaptively select denoising degree according to different area of noisy image, and better protect the texture and details of image, so as to achieve state-of-the-art denoising effect.
Ⅱ. Total variation denoising In signal processing, total variation denoising, also known as total variation regularization is a process, most often used in image processing, that has application in noise removal. It is based on the principle that signals with excessive
1
*Corresponding author Email:[email protected]
Page 1 of 8
and possibly spurious detail have high total variation (TV), that is, the integral of the absolute gradient of the signal is high. According to this principle, reducing the total variation of the signal subject to it being a close match to the original signal, removes unwanted detail whilst preserving important details such as edges. The concept of total variation was pioneered by Rudin et al. [13]. In area of image processing, f is the original
ip t
image, f0 is the noisy version of f , this relation can be mathematically expressed by Eq. (1).
f 0 ( x , y ) f ( x, y ) n ( x , y ) 2
cr
Here n ( x, y ) is random noise with zero mean and
(1)
variation. The classical TV denoising is a minimization
us
process of TV, this method explores the equal state of energy function relative to the TV norm of f and the regularization of f0 , which is expressed in Eq. (2).
an
1 ETV (| f |) ( f f 0 ) 2 ) dxdy 2
(2)
M
Here represents the definition area of image, all pixels ( x, y ) . Normally the TV of ideal image is smaller than noisy image, so minimize TV can reduce the noise of image. Based on this principle, Eq. (2) can be equaled as follows.
f x2 f y2 dxdy 0
(3)
te
2
2 ( f f 0 ) dxdy
d
The first term in Eq. (3) is data fidelity term, which can retain characteristics of the original image and reduce distortion. The second term in Eq. (3) is regularization term, which depends on noise level and balance denoising and
Ac ce p
smoothing. The Euler-Langarant equation derivate from Eq. (3) is represented by Eq. (4).
( f f0 ) (
Here
1
| f |
f | f |
)0
(4)
is a diffusion coefficient. In edge of image, large | f | will lead to small diffusion coefficient, so the
diffusion along edge is weak to preserve edge of image. In smoothing area of image, small | f | will lead to large diffusion coefficient, so the diffusion in smoothing area is strong to remove noise in image.
Ⅲ. Image denoising method based on local adaptive regularization Classical TV model described in Section Ⅱ can smooth image by energy constraint, but it is difficult to artificial set regularization factor . High will lead to over-smoothing and small can not effectively remove noise in image. In order to better handle different input noisy image, an appropriate regularization factor must be obtained to adjust different input image, so as to achieve balance of data fidelity term and regularization term. Based on above discussion, this paper proposed a novel image denoising method based on local adaptive regularization, which adaptively adjusted denoising degree according to noise level in different area of image, and accomplished by defining a space variable energy function. The detailed steps of this proposed method is described in below.
Page 2 of 8
Step 1 Input noisy image f0 ( x, y ) . '
Step 2 f0 ( x, y ) is denoised to obtain an original estimated version f 0 ( x , y ) by classical TV regularization. '
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Step 3 Residual error f r (noise) is computed by input noisy image and estimated denoising version f0 ( x, y ) , which '
Step 4 The mean value of residual error f r is computed and named M r .
Step 6 Make S ( x, y )
4 PLV
1 ||
f r ( x, y ) w( x, y ) dxdy .
us
Step 5 Local variance of residual image is computed by PLV ( x, y )
cr
is f r f 0 ( x, y ) f 0 ( x , y ) .
to get prior information of noise energy in noisy image, here is noise standard
an
deviation of input noisy image f0 ( x, y ) .
'
Step 7 Input noisy image f0 ( x, y ) is denoised to obtain a estimated denoising version f 0 ( x , y ) by classical TV
Step 8 Make Q ( f0 f0 ) div ( '
' f 0' ) | f 0' |
M
regularization.
, regularization factor is computed by ( x, y )
Q( x, y) S ( x, y )
.
gaussian point spread function. n
n 1
| , here is a small value, such as 0.001. If this condition is satisfied,
te
Step 10 Judge the end condition | f f
d
Step 9 Make ( x, y ) ( x, y ) P ( x , y ) , here represents convolution operation and P ( x , y ) is a 11 11
the convergence is over to obtain the final results; if it is not satisfied, then repeat step (7)-step (9), until it is convergent.
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Ⅳ. Experiments and discussion
In order to better verify the effectiveness of the proposed image denoising method, three different types of noise were added into test images to perform simulated noise experiments respectively. The denoising result of the proposed method is compared with other efficient denoising method, such as classical TV denoising, bilateral filtering and the denoising method proposed in Ref. [12]. All these comparable results fully demonstrate effectiveness and robustness of the denoising method proposed in this paper.
A. Salt & pepper noise
Firstly, the test image “woman” was additive with salt & pepper noise, which variance is 0.02. In order to better show different denoising result, the local area of “woman” is extracted and enlarged for experimental comparison.
(a)
(b)
(c)
(d)
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(f)
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(e) (a)Local part of original image (b)Local part of noisy image (SNR=8.1867)
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(c)Local part of classical TV denoising method (SNR=15.0410) (d)Local part of bilateral filter method (SNR=15.2426) (e)Local part of the method proposed in Ref. [12] (SNR=15.7141)
(f)Local part of the proposed method in this paper (SNR=15.9832)
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Figure 1. Comparison of salt & pepper noisy image and denosing result.
In Fig. 1, (a) is the local part extracted from the original "woman" image, it is seemed very clear. Fig. 1(b) is the local part of noisy image added with salt & pepper noise, a number of noise points can be seen in the image. Fig. 1(c) is the
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denoising result of classical TV method. Woman’s face is enlarged to highlight similar phenomena to those obtained in the case of the synthetic example, where preserved some texture in the right smoothing part of image, but it is not ideal and edge of image is smoothing and detailed information is lost. Result shown in Fig. 1(d) is fuzzy and noisy, the visual effect is bad. Fig. 1(e) can remove noise in image, but the denoising effect is over-smoothing. The denoising result in
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Fig. 1(f) is clear and details preserved well, although there are still a few grain noise, but it does not affect the overall viewing effect.
B. Gaussian noise
d
In this experiment, test image “lena” is added into Gaussian noise with 0 mean and variation is 0.06. Fig. 2 shows the lena’s face from the lena image where the textured face parts are in front of a smooth background. From this way, the
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noisy image and denoising result by different method is compared here.
(a)
(b)
(e)
(f)
(c)
(d)
(a)Local part of original “lena” image (b) Local part of noisy image (SNR=7.5325) (c) Local part of classical TV method (SNR=13.3146) (d) Local part of bilateral filter (SNR=11.9687) (e) )Local part of the method proposed in Ref. [12] (SNR=14.3413)
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(f) Local part of the proposed method in this paper (SNR=14.6835) Figure 2. Comparison of Gaussian noisy image and denoising result by different methods
In Fig. 2, (a) is local part of the original "Lena" image. Fig. 2(b) is local part of noisy image added with gaussian noise, a large number of noise can be seen in it. Fig. 2(c) is the local part of denoising result by classical TV method, the visual effect is overall smoothing. Fig. 2(d) is the denoising result of bilateral filter, it is contains quite a few noise
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in denoised result. Fig. 2(e) is the denoising result of method proposed in Ref. [12], it is well remove the noise in the image, but the detailed information is lost and overall visual effect is too smoothing. Fig. 2(f) is the local part of denoising result by the proposed method in this paper, it is can be seen that the detail information (such as feather in
cr
cap, eye and mouth edge) is keep well, the overall viewing effect is very good and noise is reduced selectively in a natural manner.
C. Random noise
us
In this experimental subsection, standard test image “lena” is added into random noise with variance is 20. From this way, a noisy image is obtained in this experiment. In order to better illustrate the effectiveness of the proposed
M
an
algorithm, the local part of denoising result is extracted for experimental show in Fig. 3.
(b)
(c)
(d)
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te
d
(a)
(e)
(f)
(a)Local part of the original “lena” image
(b)Local part of noisy image(SNR=7.1122)
(c)Local part of classical TV method (SNR=14.2720) (d)Local part of bilateral filter (SNR=11.3260)
(e)Local part of the method proposed in Ref. [12] (SNR=14.6837) (f) Local part of the proposed method in this paper (SNR=14.8018) Figure 3. Comparison of denoising result by different method
Fig. 3 (a) is the local part of original "Lena" image. Fig. 3(b) is the local part of noisy image added with random noise. Fig. 3(c) is the local part of denoising result by classical TV method, the overall visual effect is fuzzy. The result of Fig. 3(d) contains a large number of noise which can not be removed. Fig. 3(e) is the local part of denoising result by denoising method proposed in Ref. [12], it removed noise well but smoothed the details in image simultaneously. Fig. 3(f) is the local part of denoising result by the proposed method in this paper, it is can be seen that the detailed information is protected well (such as eye and the edge of mouth), the overall visual effect is also acceptable.
Page 5 of 8
D. Texture image denoising Because of advantage of the proposed denoising method is well preserved the texture of image, so the test image
(b)
(e)
(f)
(c)
(d)
(b) Local part of noisy image (SNR=6.5743)
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(a) Local part of texture image
d
M
an
(a)
us
cr
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contained redundant texture is used in this experiment.
(c) Local part of traditional TV regularization (SNR=13.2796)
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(d)Local part of bilateral filter (SNR=10.8693)
(e) Local part of the method proposed in Ref. [12] (SNR=13.6352) (f) Local part of the proposed method in this paper (SNR=13.8936) Figure 4. Comparison of different denoising method of noisy image
In Fig. 4, (a) is local part of the original texture image. Fig. 4(b) is local part of noisy texture image added with random noise. Fig. 4(c) is local part of denoising result by classical TV method, it is difficult to distinguish true texture information and noise, so overall visual effect is fuzzy. Fig. 4(d) is the local part of denoising result by bilateral filter, it still contained much noise in the denoising result. Fig. 4(e) is local part of denoising result by method proposed in Ref. [12], a lot of noise is also contained in the denoised result. Fig. 4(f) is local part of denoising result by the proposed method in this paper, noise is well removed and the texture information is well preserved in the image, the visual effect is good. In order to show superior of the proposed denoiding method further, the average SNR value comparison of different denoising method is illustrated in Table 1 additionally. From this comparison, it is clearly shown that the average SNR value of the proposed method is highest, it rise about 0.1-0.3 dB. Therefore, it is proved the effectiveness of the proposed denoising method in this paper. Note that in Table 1, random noise 1 is refer to subsection C. (experiment 3), random noise 2 is refer to subsection D. (experiment 4).
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COMPARISON OF DENOISING RESULT BY DIFFERENT METHOD IN TERMS OF AVERAGE SNR VALUE Salt&pepper
Gaussian
Random
noise
noise
noise 1
8.1867
7.5325
15.0410
Bilateral filter Method in Ref. [9]
Average SNR
7.1122
6.5743
7.3514
13.3146
14.2720
13.2796
13.9768
15.2426
11.9687
11.3260
10.8693
12.3517
15.7141
14.3413
14.6837
13.6352
15.9832
14.6835
14.8018
13.8936
Classical TV method
Method proposed
14.5936
14.8405
us
in this paper
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Noisy image
Random noise 2
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TABLE I.
Ⅴ. Conclusion
The widely used denoising algorithm based on global energy constrain can obtain good denoising effect on the
an
simple structural image. However, it is can not effectively remove noise in image contained a large number texture information and small scale detailed information. Furthermore, it is easy to lose key detail information when remove noise. For these reason, an image denoising method based on local adaptive regularization is proposed in this paper, it
M
can effectively control denoising level in different area of noisy image by constrain residual local energy of energy function, state-of-the-art denoising result can be obtained by this way. Edge, texture and detail information of image can also be well preserved at the same time. From various experiments, it is proved that the proposed denoising method is effective and robust.
d
Furthermore, we make several suggestions as the direction to inspire new insights for the further image denoising
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exploration. One of the most challenging problems is to adaptively calibrate the balance in regularization and data fidelity, which can avoid over-smoothing in denoising process.
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Acknowledgment
The work is supported by Science Research Program of Hubei provincial Science &Technology Department of china (No.2012FFC02601, No.2011cdb088), Science Research Program of Hubei Provincial Department of Education of china (No.Q20111907), Science Research Program of Sichuan Provincial Department of Education of china (No. 11ZB073 ), Science Research Program of visual computing and virtual reality key laboratory of Sichuan Province (NO: PJ201113), and the National Natural Science Foundation of China (No. 61263030), Innovation Projection of Culture Ministry (No. 201217).
References
[1] J. Weickert, “A review of nonlinear diffusion filtering,” Scale-space theory in computer vision, 1997, 1252: 1-28. [2] J. Weickert, “Cohterence-enhancing duffusion filtering,” IJCV 31(2/3), 1999:111-127. [3] Tony F. Chan and Selim Esedoglu, “Aspects of total variation regularized L1 function approximation,” Technical report, UCLA Mathematics Department, 2004, 2:20-24. [4] May K, Stathaki T, Katsaggelos A K, “Spatially adaptive intensity bounds for image restoration,” EURASIP Journal on Applied Signal Processing, 2003, 12:1167-1180. [5] Mithun Das Gupta, Sanjeev Kumar, “Non-Convex P-norm Projection for Robust Sparsity,” ICCV 2013. [6] J. B. Yan, Z. G. Liu, “Diffusion smoothing based on local structure of image,” Computer Engineering, 2008, 9:64-68. [7] C. Bo, Z. Li, “Symmetric four order PDE denoising algorithm,” Computer Engineerin，2008, 34(13): 188-189. [8] G. J. Liu, X. P. Zeng, ”Map image adaptive regularization denoising,” Journal of Chongqing University, 2012, 35(10): 63-67. [9] D. H. Xu, R. S. Wang, “Imaging denoising based on non-local regularization,” The research and application of computer, 2009, 26(12): 4830-4832. [10] X. W. Liu, L. H. Huang, “A new nonlocal total variation regularization algorithm for image denoising,” Mathematics and Computers in Simulation, 2014, 97(3): 224-233 [11] M. J.Chen, P. X. Yang, J. Wang, “Adaptive image denoising model based regularization and TV fidelity,” Journal of Chongqing post and telecommunication, 2011, 23(5): 621-625.
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te
d
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an
us
cr
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[12] J. Yan, W.S. Lu, “Imaging denoising by generalized total variation regularization and least squares fidelity,” Multimensional Systems and Signal Processing, 2013, 10, 10.1007/s11045-013-0255-2J. [13] L. Rudin, S. Osher, E. Fatemi,”Nonlinear total variation based noise removal algorithms,” Physica D60, 1992:259-268.
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S0030-4026(15)00988-2 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.221 IJLEO 56145
To appear in: Received date: Accepted date:
12-9-2014 29-8-2015
Please cite this article as: L. Guo, W. Chen, Y. Liao, H. Liao, J. Li, A texture-preserved image denoising algorithm based on local adaptive regularization, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.221 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A texture-preserved image denoising algorithm based on local adaptive regularization Li Guo1, Weilong Chen2,Yu Liao1*, Honghua Liao1,and Jun Li1 Information Engineering Department, Hubei Minzu University, Enshi, China 2
ip t
1
Digital Media College, Sichuan Normal University, Chengdu, China
cr
Abstract Denoising methods based on gradient dependent energy functional, such as Perona-Malik and total variation denoising, modify images towards piecewise constant function. In these methods, some important information such as edge sharpness and location is
us
well preserved, but encoded in image features like texture or certain detail is often compromised in the process of denoising. For this reason, we propose an image denoising method based on local adaptive regularization in this paper, which can adaptively adjust denoising degree of noisy image by adding spatial variable fidelity term, so as to better preserve fine scale features of image.
signal-noise-ratio (SNR) is also objectively improved by 0.1-0.3dB.
an
Experimental results show that the proposed denoising method can achieve state-of-the-art subjective visual effect, and the
Keywords: adaptive regularization term; texture preserved; image denoising; total-variation function
M
Ⅰ. INTRODUCTION
In the process of image acquisition and transmission, image is inevitably polluted by noise due to many factors. Noise deteriorates the quality of image and causes difficulty in image observation, feature extraction and image
d
analysis. In order to effectively remove noise, some filters such as mean filter and Gaussian filter are applied on the noisy image. It is limited that filters can lose large edge and texture information of image in denoising process. In order
te
to avoid this problem, many researchers proposed various methods. Weickert proposed a partial differential equation (PDE) method to remove noise while keep the image edge information [1, 2]. Tony proposed a total variation method
Ac ce p
based on L1 norm [3]. May proposed a spatial-adaptive image restoration method [4], this method consider the noise as image edge and produce stair effect in smoothing area. Mithun proposed a generalized total variation denoising model, which can remove false edge, but it is sensitive to the selection of P factor [5]. Jiabing Y. proposed a denoising method based on local structure and symmetric four-order partial differential equation respectively [6, 7]. These methods can improve the image denoising quality in some extent, but destroy high-frequency information of image inevitably. In recent years, Liu proposed a map image adaptive regularization denoising [8]. Xu proposed a image denoising method based non-local regularization [9, 10]. Chen proposed an adaptive image denoising model based on regularization and fidelity total variation [11]. Yan proposed an imaging denoising by generalized total variation regularization and least squares fidelity [12]. Such image denoising method described above can achieve acceptable denoising effect, but it is a problem that how to select an appropriate regularized factor and set reasonable iteration. For these analyses, this paper proposed an image denoising method based on local adaptive regularization, which can adaptively select denoising degree according to different area of noisy image, and better protect the texture and details of image, so as to achieve state-of-the-art denoising effect.
Ⅱ. Total variation denoising In signal processing, total variation denoising, also known as total variation regularization is a process, most often used in image processing, that has application in noise removal. It is based on the principle that signals with excessive
1
*Corresponding author Email:[email protected]
Page 1 of 8
and possibly spurious detail have high total variation (TV), that is, the integral of the absolute gradient of the signal is high. According to this principle, reducing the total variation of the signal subject to it being a close match to the original signal, removes unwanted detail whilst preserving important details such as edges. The concept of total variation was pioneered by Rudin et al. [13]. In area of image processing, f is the original
ip t
image, f0 is the noisy version of f , this relation can be mathematically expressed by Eq. (1).
f 0 ( x , y ) f ( x, y ) n ( x , y ) 2
cr
Here n ( x, y ) is random noise with zero mean and
(1)
variation. The classical TV denoising is a minimization
us
process of TV, this method explores the equal state of energy function relative to the TV norm of f and the regularization of f0 , which is expressed in Eq. (2).
an
1 ETV (| f |) ( f f 0 ) 2 ) dxdy 2
(2)
M
Here represents the definition area of image, all pixels ( x, y ) . Normally the TV of ideal image is smaller than noisy image, so minimize TV can reduce the noise of image. Based on this principle, Eq. (2) can be equaled as follows.
f x2 f y2 dxdy 0
(3)
te
2
2 ( f f 0 ) dxdy
d
The first term in Eq. (3) is data fidelity term, which can retain characteristics of the original image and reduce distortion. The second term in Eq. (3) is regularization term, which depends on noise level and balance denoising and
Ac ce p
smoothing. The Euler-Langarant equation derivate from Eq. (3) is represented by Eq. (4).
( f f0 ) (
Here
1
| f |
f | f |
)0
(4)
is a diffusion coefficient. In edge of image, large | f | will lead to small diffusion coefficient, so the
diffusion along edge is weak to preserve edge of image. In smoothing area of image, small | f | will lead to large diffusion coefficient, so the diffusion in smoothing area is strong to remove noise in image.
Ⅲ. Image denoising method based on local adaptive regularization Classical TV model described in Section Ⅱ can smooth image by energy constraint, but it is difficult to artificial set regularization factor . High will lead to over-smoothing and small can not effectively remove noise in image. In order to better handle different input noisy image, an appropriate regularization factor must be obtained to adjust different input image, so as to achieve balance of data fidelity term and regularization term. Based on above discussion, this paper proposed a novel image denoising method based on local adaptive regularization, which adaptively adjusted denoising degree according to noise level in different area of image, and accomplished by defining a space variable energy function. The detailed steps of this proposed method is described in below.
Page 2 of 8
Step 1 Input noisy image f0 ( x, y ) . '
Step 2 f0 ( x, y ) is denoised to obtain an original estimated version f 0 ( x , y ) by classical TV regularization. '
ip t
Step 3 Residual error f r (noise) is computed by input noisy image and estimated denoising version f0 ( x, y ) , which '
Step 4 The mean value of residual error f r is computed and named M r .
Step 6 Make S ( x, y )
4 PLV
1 ||
f r ( x, y ) w( x, y ) dxdy .
us
Step 5 Local variance of residual image is computed by PLV ( x, y )
cr
is f r f 0 ( x, y ) f 0 ( x , y ) .
to get prior information of noise energy in noisy image, here is noise standard
an
deviation of input noisy image f0 ( x, y ) .
'
Step 7 Input noisy image f0 ( x, y ) is denoised to obtain a estimated denoising version f 0 ( x , y ) by classical TV
Step 8 Make Q ( f0 f0 ) div ( '
' f 0' ) | f 0' |
M
regularization.
, regularization factor is computed by ( x, y )
Q( x, y) S ( x, y )
.
gaussian point spread function. n
n 1
| , here is a small value, such as 0.001. If this condition is satisfied,
te
Step 10 Judge the end condition | f f
d
Step 9 Make ( x, y ) ( x, y ) P ( x , y ) , here represents convolution operation and P ( x , y ) is a 11 11
the convergence is over to obtain the final results; if it is not satisfied, then repeat step (7)-step (9), until it is convergent.
Ac ce p
Ⅳ. Experiments and discussion
In order to better verify the effectiveness of the proposed image denoising method, three different types of noise were added into test images to perform simulated noise experiments respectively. The denoising result of the proposed method is compared with other efficient denoising method, such as classical TV denoising, bilateral filtering and the denoising method proposed in Ref. [12]. All these comparable results fully demonstrate effectiveness and robustness of the denoising method proposed in this paper.
A. Salt & pepper noise
Firstly, the test image “woman” was additive with salt & pepper noise, which variance is 0.02. In order to better show different denoising result, the local area of “woman” is extracted and enlarged for experimental comparison.
(a)
(b)
(c)
(d)
Page 3 of 8
(f)
ip t
(e) (a)Local part of original image (b)Local part of noisy image (SNR=8.1867)
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(c)Local part of classical TV denoising method (SNR=15.0410) (d)Local part of bilateral filter method (SNR=15.2426) (e)Local part of the method proposed in Ref. [12] (SNR=15.7141)
(f)Local part of the proposed method in this paper (SNR=15.9832)
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Figure 1. Comparison of salt & pepper noisy image and denosing result.
In Fig. 1, (a) is the local part extracted from the original "woman" image, it is seemed very clear. Fig. 1(b) is the local part of noisy image added with salt & pepper noise, a number of noise points can be seen in the image. Fig. 1(c) is the
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denoising result of classical TV method. Woman’s face is enlarged to highlight similar phenomena to those obtained in the case of the synthetic example, where preserved some texture in the right smoothing part of image, but it is not ideal and edge of image is smoothing and detailed information is lost. Result shown in Fig. 1(d) is fuzzy and noisy, the visual effect is bad. Fig. 1(e) can remove noise in image, but the denoising effect is over-smoothing. The denoising result in
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Fig. 1(f) is clear and details preserved well, although there are still a few grain noise, but it does not affect the overall viewing effect.
B. Gaussian noise
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In this experiment, test image “lena” is added into Gaussian noise with 0 mean and variation is 0.06. Fig. 2 shows the lena’s face from the lena image where the textured face parts are in front of a smooth background. From this way, the
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noisy image and denoising result by different method is compared here.
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(a)Local part of original “lena” image (b) Local part of noisy image (SNR=7.5325) (c) Local part of classical TV method (SNR=13.3146) (d) Local part of bilateral filter (SNR=11.9687) (e) )Local part of the method proposed in Ref. [12] (SNR=14.3413)
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(f) Local part of the proposed method in this paper (SNR=14.6835) Figure 2. Comparison of Gaussian noisy image and denoising result by different methods
In Fig. 2, (a) is local part of the original "Lena" image. Fig. 2(b) is local part of noisy image added with gaussian noise, a large number of noise can be seen in it. Fig. 2(c) is the local part of denoising result by classical TV method, the visual effect is overall smoothing. Fig. 2(d) is the denoising result of bilateral filter, it is contains quite a few noise
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in denoised result. Fig. 2(e) is the denoising result of method proposed in Ref. [12], it is well remove the noise in the image, but the detailed information is lost and overall visual effect is too smoothing. Fig. 2(f) is the local part of denoising result by the proposed method in this paper, it is can be seen that the detail information (such as feather in
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cap, eye and mouth edge) is keep well, the overall viewing effect is very good and noise is reduced selectively in a natural manner.
C. Random noise
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In this experimental subsection, standard test image “lena” is added into random noise with variance is 20. From this way, a noisy image is obtained in this experiment. In order to better illustrate the effectiveness of the proposed
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algorithm, the local part of denoising result is extracted for experimental show in Fig. 3.
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(a)Local part of the original “lena” image
(b)Local part of noisy image(SNR=7.1122)
(c)Local part of classical TV method (SNR=14.2720) (d)Local part of bilateral filter (SNR=11.3260)
(e)Local part of the method proposed in Ref. [12] (SNR=14.6837) (f) Local part of the proposed method in this paper (SNR=14.8018) Figure 3. Comparison of denoising result by different method
Fig. 3 (a) is the local part of original "Lena" image. Fig. 3(b) is the local part of noisy image added with random noise. Fig. 3(c) is the local part of denoising result by classical TV method, the overall visual effect is fuzzy. The result of Fig. 3(d) contains a large number of noise which can not be removed. Fig. 3(e) is the local part of denoising result by denoising method proposed in Ref. [12], it removed noise well but smoothed the details in image simultaneously. Fig. 3(f) is the local part of denoising result by the proposed method in this paper, it is can be seen that the detailed information is protected well (such as eye and the edge of mouth), the overall visual effect is also acceptable.
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D. Texture image denoising Because of advantage of the proposed denoising method is well preserved the texture of image, so the test image
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(b) Local part of noisy image (SNR=6.5743)
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(a) Local part of texture image
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contained redundant texture is used in this experiment.
(c) Local part of traditional TV regularization (SNR=13.2796)
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(d)Local part of bilateral filter (SNR=10.8693)
(e) Local part of the method proposed in Ref. [12] (SNR=13.6352) (f) Local part of the proposed method in this paper (SNR=13.8936) Figure 4. Comparison of different denoising method of noisy image
In Fig. 4, (a) is local part of the original texture image. Fig. 4(b) is local part of noisy texture image added with random noise. Fig. 4(c) is local part of denoising result by classical TV method, it is difficult to distinguish true texture information and noise, so overall visual effect is fuzzy. Fig. 4(d) is the local part of denoising result by bilateral filter, it still contained much noise in the denoising result. Fig. 4(e) is local part of denoising result by method proposed in Ref. [12], a lot of noise is also contained in the denoised result. Fig. 4(f) is local part of denoising result by the proposed method in this paper, noise is well removed and the texture information is well preserved in the image, the visual effect is good. In order to show superior of the proposed denoiding method further, the average SNR value comparison of different denoising method is illustrated in Table 1 additionally. From this comparison, it is clearly shown that the average SNR value of the proposed method is highest, it rise about 0.1-0.3 dB. Therefore, it is proved the effectiveness of the proposed denoising method in this paper. Note that in Table 1, random noise 1 is refer to subsection C. (experiment 3), random noise 2 is refer to subsection D. (experiment 4).
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COMPARISON OF DENOISING RESULT BY DIFFERENT METHOD IN TERMS OF AVERAGE SNR VALUE Salt&pepper
Gaussian
Random
noise
noise
noise 1
8.1867
7.5325
15.0410
Bilateral filter Method in Ref. [9]
Average SNR
7.1122
6.5743
7.3514
13.3146
14.2720
13.2796
13.9768
15.2426
11.9687
11.3260
10.8693
12.3517
15.7141
14.3413
14.6837
13.6352
15.9832
14.6835
14.8018
13.8936
Classical TV method
Method proposed
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14.8405
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in this paper
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Noisy image
Random noise 2
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TABLE I.
Ⅴ. Conclusion
The widely used denoising algorithm based on global energy constrain can obtain good denoising effect on the
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simple structural image. However, it is can not effectively remove noise in image contained a large number texture information and small scale detailed information. Furthermore, it is easy to lose key detail information when remove noise. For these reason, an image denoising method based on local adaptive regularization is proposed in this paper, it
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can effectively control denoising level in different area of noisy image by constrain residual local energy of energy function, state-of-the-art denoising result can be obtained by this way. Edge, texture and detail information of image can also be well preserved at the same time. From various experiments, it is proved that the proposed denoising method is effective and robust.
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Furthermore, we make several suggestions as the direction to inspire new insights for the further image denoising
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exploration. One of the most challenging problems is to adaptively calibrate the balance in regularization and data fidelity, which can avoid over-smoothing in denoising process.
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Acknowledgment
The work is supported by Science Research Program of Hubei provincial Science &Technology Department of china (No.2012FFC02601, No.2011cdb088), Science Research Program of Hubei Provincial Department of Education of china (No.Q20111907), Science Research Program of Sichuan Provincial Department of Education of china (No. 11ZB073 ), Science Research Program of visual computing and virtual reality key laboratory of Sichuan Province (NO: PJ201113), and the National Natural Science Foundation of China (No. 61263030), Innovation Projection of Culture Ministry (No. 201217).
References
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[12] J. Yan, W.S. Lu, “Imaging denoising by generalized total variation regularization and least squares fidelity,” Multimensional Systems and Signal Processing, 2013, 10, 10.1007/s11045-013-0255-2J. [13] L. Rudin, S. Osher, E. Fatemi,”Nonlinear total variation based noise removal algorithms,” Physica D60, 1992:259-268.
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