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Wavelet-based image denoising using variance field diffusion
Optics Communications 285 (2012) 1744–1747
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Wavelet-based image denoising using variance field diffusion Zhenyu Liu a, Jing Tian b,⁎, Li Chen b, Yongtao Wang c a b c
School of Mathematics and Statistics, Zaozhuang University, 277160, PR China School of Computer Science and Technology, Wuhan University of Science and Technology, 430081, PR China Institute of Computer Science & Technology of Peking University, Peking University, Beijing, 100871, PR China
a r t i c l e
i n f o
Article history: Received 4 July 2011 Received in revised form 29 November 2011 Accepted 5 December 2011 Available online 17 December 2011 Keywords: Image restoration Wavelet Diffusion
a b s t r a c t Wavelet shrinkage is an image restoration technique based on the concept of thresholding the wavelet coefficients. The key challenge of wavelet shrinkage is to find an appropriate threshold value, which is typically controlled by the signal variance. To tackle this challenge, a new image restoration approach is proposed in this paper by using a variance field diffusion, which can provide more accurate variance estimation. Experimental results are provided to demonstrate the superior performance of the proposed approach. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Wavelet-based algorithms have been proved to be effective for remove noises from the image, which is often corrupted during image acquisition and image transmission. The signal variance estimation plays a key role in controlling the degree of shrinkage, and consequently controls the quality of the denoised image. The signal variance value is usually estimated based on a local neighborhood of the wavelet coefficient; this local neighborhood could be all of the coefficients in its neighborhood in the same subband [1,2], or the local neighborhood of the wavelet coefficient within the same subband [3–5]. The above-mentioned algorithms assume that the wavelet coefficients are locally independent and identically distributed; therefore, the energy distribution of the image in each subband is isotropic. Non-exact estimation of signal values would result in non-exact estimation of the denoised signal value; consequently, the denoised image would suffer from non-satisfied perceptual visual quality [6–8]. The signal variance fields reflect an important property of the image. The large signal variation values tend to yield at the location of edges, since the wavelet coefficients of the high-pass filtered subbands have large values at these locations. This results in a softer shrinkage of these coefficients. Furthermore, for those pixels located in the vicinity of an edge (say, few pixels away from the edge), even though they might not necessarily fall on an edge, if a softer shrinkage is performed at these locations, the denoised image will yield a better result with more clear edges, and thus a sharper image. In view of ⁎ Corresponding author. E-mail addresses: [email protected] (Z. Liu), [email protected] (J. Tian), [email protected] (L. Chen), [email protected] (Y. Wang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.026
this, a new approach is proposed in this paper to provide a more accurate variance estimation using variance field diffusion. The rest of this paper is organized as follows. A problem formulation is provided in Section 2, followed by the development of the proposed image shrinkage approach. Experimental results are presented in Section 3. Finally, Section 4 concludes this paper. 2. Proposed image shrinkage approach 2.1. Problem formulation A noisy image in a wavelet domain can be mathematically modeled as [4] yði; jÞ ¼ sði; jÞ þ nði; jÞ;
ð1Þ
where y(i, j) is the observed noisy coefficient, s(i, j) is the unknown original (noise-free) coefficient, and n(i, j) is assumed to be a white Gaussian noise with a zero mean and a variance σn2. The goal of image denoising is to recover the signal s(i, j) from the noisy observation y(i, j). Given the signal variance σ 2(i, j) for a wavelet coefficient s(i, j), which is assumed to be an independent Gaussian variable, the minimum mean square error (MMSE) estimator of s(i, j) is given by [4]. ^s ði; jÞ ¼
2 σ ði; jÞ yði; jÞ: σ ði; jÞ þ σ 2n 2
ð2Þ
The key issue of the above method is to estimate the signal variance (i.e., σ 2(i, j) in (2)). Non-exact estimation of signal values would result in non-exact estimation of the denoised signal value; consequently, the denoised image would suffer from non-satisfied
Z. Liu et al. / Optics Communications 285 (2012) 1744–1747
(a). Barbara
(b). W indow
1745
(c). Lighthouse
Fig. 1. Three test images used in this paper.
2.2. Diffusion of the variance field
ð3Þ
j
where (i, j) is the coordinate index of the variance field. The cost function consists of two terms. First, the data term Cd(i, j) reflects the fidelity between the diffused variance field (denoted as f) and the initial estimated field (denoted as g). It is defined as 2 C d ði; jÞ ¼ g ði; jÞ−f ði; jÞ :
j
ð4Þ
Second, the smoothness term Cs(i, j) reflects the smoothness of the diffused variance field f, λ is a regularization parameter that provides the tradeoff between two terms. It is defined as C s ði; jÞ ¼ jf i ði; jÞj þ jf j ði; jÞj;
ð5Þ
where fi and fj denote the partial derivatives of f with respect to the directional variables i and j, respectively. Note that the initial signal variance field g is estimated as g ði; jÞ ¼
i
! ;
ð6Þ
þ
where c(i, j) is defined as the coefficients within a local square window that is centered at the coefficient y(i, j), and |c(i, j)| is the
j
j
j
ð7Þ
By using the calculus [10], the closed-form solution can be obtained iteratively as follows f
ðnþ1Þ
¼ 2f
ðnÞ
2 ðnÞ
−g−λ∇ f
;
ð8Þ
where n is the iteration index, ∇ 2 is the Laplacian operator, that is, 2 ðn Þ
C ¼ ∑ ∑ ðC d ði; jÞ þ λC s ði; jÞÞ;
1 2 2 ∑ y ði; jÞ−σ^ n jcði; jÞj ym;n ∈cði;jÞ
2 C ¼ ∑ ∑ g ði; jÞ−f ði; jÞ þ λf i ði; jÞ þ λf j ði; jÞ :
∇ f
Motivated by the above observations, our approach is to diffuse the large variance values. For that, an intuitive approach is to apply a smoothing (or low-pass) filter to the variance fields. However, this simple solution will inevitably smooth out the large variations on the edges, which is not desirable. Instead of doing so, the variance field is proposed to be adaptively filtered by minimizing the following cost function. i
cardinality (i.e., the number of coefficients) of c(i, j), the output of the function (x)+ is 1, if x > 0; otherwise, the output is 0. Furthermore, the noise variance σn2 is estimated via σ^ n ¼ medianyHH =0:6745 [9], HH where the coefficient y belongs to the highest HH subband. Finally, substituting Eqs. (4) and (5) into Eq. (3), we get the final expression of the cost function as
¼f
ðnÞ
ði þ 1; jÞ þ f
ðnÞ
ði; j þ 1Þ þ f
ðnÞ
ði−1; jÞf
ðnÞ
ði; j−1Þ−4f
ðnÞ
ði; jÞ: ð9Þ
j
The above iteration will be terminated until f^ðnþ1Þ −^f ðnÞ
j j ^f ðnÞ j≤10−4 . =
perceptual visual quality. In view of this, a new approach is proposed in this paper to provide more accurate variance estimation using variance field diffusion. The signal variance fields reflect an important property: The large signal variation values tend to yield at the location of edges, since the wavelet coefficients of the high-pass filtered subbands have large values at these locations. Consequently, σ 2(i, j) ≫ σn2 in Eq. (2); this results in a negligible shrinkage of these coefficients. On the other hand, for a noise-dominated coefficient, that is, σ 2(i, j) ≪ σn2; that results in a large shrinkage of the corresponding coefficient. Furthermore, for those pixels located in the vicinity of an edge (say, few pixels away from the edge), even though they might not necessarily fall on an edge, if a softer shrinkage is performed at these locations, the denoised image will yield a better result with more clear edges, and thus a sharper image.
3. Experimental results Experiments are conducted to evaluate the performance of the proposed method using the well-known test images (as shown in Fig. 1) 512 × 512 Barbara, 512 × 512 Window and 512 × 512 Lighthouse, which serve as the ground truth and compared with the denoised images for Table 1 The PSNR (in dB) performance comparison. Method
σ = 10
σ = 15
σ = 20
σ = 25
Test image Barbara Ref. [11] Ref. [1] Ref. [12] Ref. [13] Proposed approach
30.97 30.85 31.55 31.48 32.50
28.65 28.48 29.30 29.03 30.24
27.01 27.03 27.73 27.64 28.55
25.88 25.99 26.61 26.62 27.32
Test image Window Ref. [11] Ref. [1] Ref. [12] Ref. [13] Proposed approach
30.15 30.30 30.58 29.92 31.02
27.53 27.78 28.19 27.93 28.53
25.90 26.08 26.56 26.26 26.83
24.72 24.84 25.39 25.03 25.58
Test image Lighthouse Ref. [11] Ref. [1] Ref. [12] Ref. [13] Proposed approach
31.98 31.77 32.91 32.88 33.20
29.50 29.43 30.53 30.58 30.83
27.92 27.85 28.92 29.00 29.20
26.76 26.62 27.73 27.74 27.95
1746
Z. Liu et al. / Optics Communications 285 (2012) 1744–1747
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Various results of test image Barbara: (a). noisy image (σ = 20); (b)–(e). Refs. [11,1,12,13], respectively; (f). proposed approach.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. Various results of test image Window: (a). noisy image (σ = 20); (b)–(e). Refs. [11,1,12,13], respectively; (f). proposed approach.
performance comparison, respectively. The noisy images are generated by adding the ground truth image with an additive white Gaussian noise with a zero mean and a standard deviation σn, respectively. The wavelet decomposition is implemented via a five-level decomposition using a Daubechies's wavelet with 8 vanishing moments. The window size used in Eq. (6) for estimating the signal variance is set to be 5 × 5. The regularization parameter λ is experimentally chosen as 0.05, and the number of iterations in diffusion is set to be 5. The first experiment is to compare the proposed method with other four denoising methods [11,1,12,13]. Table 1 compares the
PSNR performances of the above mentioned methods, where the output PSNRs have been averaged over ten noise realizations. Figs. 2, 3, and 4 provide the subjective performance comparison. As seen from the above Table and Figures, the proposed algorithm always outperforms the above four denoising methods to yield the best objective performance as well as the best subjective performance. The second experiment is to explore the computational complexity of the proposed approach. The above denoising approaches are implemented using the Matlab programming language and run on a PC with a Pentium 2.3 GHz CPU and a 3 GB RAM. Ten experiments
Z. Liu et al. / Optics Communications 285 (2012) 1744–1747
(a)
(b)
(c)
(d)
(e)
(f)
1747
Fig. 4. Various results of test image Lighthouse: (a). noisy image (σ = 20); (b)–(e). Refs. [11,1,12,13], respectively; (f). proposed approach.
Acknowledgment
Table 2 The run time performance (in seconds) comparison. Method
Ref. [11]
Ref. [1]
Ref. [12]
Ref. [13]
Proposed approach
Run time
0.98
0.96
0.90
1.06
2.06
are conducted for each of the above-mentioned approaches. Furthermore, the computational complexity (in terms of the run time) of the proposed approach is compared with that of the above six denoising approaches, and their respective average run-times are presented in Table 2.
4. Conclusions A new wavelet-domain image denoising approach has been proposed in this paper. The proposed approach can provides more accurate variance field estimation for providing better denoised image. The proposed approach outperforms the conventional approaches, as verified in our experiments.
This work was supported by National Natural Science Foundation of China (No. 61105010), Hubei Provincial Natural Science Funds for Distinguished Young Scholar of China (No. 2010CDA090). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S.G. Chang, B. Yu, M. Vetterli, IEEE Trans. Image Process. 9 (Sep. 2000) 1522. L. Sendur, I.W. Selesnick, IEEE Signal Process. Lett. 9 (Dec. 2002) 438. I.K. Eom, Y.S. Kim, IEEE Signal Process. Lett. 11 (Dec. 2004) 937. M.K. Mihcak, I. Kozintsev, K. Ramchandran, P. Moulin, IEEE Signal Process. Lett. 6 (Dec. 1999) 300. P.-L. Shui, IEEE Signal Process. Lett. 12 (Oct. 2005) 681. J. Tian, L. Chen, L. Ma, IEICE Electron. Express 7 (Sep. 2010) 1409. J. Tian, W. Yu, L. Ma, Pattern Recognit. Lett. 13 (Oct. 2010) 1751. J. Tian, L. Chen, Biomed. Signal Process. Control 6 (2011). D.L. Donoho, I.M. Johnstone, J. Amer. Statist. Assoc. 90 (1995) 1200. T. Chan, J. Shen, Image processing and analysis: variational, PDE, wavelet, and stochastic methods, Society for Industrial Mathematics, 2005. L. Kaur, S. Gupta, R.C. Chauhan, Proc. Int. Conf. on Computer Vision, Graphics and Image Processing, Dec. 2002, (Ahmadabad, India). F. Luisier, T. Blu, M. Unser, IEEE Trans. Image Process. 16 (Mar. 2007) 593. A. Pizurica, W. Philips, IEEE Trans. Image Process. 15 (Mar. 2006) 654.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Wavelet-based image denoising using variance field diffusion Zhenyu Liu a, Jing Tian b,⁎, Li Chen b, Yongtao Wang c a b c
School of Mathematics and Statistics, Zaozhuang University, 277160, PR China School of Computer Science and Technology, Wuhan University of Science and Technology, 430081, PR China Institute of Computer Science & Technology of Peking University, Peking University, Beijing, 100871, PR China
a r t i c l e
i n f o
Article history: Received 4 July 2011 Received in revised form 29 November 2011 Accepted 5 December 2011 Available online 17 December 2011 Keywords: Image restoration Wavelet Diffusion
a b s t r a c t Wavelet shrinkage is an image restoration technique based on the concept of thresholding the wavelet coefficients. The key challenge of wavelet shrinkage is to find an appropriate threshold value, which is typically controlled by the signal variance. To tackle this challenge, a new image restoration approach is proposed in this paper by using a variance field diffusion, which can provide more accurate variance estimation. Experimental results are provided to demonstrate the superior performance of the proposed approach. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Wavelet-based algorithms have been proved to be effective for remove noises from the image, which is often corrupted during image acquisition and image transmission. The signal variance estimation plays a key role in controlling the degree of shrinkage, and consequently controls the quality of the denoised image. The signal variance value is usually estimated based on a local neighborhood of the wavelet coefficient; this local neighborhood could be all of the coefficients in its neighborhood in the same subband [1,2], or the local neighborhood of the wavelet coefficient within the same subband [3–5]. The above-mentioned algorithms assume that the wavelet coefficients are locally independent and identically distributed; therefore, the energy distribution of the image in each subband is isotropic. Non-exact estimation of signal values would result in non-exact estimation of the denoised signal value; consequently, the denoised image would suffer from non-satisfied perceptual visual quality [6–8]. The signal variance fields reflect an important property of the image. The large signal variation values tend to yield at the location of edges, since the wavelet coefficients of the high-pass filtered subbands have large values at these locations. This results in a softer shrinkage of these coefficients. Furthermore, for those pixels located in the vicinity of an edge (say, few pixels away from the edge), even though they might not necessarily fall on an edge, if a softer shrinkage is performed at these locations, the denoised image will yield a better result with more clear edges, and thus a sharper image. In view of ⁎ Corresponding author. E-mail addresses: [email protected] (Z. Liu), [email protected] (J. Tian), [email protected] (L. Chen), [email protected] (Y. Wang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.026
this, a new approach is proposed in this paper to provide a more accurate variance estimation using variance field diffusion. The rest of this paper is organized as follows. A problem formulation is provided in Section 2, followed by the development of the proposed image shrinkage approach. Experimental results are presented in Section 3. Finally, Section 4 concludes this paper. 2. Proposed image shrinkage approach 2.1. Problem formulation A noisy image in a wavelet domain can be mathematically modeled as [4] yði; jÞ ¼ sði; jÞ þ nði; jÞ;
ð1Þ
where y(i, j) is the observed noisy coefficient, s(i, j) is the unknown original (noise-free) coefficient, and n(i, j) is assumed to be a white Gaussian noise with a zero mean and a variance σn2. The goal of image denoising is to recover the signal s(i, j) from the noisy observation y(i, j). Given the signal variance σ 2(i, j) for a wavelet coefficient s(i, j), which is assumed to be an independent Gaussian variable, the minimum mean square error (MMSE) estimator of s(i, j) is given by [4]. ^s ði; jÞ ¼
2 σ ði; jÞ yði; jÞ: σ ði; jÞ þ σ 2n 2
ð2Þ
The key issue of the above method is to estimate the signal variance (i.e., σ 2(i, j) in (2)). Non-exact estimation of signal values would result in non-exact estimation of the denoised signal value; consequently, the denoised image would suffer from non-satisfied
Z. Liu et al. / Optics Communications 285 (2012) 1744–1747
(a). Barbara
(b). W indow
1745
(c). Lighthouse
Fig. 1. Three test images used in this paper.
2.2. Diffusion of the variance field
ð3Þ
j
where (i, j) is the coordinate index of the variance field. The cost function consists of two terms. First, the data term Cd(i, j) reflects the fidelity between the diffused variance field (denoted as f) and the initial estimated field (denoted as g). It is defined as 2 C d ði; jÞ ¼ g ði; jÞ−f ði; jÞ :
j
ð4Þ
Second, the smoothness term Cs(i, j) reflects the smoothness of the diffused variance field f, λ is a regularization parameter that provides the tradeoff between two terms. It is defined as C s ði; jÞ ¼ jf i ði; jÞj þ jf j ði; jÞj;
ð5Þ
where fi and fj denote the partial derivatives of f with respect to the directional variables i and j, respectively. Note that the initial signal variance field g is estimated as g ði; jÞ ¼
i
! ;
ð6Þ
þ
where c(i, j) is defined as the coefficients within a local square window that is centered at the coefficient y(i, j), and |c(i, j)| is the
j
j
j
ð7Þ
By using the calculus [10], the closed-form solution can be obtained iteratively as follows f
ðnþ1Þ
¼ 2f
ðnÞ
2 ðnÞ
−g−λ∇ f
;
ð8Þ
where n is the iteration index, ∇ 2 is the Laplacian operator, that is, 2 ðn Þ
C ¼ ∑ ∑ ðC d ði; jÞ þ λC s ði; jÞÞ;
1 2 2 ∑ y ði; jÞ−σ^ n jcði; jÞj ym;n ∈cði;jÞ
2 C ¼ ∑ ∑ g ði; jÞ−f ði; jÞ þ λf i ði; jÞ þ λf j ði; jÞ :
∇ f
Motivated by the above observations, our approach is to diffuse the large variance values. For that, an intuitive approach is to apply a smoothing (or low-pass) filter to the variance fields. However, this simple solution will inevitably smooth out the large variations on the edges, which is not desirable. Instead of doing so, the variance field is proposed to be adaptively filtered by minimizing the following cost function. i
cardinality (i.e., the number of coefficients) of c(i, j), the output of the function (x)+ is 1, if x > 0; otherwise, the output is 0. Furthermore, the noise variance σn2 is estimated via σ^ n ¼ medianyHH =0:6745 [9], HH where the coefficient y belongs to the highest HH subband. Finally, substituting Eqs. (4) and (5) into Eq. (3), we get the final expression of the cost function as
¼f
ðnÞ
ði þ 1; jÞ þ f
ðnÞ
ði; j þ 1Þ þ f
ðnÞ
ði−1; jÞf
ðnÞ
ði; j−1Þ−4f
ðnÞ
ði; jÞ: ð9Þ
j
The above iteration will be terminated until f^ðnþ1Þ −^f ðnÞ
j j ^f ðnÞ j≤10−4 . =
perceptual visual quality. In view of this, a new approach is proposed in this paper to provide more accurate variance estimation using variance field diffusion. The signal variance fields reflect an important property: The large signal variation values tend to yield at the location of edges, since the wavelet coefficients of the high-pass filtered subbands have large values at these locations. Consequently, σ 2(i, j) ≫ σn2 in Eq. (2); this results in a negligible shrinkage of these coefficients. On the other hand, for a noise-dominated coefficient, that is, σ 2(i, j) ≪ σn2; that results in a large shrinkage of the corresponding coefficient. Furthermore, for those pixels located in the vicinity of an edge (say, few pixels away from the edge), even though they might not necessarily fall on an edge, if a softer shrinkage is performed at these locations, the denoised image will yield a better result with more clear edges, and thus a sharper image.
3. Experimental results Experiments are conducted to evaluate the performance of the proposed method using the well-known test images (as shown in Fig. 1) 512 × 512 Barbara, 512 × 512 Window and 512 × 512 Lighthouse, which serve as the ground truth and compared with the denoised images for Table 1 The PSNR (in dB) performance comparison. Method
σ = 10
σ = 15
σ = 20
σ = 25
Test image Barbara Ref. [11] Ref. [1] Ref. [12] Ref. [13] Proposed approach
30.97 30.85 31.55 31.48 32.50
28.65 28.48 29.30 29.03 30.24
27.01 27.03 27.73 27.64 28.55
25.88 25.99 26.61 26.62 27.32
Test image Window Ref. [11] Ref. [1] Ref. [12] Ref. [13] Proposed approach
30.15 30.30 30.58 29.92 31.02
27.53 27.78 28.19 27.93 28.53
25.90 26.08 26.56 26.26 26.83
24.72 24.84 25.39 25.03 25.58
Test image Lighthouse Ref. [11] Ref. [1] Ref. [12] Ref. [13] Proposed approach
31.98 31.77 32.91 32.88 33.20
29.50 29.43 30.53 30.58 30.83
27.92 27.85 28.92 29.00 29.20
26.76 26.62 27.73 27.74 27.95
1746
Z. Liu et al. / Optics Communications 285 (2012) 1744–1747
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Various results of test image Barbara: (a). noisy image (σ = 20); (b)–(e). Refs. [11,1,12,13], respectively; (f). proposed approach.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. Various results of test image Window: (a). noisy image (σ = 20); (b)–(e). Refs. [11,1,12,13], respectively; (f). proposed approach.
performance comparison, respectively. The noisy images are generated by adding the ground truth image with an additive white Gaussian noise with a zero mean and a standard deviation σn, respectively. The wavelet decomposition is implemented via a five-level decomposition using a Daubechies's wavelet with 8 vanishing moments. The window size used in Eq. (6) for estimating the signal variance is set to be 5 × 5. The regularization parameter λ is experimentally chosen as 0.05, and the number of iterations in diffusion is set to be 5. The first experiment is to compare the proposed method with other four denoising methods [11,1,12,13]. Table 1 compares the
PSNR performances of the above mentioned methods, where the output PSNRs have been averaged over ten noise realizations. Figs. 2, 3, and 4 provide the subjective performance comparison. As seen from the above Table and Figures, the proposed algorithm always outperforms the above four denoising methods to yield the best objective performance as well as the best subjective performance. The second experiment is to explore the computational complexity of the proposed approach. The above denoising approaches are implemented using the Matlab programming language and run on a PC with a Pentium 2.3 GHz CPU and a 3 GB RAM. Ten experiments
Z. Liu et al. / Optics Communications 285 (2012) 1744–1747
(a)
(b)
(c)
(d)
(e)
(f)
1747
Fig. 4. Various results of test image Lighthouse: (a). noisy image (σ = 20); (b)–(e). Refs. [11,1,12,13], respectively; (f). proposed approach.
Acknowledgment
Table 2 The run time performance (in seconds) comparison. Method
Ref. [11]
Ref. [1]
Ref. [12]
Ref. [13]
Proposed approach
Run time
0.98
0.96
0.90
1.06
2.06
are conducted for each of the above-mentioned approaches. Furthermore, the computational complexity (in terms of the run time) of the proposed approach is compared with that of the above six denoising approaches, and their respective average run-times are presented in Table 2.
4. Conclusions A new wavelet-domain image denoising approach has been proposed in this paper. The proposed approach can provides more accurate variance field estimation for providing better denoised image. The proposed approach outperforms the conventional approaches, as verified in our experiments.
This work was supported by National Natural Science Foundation of China (No. 61105010), Hubei Provincial Natural Science Funds for Distinguished Young Scholar of China (No. 2010CDA090). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S.G. Chang, B. Yu, M. Vetterli, IEEE Trans. Image Process. 9 (Sep. 2000) 1522. L. Sendur, I.W. Selesnick, IEEE Signal Process. Lett. 9 (Dec. 2002) 438. I.K. Eom, Y.S. Kim, IEEE Signal Process. Lett. 11 (Dec. 2004) 937. M.K. Mihcak, I. Kozintsev, K. Ramchandran, P. Moulin, IEEE Signal Process. Lett. 6 (Dec. 1999) 300. P.-L. Shui, IEEE Signal Process. Lett. 12 (Oct. 2005) 681. J. Tian, L. Chen, L. Ma, IEICE Electron. Express 7 (Sep. 2010) 1409. J. Tian, W. Yu, L. Ma, Pattern Recognit. Lett. 13 (Oct. 2010) 1751. J. Tian, L. Chen, Biomed. Signal Process. Control 6 (2011). D.L. Donoho, I.M. Johnstone, J. Amer. Statist. Assoc. 90 (1995) 1200. T. Chan, J. Shen, Image processing and analysis: variational, PDE, wavelet, and stochastic methods, Society for Industrial Mathematics, 2005. L. Kaur, S. Gupta, R.C. Chauhan, Proc. Int. Conf. on Computer Vision, Graphics and Image Processing, Dec. 2002, (Ahmadabad, India). F. Luisier, T. Blu, M. Unser, IEEE Trans. Image Process. 16 (Mar. 2007) 593. A. Pizurica, W. Philips, IEEE Trans. Image Process. 15 (Mar. 2006) 654.