- Email: [email protected]
Wiener filter-based wavelet domain denoising
Accepted Manuscript Wiener Filter-based Wavelet Domain Denoising Jin Wang, Jiaji Wu, Zhensen Wu, Jechang Jeong, Gwanggil Jeon PII: DOI: Reference:
S0141-9382(16)30164-0 http://dx.doi.org/10.1016/j.displa.2016.12.003 DISPLA 1814
To appear in:
Displays
Received Date: Accepted Date:
12 September 2016 23 December 2016
Please cite this article as: J. Wang, J. Wu, Z. Wu, J. Jeong, G. Jeon, Wiener Filter-based Wavelet Domain Denoising, Displays (2016), doi: http://dx.doi.org/10.1016/j.displa.2016.12.003
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.
Wiener Filter-based Wavelet Domain Denoising Jin Wang1,3, Jiaji Wu2, Zhensen Wu1, Jechang Jeong3, and Gwanggil Jeon4 1
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, Shaanxi, China 2 School of Electronic Engineering, Xidian University, Xi’an, Shaanxi, China 3 Department of Electronics and Computer Engineering, Hanyang University, Seoul, Korea 4 Department of Embedded Systems Engineering, Incheon National University, Incheon, Korea email: [email protected], [email protected], [email protected], [email protected], [email protected] Tel: +82-32-835-8946, Fax: +82-32-835-0782
Abstract: The wavelet domain Wiener filter has been widely adopted as an effective image denoising method that has low complexity. In this paper we propose a novel Wiener filter with high-resolution estimation that determines the signal power while preserving the edge information. We assume that a noisy image is composed of noise and the original image, which are mutually orthogonal. Based on this assumption, we utilize the local covariance to obtain high-resolution coefficients from the low-resolution coefficients and to estimate the signal variance in the Wiener filter by using the high resolution values. The experimental results show that the proposed algorithm improves the objective and subjective performance significantly. Key words: Wiener filter, image denoising, high resolution estimation. 1. Introduction Image denoising is the recovery of a digital image that has been contaminated by additive zero-mean white Gaussian noise during its acquisition and transmission. The need for effective image restoration algorithms has grown with the massive production of digital images and movies of all kinds, which often are acquired under poor conditions. The goal of image denoising is to eliminate the noise and preserve the edge and texture detail to remove artifacts visible to the human eye by reconstructing the original image from the distorted image. Over the last decades, many algorithms have been proposed for high-quality denoising performance. Among them, discrete wavelet transform (DWT) has been widely used for noise removal in the field of image processing [1-13]. The success of DWT is due to the fact that DWT can be applied to a sparse representation of the image, which means that many of the coefficients are close to zero. In the wavelet domain, the noise is uniformly spread throughout the high frequency coefficients, while, most of the image information 1
is concentrated in few significant low frequency coefficients. Therefore, one straightforward denoising method is to threshold the wavelet coefficients in the wavelet domain in order to discriminate the real information from the noise. For example, Donoho proposed a soft-thresholding method that is the most popular and has been theoretically justified [1]. In [2,3], Sendur and Selesnick assumed that the wavelet coefficients exhibit strong intrascale and interscale dependency and that the signal variance can be estimated from its adjacent coefficients. Meanwhile, Wiener filtering in the wavelet domain has received attention for image denoising. The Wiener filter (WF) requires knowledge of the variances of the unknown original signals, and assumptions are made to estimate the variance from the noisy data. In [4], Kazubek proposed a thresholding-based WF, which is an approximate analysis of the error occurring in the empirical WF. In [5], the authors proposed an algorithm for determining the variable size of the locally adaptive window using a region-based approach. In [6], Mihçak et al. presented a locally adaptive window-based denoising algorithm using MAP (LAWMAP) by learning the signal variance which is determined by maximizing the a posteriori estimate [6]. Meanwhile, many models have been studied in order to obtain accurate and efficient wavelet coefficients. For example, the hidden Markov model (HMT) based wavelet domain framework has been proposed to capture the statistical dependencies [7]. As described above, the estimation of the signal variance in a noisy environment is a critical issue in WF-based denoising. We estimate the variance of every wavelet coefficient based on the local neighborhood of the wavelet coefficient. This assumption is based on the knowledge that the wavelet coefficients are locally independent and identically distributed (IID), and the locally IID assumption becomes inaccurate as the neighborhood grows [6]. Meanwhile, window size is another important issue in the denoising problem. The method proposed in [5] is wavelet-based WF with a nearly arbitrarily shaped window (NASW). The NASW has a drawback in that it uses a large window size which degrades the image detail, especially in the edge area. We introduce high resolution estimation to alleviate this problem. By applying the proposed algorithm, the edges and the textures in the images are preserved. The rest of the paper is organized as follows. The locally adaptive WF is introduced in Section 2. In Section 3, we describe the proposed algorithm. In Section 4, the experimental results and the corresponding discussion are provided. Finally, conclusions are drawn in Section 5.
2. The Conventional Wiener Filter Let us assume a noisy image in a wavelet domain with the following degradation process: 2
X S N,
(1)
where X is given by the observed noisy image with size W×H and X=[xi,j]W×H; i=0,…,H-1; j=0, …, W-1; S is the noise-free clean image S=[si,j]W×H; and N denotes the additive white Gaussian noise with zero mean and a finite variance σ2n. The noise, N, is an independent Gaussian variable that is not correlated with the signal S, and our goal is to obtain the best estimate of S from X. The form of the WF simplifies to the following scalar relation: si , j i , j xi , j
i, j
,
(2)
E xi2, j
(3)
E si2, j
.
Equation (2) shows that we can represent S as a linear combination of {xi,j}, which is formulated as S X ,
(4)
where the operation “·” is point to point multiplication, and Ψ=[ψij]W×H is shown in Eq. (3). Since the noise-free signal S is uncorrelated with the noise N, we can write
E si2, j E xi2, j n2 .
is estimated from the neighborhood of the coefficient x E x as follows:
2 The expected value E xi , j
We approximate
(5)
i,j.
2 i, j
E xi2, j
1 Ci , j
xm ,n Ci , j
xm2 ,n ,
(6)
where Ci,j is defined as the set of all coefficients within a local window of size MH×MV that is centered at location (i, j), and |Ci,j| is the cardinality of Ci,j.
i, j
E x 2 i, j
E x
2 i, j
3
2 n
,
(7)
where
E x 2 i, j
2 n
max E xi2, j n2 ,0 . Thus we can see that the estimation of
E xi2, j is a key issue in the WF-based denoising method.
3. The Proposed Algorithm
(a) (b) Fig. 1 Comparison of the PSNR of different window sizes for (a) LAWMAP and (b) WF. A wavelet-based WF with NASW denoising was presented in [5]. The NASW has a drawback because it employs a large window size, which degrades the image detail. The dependence between the wavelet coefficients decreases when the window size increases. Therefore, the size of the locally adaptive window is also an important factor in estimating the signal power. In most WF-based algorithms, a large window size causes poor performance. This is because the relationship between the pixels is weakened in a large window, especially for the complex region. However, a small window size is not appropriate either because a larger region to estimate the signal variance results in a more reliable estimation. The peak signal-to-noise ratio (PSNR) differences of LAWMAP [6] and WF are shown in Fig. 1(a) and Fig. 1(b), where the different window size (3×3, 5×5, and 7×7) are compared. As shown, the PSNR results are the best for a 5×5 window size. To maintain consistency with NASW and to preserve the edge information, we introduce a new method called high-resolution estimation WF (HRE-WF), and we estimate the image variance from the high-resolution values that are expanded from the low-resolution image. For X, from low resolution to high-resolution, we utilize the traditional wavelet coefficient technique, which is 2-D uploading. Then we can expand X
~
to be X as shown in Eq. (8) and Eq. (9). 4
x i i , if i is even and j is even , xi , j 2 2 , 0, otherwise x0,0 x 1,0 x2,0 X x2 H 3,0 x2 H 2,0 x0,0 0 x1,0 0 xH 1,0
x0,1 x1,1 x2,1
x0,2 x1,2 x2,2
x0,2W 3 x1,2W 3 x2,2W 3
x2 H 3,1 x2 H 2,1
x2 H 3,2 x2 H 2,2
x2 H 3,2W 3 x2 H 2,2W 3
0 0 0
x0,1 0 x1,1
0 0 0 xH 1,1
0 0 0 xH 1,W 1 0 0 0
x0,W 1 0 x1,W 1
(8)
x2 H 3,2W 2 x2 H 2,2W 2 x0,2W 2 x1,2W 2 x2,2W 2
.
(9)
The conventional WF represents S as a linear combination of X. However, in general, there is no Ψ for which equation Eq. (4) has solutions because S span{xi , j } . Suppose we allow a small perturbation Y such that S X Y .
(10)
Fig. 2 The relationship between the noisy signal X, the original signal S, the noise N, and the perturbation Y. 5
We set X and Y to be orthogonal. The relationship in the vector space is shown in Fig. 2.
~
Based on the above discussion, we expand X to X . To maintain consistency, Y should be
~
~
the same size as X. Therefore, Y is expanded to Y . The simplest way to ensure Y is
~
orthogonal to X , is to set yi , j 0 , where i and j are all even numbers. Then no matter
~
~
~
~
what the other coefficients of Y are Y X . Now, we can write Y in matrix form as shown in Eq. (11).
0 y 1,0 0 Y y2 H 3,0 0
y0,1 y1,1 y2,1
0 y1,2 0
y2 H 3,1 y2 H 2,1
y2 H 3,2 0
y0,2W 3 y1,2W 3 y2,2W 3 y2 H 3,2W 3 y2 H 2,2W 3
y1,2W 2 0 y2 H 3,2W 2 0 . 0
~
(11)
Equation (11) shows that deriving Y is critical in the proposed algorithm. However, in the denoising problem, only the noisy image data is available. Therefore Y should be estimated from X. We set up seven models to investigate the relationship between X and Y. A brief introduction of these models is given in Table 1. Table 1 Representation of Different Methods to Estimate Y Model # M#: Description Model 1 (zero padding) M1: No perturbation, i.e., Y = 0 Model 2 (repeating) M2: Y repeats the value of X Model 3 (average) M3: Y is average of the four neighborhood values Model 4 (median) M4: Y is the median of the five neighborhood values Model 5 (bilinear) M5: Bilinear is an interpolation technique Model 6 (bicubic) M6: Bicubic is an interpolation technique Model 7 (MMSE) M7: Y is a linear combination of four nearest neighbors
6
Fig. 3 Comparison of the PSNR of different models for the Lena image. To obtain the optimal estimate and preserve the edge information, we use MMSE [14] to
~
derive Y as shown in Fig. 3. Moreover, we make a small modification for the threshold Tk used in NASW. In our algorithm, Tk is set to Tk 2L J ,
(12)
where κ=0.1, L=log2W, scale factor λ = 0,…,J, from the finest scale to the coarsest scale, and J is the coarsest scale number. The flowchart of the proposed algorithm is shown in Fig. 4.
7
Noisy Image
Wavelet Transform
Low Frequency
High Frequency
MMSE
High Resolution
NASW
Update High Frequency
Denoised Image
Fig. 4 Flowchart of the proposed algorithm.
4. Experimental Results The simulation study was performed with standard test images Lena and Barbara, and WaveLab library [15] to verify the superiority compared to the existing algorithms: DTWF [4], NASW [5], HMT [7], EWD [8], and adaptive nonlinear diffusion (AND) based image denoising [16]. In our algorithm, a five-scale orthogonal wavelet transform by utilizing “symmlet” with eight vanishing moments was applied. Table 2 provides the PSNR results for different noise standard deviations for our proposed method compared to those of conventional methods. The proposed algorithm performs better than the other methods. It also provides the best results for both images with a variety of noise standard deviations. For a subjective performance evaluation of the denoised images in terms of the visual effect, we also show part of the perceived image quality of Lena in Fig. 5 [17-19]. The original Lena image is contaminated by noise with σn=20. It is obvious that the proposed algorithm is effective at reducing the image noises while preserving the edge information. It is also yields better visual quality for the details of the Lena image than the conventional methods. The simulation results demonstrate the superiority of the objective and perceived quality of the proposed algorithm.
8
Table 2 PSNR Results for Different Noise Standard Deviations Compared with Existing Algorithms Image σn NoisyIm WF DTWF [4] NASW [5] HMT [7] EWD [8] AND [16] Proposed
10 28.13 32.58
15 24.62 30.43
Lena 20 22.15 29.62
25 14.62 28.12
avg. 22.38 30.19
10 28.16 30.36
15 24.67 28.01
Barbara 20 22.17 26.88
25 20.25 25.21
avg. 23.81 27.62
34.08
32.01
30.18
29.52
31.45
31.38
29.01
27.35
26.22
28.49
34.09
32.02
30.16
29.54
31.45
31.4
29.03
27.46
26.29
28.56
33.89 32.94
31.82 31.01
30.41 29.73
29.36 28.76
31.37 30.61
31.36 31.13
29.23 29.11
27.8 27.25
25.99 26.06
28.60 28.39
32.70
30.56
29.79
28.16
30.30
26.64
26.28
25.28
24.96
25.79
34.4
32.59
31.3
30.27
32.14
32.06
30.03
28.36
27.03
29.37
5. Conclusions The wavelet-based WF used in [5] does not adequately preserve the edges in an image and requires a large window size. To overcome these drawbacks, we proposed a highresolution estimation WF in the wavelet domain that determines the signal power while preserving the edge information. We assumed that a noisy image is composed of noise and an original image that are mutually orthogonal. Based on this assumption, we utilized the local covariance to acquire high resolution coefficients from the low resolution coefficients, and determined the signal variance in the Wiener filter by using the highresolution signals. The proposed algorithm is effective at reducing the image noise while preserving the edge information. The empirical results demonstrate the superior quality, both objective and perceived, of the proposed algorithm. Acknowledgment This research was supported by Post-Doctor Research Program (2015) through the Incheon National University (INU), Incheon, South Korea. Reference [1] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inform. Theory 41 (1995) 613-627. [2] L. Sendur, I. W. Selesnick, Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency, IEEE Trans. Signal Process. 50 (2002) 2744-2756. [3] L. Sendur, I.W. Selesnick, Bivariate shrinkage with local variance estimation, IEEE Signal Process. Lett. 9 (2002) 438–441. [4] J. Khan, S. M. A. Bhuiyan, G. Murphy, M. Arline, Embedded-zerotree-wavelet-based data denoising and compression for smart grid, IEEE Trans. Industry Applications 51 (2015) 4190-4200. 9
[5] S. Gaci, The use of wavelet-based denoising techniques to enhance the first-arrival picking on seismic traces, IEEE Trans. Geoscience and Remote Sensing 52 (2014) 45884563. [6] A. Varsha, P. Basu, An improved dual tree complex wavelet transform based image denoising using GCV thresholding, in Proc. Int’l Conf. Computational Systems and Communications (ICCSC), 2014. [7] M. S. Crouse, R. D. Nowak, R. G. Baraniuk, Wavelet-based statistical signal processing using hidden Markov models, IEEE Trans. Signal Process. 46 (1998) 886-902. [8] J. M. Parmar, S. A. Patil, Performance evaluation and comparison of modified denoising method and the local adaptive wavelet image denoising method, in Proc. Int’l Conf. Intelligent Systems and Signal Processing (ISSP), 2013. [9] S. G. Chang, B. Yu, M. Vetterli, Adaptive wavelet thresholding for image denoising and compression, IEEE Trans. Image Process. 9 (2000) 1532-1546. [10] F. Luisier, T. Blu, M. Unser, A new SURE approach to image denoising: interscale orthonormal wavelet thresholding, IEEE Trans. Image Process. 16 (2007) 593-606. [11] C. Jung, L. Jiao, H. Kim, J. Kim, Spatial-gradient-local-inhomogeneity: an efficient image-denoising prior, J. Electron. Imaging, 19 (2010) 033005. [12] C. Y. Dang, J. M. Gao, Z. Wang, F. M. Chen, Y. L. Xiao, Optimized wavelet denoising algorithm using hybrid noise model for radiographic images, in Proc. Int’l Conf. of Computational Intelligence and Design (ISCID), 2014. [13] M. Nikpour, H. Hassanpour, Using diffusion equations for improving performance of wavelet-based image denoising techniques, IET Image Processing, 4 (2010) 452-462. [14] X. Li, M. T. Orchard, New edge-directed interpolation, IEEE Tran. Image Process. 10 (2001) 1521-1527. [15] J. B. Bucheit, S. Chen, D. L. Donoho, I. M. Johnston, J. D. Scargle. WaveLab Toolkit. [Online]. Available: http://www-stat.stanford.edu/~wavelab/ [16] A. K. Mandava, E, E. Regentova, Image denoising based on adaptive nonlinear diffusion in wavelet domain, J. Electron. Imaging 20 (2011) 033016. [17] G. Jeon, Denoising in contrast-enhanced X-ray images, Sensing and Imaging, 17 (2016) 1-11. [18] J. Wang, G. Jeon, J. Jeong, Image de-noising using adaptive directional Wiener filter in wavelet domain, in Proc. ICEIC2012, Jeongseon, Korea, 2012. [19] T. Bai, H. Wang, Y. Pang, G. Li, J. Lin, Q. Zhou, G. Jeon, A PPG signal de-noising method based on the DTCWT and the morphological filtering, in Proc. IEEE SITIS 2016, Nov. 2016.
10
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 5 (a) Original Lena image and (b) noisy Lena image. Perceived image quality using various denosing methods: (c) WF, (d) DTWF, (e) NASW, (f) HMT, (g) BSF, (h) AND, and (i) the proposed method.
11
We propose a novel Wiener filter with high-resolution estimation that determines the signal power while preserving the edge information. We assume that a noisy image is composed of noise and the original image, which are mutually orthogonal. We utilize the local covariance to obtain high-resolution coefficient from the low-resolution coefficients. We also utilize the local covariance to estimate the signal variance in the Wiener filter by using the high resolution values.
S0141-9382(16)30164-0 http://dx.doi.org/10.1016/j.displa.2016.12.003 DISPLA 1814
To appear in:
Displays
Received Date: Accepted Date:
12 September 2016 23 December 2016
Please cite this article as: J. Wang, J. Wu, Z. Wu, J. Jeong, G. Jeon, Wiener Filter-based Wavelet Domain Denoising, Displays (2016), doi: http://dx.doi.org/10.1016/j.displa.2016.12.003
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.
Wiener Filter-based Wavelet Domain Denoising Jin Wang1,3, Jiaji Wu2, Zhensen Wu1, Jechang Jeong3, and Gwanggil Jeon4 1
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, Shaanxi, China 2 School of Electronic Engineering, Xidian University, Xi’an, Shaanxi, China 3 Department of Electronics and Computer Engineering, Hanyang University, Seoul, Korea 4 Department of Embedded Systems Engineering, Incheon National University, Incheon, Korea email: [email protected], [email protected], [email protected], [email protected], [email protected] Tel: +82-32-835-8946, Fax: +82-32-835-0782
Abstract: The wavelet domain Wiener filter has been widely adopted as an effective image denoising method that has low complexity. In this paper we propose a novel Wiener filter with high-resolution estimation that determines the signal power while preserving the edge information. We assume that a noisy image is composed of noise and the original image, which are mutually orthogonal. Based on this assumption, we utilize the local covariance to obtain high-resolution coefficients from the low-resolution coefficients and to estimate the signal variance in the Wiener filter by using the high resolution values. The experimental results show that the proposed algorithm improves the objective and subjective performance significantly. Key words: Wiener filter, image denoising, high resolution estimation. 1. Introduction Image denoising is the recovery of a digital image that has been contaminated by additive zero-mean white Gaussian noise during its acquisition and transmission. The need for effective image restoration algorithms has grown with the massive production of digital images and movies of all kinds, which often are acquired under poor conditions. The goal of image denoising is to eliminate the noise and preserve the edge and texture detail to remove artifacts visible to the human eye by reconstructing the original image from the distorted image. Over the last decades, many algorithms have been proposed for high-quality denoising performance. Among them, discrete wavelet transform (DWT) has been widely used for noise removal in the field of image processing [1-13]. The success of DWT is due to the fact that DWT can be applied to a sparse representation of the image, which means that many of the coefficients are close to zero. In the wavelet domain, the noise is uniformly spread throughout the high frequency coefficients, while, most of the image information 1
is concentrated in few significant low frequency coefficients. Therefore, one straightforward denoising method is to threshold the wavelet coefficients in the wavelet domain in order to discriminate the real information from the noise. For example, Donoho proposed a soft-thresholding method that is the most popular and has been theoretically justified [1]. In [2,3], Sendur and Selesnick assumed that the wavelet coefficients exhibit strong intrascale and interscale dependency and that the signal variance can be estimated from its adjacent coefficients. Meanwhile, Wiener filtering in the wavelet domain has received attention for image denoising. The Wiener filter (WF) requires knowledge of the variances of the unknown original signals, and assumptions are made to estimate the variance from the noisy data. In [4], Kazubek proposed a thresholding-based WF, which is an approximate analysis of the error occurring in the empirical WF. In [5], the authors proposed an algorithm for determining the variable size of the locally adaptive window using a region-based approach. In [6], Mihçak et al. presented a locally adaptive window-based denoising algorithm using MAP (LAWMAP) by learning the signal variance which is determined by maximizing the a posteriori estimate [6]. Meanwhile, many models have been studied in order to obtain accurate and efficient wavelet coefficients. For example, the hidden Markov model (HMT) based wavelet domain framework has been proposed to capture the statistical dependencies [7]. As described above, the estimation of the signal variance in a noisy environment is a critical issue in WF-based denoising. We estimate the variance of every wavelet coefficient based on the local neighborhood of the wavelet coefficient. This assumption is based on the knowledge that the wavelet coefficients are locally independent and identically distributed (IID), and the locally IID assumption becomes inaccurate as the neighborhood grows [6]. Meanwhile, window size is another important issue in the denoising problem. The method proposed in [5] is wavelet-based WF with a nearly arbitrarily shaped window (NASW). The NASW has a drawback in that it uses a large window size which degrades the image detail, especially in the edge area. We introduce high resolution estimation to alleviate this problem. By applying the proposed algorithm, the edges and the textures in the images are preserved. The rest of the paper is organized as follows. The locally adaptive WF is introduced in Section 2. In Section 3, we describe the proposed algorithm. In Section 4, the experimental results and the corresponding discussion are provided. Finally, conclusions are drawn in Section 5.
2. The Conventional Wiener Filter Let us assume a noisy image in a wavelet domain with the following degradation process: 2
X S N,
(1)
where X is given by the observed noisy image with size W×H and X=[xi,j]W×H; i=0,…,H-1; j=0, …, W-1; S is the noise-free clean image S=[si,j]W×H; and N denotes the additive white Gaussian noise with zero mean and a finite variance σ2n. The noise, N, is an independent Gaussian variable that is not correlated with the signal S, and our goal is to obtain the best estimate of S from X. The form of the WF simplifies to the following scalar relation: si , j i , j xi , j
i, j
,
(2)
E xi2, j
(3)
E si2, j
.
Equation (2) shows that we can represent S as a linear combination of {xi,j}, which is formulated as S X ,
(4)
where the operation “·” is point to point multiplication, and Ψ=[ψij]W×H is shown in Eq. (3). Since the noise-free signal S is uncorrelated with the noise N, we can write
E si2, j E xi2, j n2 .
is estimated from the neighborhood of the coefficient x E x as follows:
2 The expected value E xi , j
We approximate
(5)
i,j.
2 i, j
E xi2, j
1 Ci , j
xm ,n Ci , j
xm2 ,n ,
(6)
where Ci,j is defined as the set of all coefficients within a local window of size MH×MV that is centered at location (i, j), and |Ci,j| is the cardinality of Ci,j.
i, j
E x 2 i, j
E x
2 i, j
3
2 n
,
(7)
where
E x 2 i, j
2 n
max E xi2, j n2 ,0 . Thus we can see that the estimation of
E xi2, j is a key issue in the WF-based denoising method.
3. The Proposed Algorithm
(a) (b) Fig. 1 Comparison of the PSNR of different window sizes for (a) LAWMAP and (b) WF. A wavelet-based WF with NASW denoising was presented in [5]. The NASW has a drawback because it employs a large window size, which degrades the image detail. The dependence between the wavelet coefficients decreases when the window size increases. Therefore, the size of the locally adaptive window is also an important factor in estimating the signal power. In most WF-based algorithms, a large window size causes poor performance. This is because the relationship between the pixels is weakened in a large window, especially for the complex region. However, a small window size is not appropriate either because a larger region to estimate the signal variance results in a more reliable estimation. The peak signal-to-noise ratio (PSNR) differences of LAWMAP [6] and WF are shown in Fig. 1(a) and Fig. 1(b), where the different window size (3×3, 5×5, and 7×7) are compared. As shown, the PSNR results are the best for a 5×5 window size. To maintain consistency with NASW and to preserve the edge information, we introduce a new method called high-resolution estimation WF (HRE-WF), and we estimate the image variance from the high-resolution values that are expanded from the low-resolution image. For X, from low resolution to high-resolution, we utilize the traditional wavelet coefficient technique, which is 2-D uploading. Then we can expand X
~
to be X as shown in Eq. (8) and Eq. (9). 4
x i i , if i is even and j is even , xi , j 2 2 , 0, otherwise x0,0 x 1,0 x2,0 X x2 H 3,0 x2 H 2,0 x0,0 0 x1,0 0 xH 1,0
x0,1 x1,1 x2,1
x0,2 x1,2 x2,2
x0,2W 3 x1,2W 3 x2,2W 3
x2 H 3,1 x2 H 2,1
x2 H 3,2 x2 H 2,2
x2 H 3,2W 3 x2 H 2,2W 3
0 0 0
x0,1 0 x1,1
0 0 0 xH 1,1
0 0 0 xH 1,W 1 0 0 0
x0,W 1 0 x1,W 1
(8)
x2 H 3,2W 2 x2 H 2,2W 2 x0,2W 2 x1,2W 2 x2,2W 2
.
(9)
The conventional WF represents S as a linear combination of X. However, in general, there is no Ψ for which equation Eq. (4) has solutions because S span{xi , j } . Suppose we allow a small perturbation Y such that S X Y .
(10)
Fig. 2 The relationship between the noisy signal X, the original signal S, the noise N, and the perturbation Y. 5
We set X and Y to be orthogonal. The relationship in the vector space is shown in Fig. 2.
~
Based on the above discussion, we expand X to X . To maintain consistency, Y should be
~
~
the same size as X. Therefore, Y is expanded to Y . The simplest way to ensure Y is
~
orthogonal to X , is to set yi , j 0 , where i and j are all even numbers. Then no matter
~
~
~
~
what the other coefficients of Y are Y X . Now, we can write Y in matrix form as shown in Eq. (11).
0 y 1,0 0 Y y2 H 3,0 0
y0,1 y1,1 y2,1
0 y1,2 0
y2 H 3,1 y2 H 2,1
y2 H 3,2 0
y0,2W 3 y1,2W 3 y2,2W 3 y2 H 3,2W 3 y2 H 2,2W 3
y1,2W 2 0 y2 H 3,2W 2 0 . 0
~
(11)
Equation (11) shows that deriving Y is critical in the proposed algorithm. However, in the denoising problem, only the noisy image data is available. Therefore Y should be estimated from X. We set up seven models to investigate the relationship between X and Y. A brief introduction of these models is given in Table 1. Table 1 Representation of Different Methods to Estimate Y Model # M#: Description Model 1 (zero padding) M1: No perturbation, i.e., Y = 0 Model 2 (repeating) M2: Y repeats the value of X Model 3 (average) M3: Y is average of the four neighborhood values Model 4 (median) M4: Y is the median of the five neighborhood values Model 5 (bilinear) M5: Bilinear is an interpolation technique Model 6 (bicubic) M6: Bicubic is an interpolation technique Model 7 (MMSE) M7: Y is a linear combination of four nearest neighbors
6
Fig. 3 Comparison of the PSNR of different models for the Lena image. To obtain the optimal estimate and preserve the edge information, we use MMSE [14] to
~
derive Y as shown in Fig. 3. Moreover, we make a small modification for the threshold Tk used in NASW. In our algorithm, Tk is set to Tk 2L J ,
(12)
where κ=0.1, L=log2W, scale factor λ = 0,…,J, from the finest scale to the coarsest scale, and J is the coarsest scale number. The flowchart of the proposed algorithm is shown in Fig. 4.
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Noisy Image
Wavelet Transform
Low Frequency
High Frequency
MMSE
High Resolution
NASW
Update High Frequency
Denoised Image
Fig. 4 Flowchart of the proposed algorithm.
4. Experimental Results The simulation study was performed with standard test images Lena and Barbara, and WaveLab library [15] to verify the superiority compared to the existing algorithms: DTWF [4], NASW [5], HMT [7], EWD [8], and adaptive nonlinear diffusion (AND) based image denoising [16]. In our algorithm, a five-scale orthogonal wavelet transform by utilizing “symmlet” with eight vanishing moments was applied. Table 2 provides the PSNR results for different noise standard deviations for our proposed method compared to those of conventional methods. The proposed algorithm performs better than the other methods. It also provides the best results for both images with a variety of noise standard deviations. For a subjective performance evaluation of the denoised images in terms of the visual effect, we also show part of the perceived image quality of Lena in Fig. 5 [17-19]. The original Lena image is contaminated by noise with σn=20. It is obvious that the proposed algorithm is effective at reducing the image noises while preserving the edge information. It is also yields better visual quality for the details of the Lena image than the conventional methods. The simulation results demonstrate the superiority of the objective and perceived quality of the proposed algorithm.
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Table 2 PSNR Results for Different Noise Standard Deviations Compared with Existing Algorithms Image σn NoisyIm WF DTWF [4] NASW [5] HMT [7] EWD [8] AND [16] Proposed
10 28.13 32.58
15 24.62 30.43
Lena 20 22.15 29.62
25 14.62 28.12
avg. 22.38 30.19
10 28.16 30.36
15 24.67 28.01
Barbara 20 22.17 26.88
25 20.25 25.21
avg. 23.81 27.62
34.08
32.01
30.18
29.52
31.45
31.38
29.01
27.35
26.22
28.49
34.09
32.02
30.16
29.54
31.45
31.4
29.03
27.46
26.29
28.56
33.89 32.94
31.82 31.01
30.41 29.73
29.36 28.76
31.37 30.61
31.36 31.13
29.23 29.11
27.8 27.25
25.99 26.06
28.60 28.39
32.70
30.56
29.79
28.16
30.30
26.64
26.28
25.28
24.96
25.79
34.4
32.59
31.3
30.27
32.14
32.06
30.03
28.36
27.03
29.37
5. Conclusions The wavelet-based WF used in [5] does not adequately preserve the edges in an image and requires a large window size. To overcome these drawbacks, we proposed a highresolution estimation WF in the wavelet domain that determines the signal power while preserving the edge information. We assumed that a noisy image is composed of noise and an original image that are mutually orthogonal. Based on this assumption, we utilized the local covariance to acquire high resolution coefficients from the low resolution coefficients, and determined the signal variance in the Wiener filter by using the highresolution signals. The proposed algorithm is effective at reducing the image noise while preserving the edge information. The empirical results demonstrate the superior quality, both objective and perceived, of the proposed algorithm. Acknowledgment This research was supported by Post-Doctor Research Program (2015) through the Incheon National University (INU), Incheon, South Korea. Reference [1] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inform. Theory 41 (1995) 613-627. [2] L. Sendur, I. W. Selesnick, Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency, IEEE Trans. Signal Process. 50 (2002) 2744-2756. [3] L. Sendur, I.W. Selesnick, Bivariate shrinkage with local variance estimation, IEEE Signal Process. Lett. 9 (2002) 438–441. [4] J. Khan, S. M. A. Bhuiyan, G. Murphy, M. Arline, Embedded-zerotree-wavelet-based data denoising and compression for smart grid, IEEE Trans. Industry Applications 51 (2015) 4190-4200. 9
[5] S. Gaci, The use of wavelet-based denoising techniques to enhance the first-arrival picking on seismic traces, IEEE Trans. Geoscience and Remote Sensing 52 (2014) 45884563. [6] A. Varsha, P. Basu, An improved dual tree complex wavelet transform based image denoising using GCV thresholding, in Proc. Int’l Conf. Computational Systems and Communications (ICCSC), 2014. [7] M. S. Crouse, R. D. Nowak, R. G. Baraniuk, Wavelet-based statistical signal processing using hidden Markov models, IEEE Trans. Signal Process. 46 (1998) 886-902. [8] J. M. Parmar, S. A. Patil, Performance evaluation and comparison of modified denoising method and the local adaptive wavelet image denoising method, in Proc. Int’l Conf. Intelligent Systems and Signal Processing (ISSP), 2013. [9] S. G. Chang, B. Yu, M. Vetterli, Adaptive wavelet thresholding for image denoising and compression, IEEE Trans. Image Process. 9 (2000) 1532-1546. [10] F. Luisier, T. Blu, M. Unser, A new SURE approach to image denoising: interscale orthonormal wavelet thresholding, IEEE Trans. Image Process. 16 (2007) 593-606. [11] C. Jung, L. Jiao, H. Kim, J. Kim, Spatial-gradient-local-inhomogeneity: an efficient image-denoising prior, J. Electron. Imaging, 19 (2010) 033005. [12] C. Y. Dang, J. M. Gao, Z. Wang, F. M. Chen, Y. L. Xiao, Optimized wavelet denoising algorithm using hybrid noise model for radiographic images, in Proc. Int’l Conf. of Computational Intelligence and Design (ISCID), 2014. [13] M. Nikpour, H. Hassanpour, Using diffusion equations for improving performance of wavelet-based image denoising techniques, IET Image Processing, 4 (2010) 452-462. [14] X. Li, M. T. Orchard, New edge-directed interpolation, IEEE Tran. Image Process. 10 (2001) 1521-1527. [15] J. B. Bucheit, S. Chen, D. L. Donoho, I. M. Johnston, J. D. Scargle. WaveLab Toolkit. [Online]. Available: http://www-stat.stanford.edu/~wavelab/ [16] A. K. Mandava, E, E. Regentova, Image denoising based on adaptive nonlinear diffusion in wavelet domain, J. Electron. Imaging 20 (2011) 033016. [17] G. Jeon, Denoising in contrast-enhanced X-ray images, Sensing and Imaging, 17 (2016) 1-11. [18] J. Wang, G. Jeon, J. Jeong, Image de-noising using adaptive directional Wiener filter in wavelet domain, in Proc. ICEIC2012, Jeongseon, Korea, 2012. [19] T. Bai, H. Wang, Y. Pang, G. Li, J. Lin, Q. Zhou, G. Jeon, A PPG signal de-noising method based on the DTCWT and the morphological filtering, in Proc. IEEE SITIS 2016, Nov. 2016.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 5 (a) Original Lena image and (b) noisy Lena image. Perceived image quality using various denosing methods: (c) WF, (d) DTWF, (e) NASW, (f) HMT, (g) BSF, (h) AND, and (i) the proposed method.
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We propose a novel Wiener filter with high-resolution estimation that determines the signal power while preserving the edge information. We assume that a noisy image is composed of noise and the original image, which are mutually orthogonal. We utilize the local covariance to obtain high-resolution coefficient from the low-resolution coefficients. We also utilize the local covariance to estimate the signal variance in the Wiener filter by using the high resolution values.