A New Interscale and Intrascale Orthonormal Wavelet Thresholding for
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
139
A New Interscale and Intrascale Orthonormal Wavelet Thresholding for SURE-Based Image Denoising Fengxia Yan, Lizhi Cheng, and Silong Peng
Abstract—The interscale Stein’s unbiased risk estimator (SURE)-based approach introduced by Luisier is a recent state of the art in orthonormal wavelet denoising, but it is not very effective for those images that have substantial high-frequency contents. To solve this problem, we introduce an effective integration of the intrascale correlations within the interscale SURE-based approach. We show that the consideration of both the intrascale and interscale dependencies of wavelet coefficients brings more denoising gains than those obtained with the interscale SURE-based approach, especially for denoising of images that have substantial textures such as the Barbara image. Index Terms—Image denoising, intrascale and interscale dependencies, Stein’s unbiased risk estimator (SURE) minimization.
I. INTRODUCTION N IMAGE is often corrupted by noise in its acquisition or transmission and noise elimination is still one of the most fundamental, widely studied, and largely unsolved problems in computer vision and image processing. In image denoising, a compromise has to be found between noise suppression and the preservation of the important image features. To achieve a good performance in this respect, a denoising algorithm has to adapt to image discontinuities. The wavelet representation naturally facilitates the construction of such spatially adaptive algorithms. In particular, Donoho and Johnstone [1] developed a very simple denoising procedure consisting in thresholding the noisy wavelet coefficients (shrinkage). Indeed, the wavelet transform is good at energy compaction, small coefficients are more likely due to noise, and large coefficients due to important signal features (such as edges). Wavelet shrinkage is, thus, effective for signals with sparse wavelet representations. In this context, the key issue is the threshold value selection. From an asymptotically minimax analysis, Donoho and Johostone have proposed to apply the universal threshold in the VisuShrink method. Subsequently, they suggested to choose
A
Manuscript received July 8, 2007; revised October 31, 2007. This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 60573027 and Grant No. 10601068. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Philippe Salembier. F.-X. Yan and L.-Z. Cheng are with the Department of Mathematics and System, School of Sciences, National University of Defense Technology, Changsha 410073, China (e-mail: [email protected]; [email protected]). S.-L. Peng is with the Institute of Automation, Chinese Academy of Sciences, Beijing 100080, China (e-mail: [email protected]). Digital Object Identifier 10.1109/LSP.2007.914790
the optimal threshold by minimizing Stein’s unbiased risk estimator (SURE) [2] and called the corresponding denosing approach SureShrink [3]. The SURE principle was also used by Pesquet et al. to develop sophisticated multivariate estimates for multicomponent image denoising [4]–[6]. Recently, by exploiting the Stein’s principle, Luisier et al. introduced a new interscale SURE-based approach to orthonormal wavelet image denoising that does not need any prior statistical modelization of the wavelet coefficients [7]. Instead of postulating a statistical model for the wavelet coefficients, they directly parameterized the denoising process as a sum of elementary nonlinear processes with unknown weights. The key point of their techniques is to take advantage of Stein’s MSE estimate, which depends on the noisy image alone, not on the clean one. Using this approach, they designed an image denoising algorithm that takes into account interscale dependencies, but discards intrascale correlations. In order to compensate for feature misalignment, they developed a rigorous procedure based on the relative group delay between the scaling and wavelet filters-group delay compensation. Their denoising results demonstrated that, for most of the images, the interscale SURE-based approach is competitive with the best techniques available that consider nonredundant orthonormal transforms, but with the noteworthy exception for images that have substantial high-frequency contents such as the Barbara image. The main reason for that is that the interscale correlations may be too weak for such images, which indicates that an efficient denoising process may require intrascale information as well. In this letter, we will create an orthonormal wavelet thresholding which considering both the intrascale and interscale dependencies of the wavelet coefficients. Specifically, we define a local spatial predictor to capture the intrascale correlations and propose the intrascale and interscale wavelet thresholding within the SURE-based approach. Experiments results show that our algorithm is superior to the interscale SURE-based denoising approach, and especially efficient for those images that have substantial textures such as the Barbara image. II. SURE-BASED INTRASCALE AND INTERSCALE THRESHOLDING In this letter, the denoising of an image corrupted by additive will be considered, i.e., white Gaussian noise with variance
(1) where are independent and identically distributed and independent of . The (i.i.d.) as normal
1070-9908/$25.00 © 2008 IEEE
140
goal is to obtain an estimate of mizes the mean-squared error (MSE)
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
which mini-
(2) An orthogonal wavelet transformation of the noisy input yields an equivalent additive white noise model in the wavelet domain. In each wavelet subband at a given scale (there are three subbands in each scale) and orientation, we have (3) where are noisy wavelet coefficients, are noise-free are i.i.d. normal random variables, wavelet coefficients, is the which are statistically independent from , and number of coefficients in a subband. Here, we will only consider orthonormal wavelet transform, so the MSE in the space domain defined in (2) is a weighted sum of the MSE of each individual wavelet subband. Different from Luisier’s bivariate denoising function which only considers interscale dependencies, in this letter, we choose to estimate each by the following pointwise denoising function which takes into account both interscale and intrascale dependencies: (4) where is a (weakly) differentiable function from to is the interscale predictor of , and is the intrascale predictor of . In order to compensate for feature misalignment, we also use the group delay compensation procedure [7] which is based on the relative group delay between the scaling and wavelet filters to obtain the interscale predictor . Then, we introduce a local spatial activity indicator [9] (LSAI) as an intrascale predictor as follows:
(5)
In practice, we only have access to the noisy signal and not to the original signal . In (6), we, thus, need to remove the explicit dependence on . Since has no influence in the minimization process, we do not need to estimate it. The remaining problematic term . However, for Gaussian adis only ditive noise, Stein’s formula applied to our multivariate model in terms of oballows to express served data only. Recall that the randomness of only results from the Gaussian white noise , because no statistical model is assumed on the noise-free data ; and here, we only consider an orthonormal transform—which transforms Gaussian white noise into Gaussian white noise, that is, any wavelet coefficient other than is independent of . As defined in (5), the intrascale predictor is a weighted sum of the magnitude of the coefficients in the neighborhood of . Thus, the interscale predictor (which is built out of the lowpass subband at the same scale as ) and the intrascale predictor are both independent of . Consequently, according to Stein’s theorem [2], we have
(7)
is an unbiased estimator of the MSE defined in (6), i.e.,
(8)
and where is expectation operator. Note that variables are both independent of ; it is easy to proof (8), and the concrete proof procedure is similar with [7, proof of Theorem 1]. and the LSAI into the To suitably integrate the parent pointwise denoising function, we use them as discriminators between high SNR wavelet coefficients and low SNR wavelet coefficients and choose to build a linearly parameterized denoising function of the following form:
Here, represents the locally averaged magnitude of the coof a fixed size efficients in a relatively small square window , and to be derived with respect to its center component (but is not included). Our goal is to find a function that minimizes the MSE of each wavelet subband (9)
(6)
In (9), the function depends linearly on parameters, , and —degree of freedom—which we will determine exactly by minimizing . For the sake of convenience, we put the paand together into a vector rameters , and then for ; at the same time, we define the weight variables
YAN et al.: NEW INTERSCALE AND INTRASCALE ORTHONORMAL WAVELET THRESHOLDING
141
TABLE I COMPARISON OF SEVERAL DENOISING ALGORITHMS
as for and for and for . If we introduce (9) into the estimate of the MSE given in (8), according to (7), perform differentiations over the parameters , we have for all
(10) It is equivalent to
(11)
These equations can be summarized in matrix form as , where and are is a matrix of vectors of size 6 1, and size 6 6. This linear system is solved for by (12) Formulation (9) is a general form of denoising function. Now, and the decision we should choose suitable basis functions function . As in Luisier’s work, we choose the basis functions as derivatives of Gaussian (DOG) as follows:
(13) , and these basis functions can guarantee that where the denoising function in (9) satisfies differentiability, antisymmetry, and linear behavior of large coefficients [7]. Then, we choose the decision function in (9) as
(14)
Substituting the formulations (14) and (13) into the formulation (9), we can obtain our interscale and intrascale-dependent denoising function. Then, the procedure of our proposed intrascale and interscale SURE-based orthonormal wavelet denoising can be summed up as follows. 1) Perform a level discrete wavelet transform (DWT) to the noisy data and obtain noisy wavelet . subimages 2) For each wavelet subimages, compute the interscale preusing group delay compensation and dictor compute the intrascale predictor using (5). , and 3) Solve the linear system to obtain the parameters according to (14) and (12). estimates of the noise-free high4) Compute using (4) and (14). pass subbands 5) Reconstruct the denoised image by applying the inverse discrete wavelet transform (IDWT) on the processed highpass wavelet subimages to obtain an esof the noise-free data timate .
III. EXPERIMENTAL RESULTS We tested our algorithm on “Barbara,” which represent those images that have substantial textures, and “Boat,” which represent those images that have less textures, to make a comparison with some other related algorithms [7]–[9]. To compare with Luisier’s interscale SURE-based method, we also use a critically sampled orthornormal wavelet basis with eight vanishing moments (sym8) over five decomposition stages. For the tested Barbara image, when integrating the intrascale dependencies, the window size 5 5 yielded maximum PSNR. For those tested imaged with less texture, the window size 3 3 yielded maximum PSNR. Experimental results in Table I show that the resulting SURE-based method which consider the interscale and intrascale dependencies always yields an improved PSNR as compared to the interscale version [7], and also, superior to some selected techniques reflecting the state of the art in orthonormal wavelet denoising [8], [9]. When compared to the interscale SURE-based approach [7], db, and for textural images, the PSNR gains are up to for those images with less texture, the PSNR gains are about db. Even though these PSNR gains may seem marginal, the differences can be seen visually. The visual improvement
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IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
Fig. 1. (a) Part of the noisy Barbara image, PSNR = 22:11 dB. (b) Denoised result using the interscale SURE-based approach, PSNR = 27:96 dB. (c) Denoised result using our interscale and intrascale-dependent thresholding function (9): PSNR = 28:88 dB.
mainly consists of better suppressing noise in uniform areas as can be seen from Fig. 1. IV. CONCLUSION In this letter, improvements to the original interscale SURE orthonormal wavelet image denoising method introduced in [7] were proposed. In order to efficiently integrate the intrascale correlations within the SURE-based approach, we introduce a LSAI to describe the interscale dependencies. Experimental results show that the proposed algorithm gives better denoising performance, especially for images that have substantial highfrequency contents. For the next step, we will work on a more effective integration of the intrascale and interscale correlations within the SURE-based approach. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive suggestions that improved the presentation of this letter. The authors also would like to thank F. Luisier, T. Blu, M. Unser, and their Biomedical Imaging Group (BIG) for generously sharing the source codes for the interscale SUREbased denoising.
REFERENCES [1] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, 1994. [2] C. Stein, “Estimation of the mean of a multivariate normal distribution,” Ann. Statist., vol. 9, pp. 1135–1151, 1981. [3] D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Amer. Statist. Assoc., vol. 90, no. 432, pp. 1200–1224, Dec. 1995. [4] A. Benazza-Benyahia and J.-C. Pesquet, “An extended SURE approach for multicomponent image denoising,” in Proc. ICASSP’2004, Montreal, QC, Canada, May 2004. [5] C. Chaux, A. Benazza-Benyahia, and J. C. Pesquet, “Using Stein’s principle for multichannel image denoising,” in Proc. IEEE EURASIP Int. Symp. Control, Communications, and Signal Processing (ISCCSP 2006), Marrakech, Morocco, Mar. 2006. [6] A. Benazza-Benyahia and J. C. Pesquet, “Building robust wavelet estimators for multicomponent image using Stein’s principle,” IEEE Trans. Image Process., vol. 14, no. 11, pp. 1814–1830, Nov. 2005. [7] F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: Interscale ortonormal wavelet thresholding,” IEEE Trans. Image Process., vol. 16, no. 3, pp. 593–606, Mar. 2007. [8] L. Sendur and I. W. Selesnick, “Bivariate shrinkage with local variance estimation,” IEEE Signal Process. Lett., vol. 9, no. 12, pp. 438–441, Dec. 2002. [9] A. Pizurica and W. Philips, “Estimating the probability of the presence of a signal of interest in multiresolution single—and multiband image denoising,” IEEE Trans. Image Process., vol. 15, no. 3, pp. 645–665, Mar. 2006.
139
A New Interscale and Intrascale Orthonormal Wavelet Thresholding for SURE-Based Image Denoising Fengxia Yan, Lizhi Cheng, and Silong Peng
Abstract—The interscale Stein’s unbiased risk estimator (SURE)-based approach introduced by Luisier is a recent state of the art in orthonormal wavelet denoising, but it is not very effective for those images that have substantial high-frequency contents. To solve this problem, we introduce an effective integration of the intrascale correlations within the interscale SURE-based approach. We show that the consideration of both the intrascale and interscale dependencies of wavelet coefficients brings more denoising gains than those obtained with the interscale SURE-based approach, especially for denoising of images that have substantial textures such as the Barbara image. Index Terms—Image denoising, intrascale and interscale dependencies, Stein’s unbiased risk estimator (SURE) minimization.
I. INTRODUCTION N IMAGE is often corrupted by noise in its acquisition or transmission and noise elimination is still one of the most fundamental, widely studied, and largely unsolved problems in computer vision and image processing. In image denoising, a compromise has to be found between noise suppression and the preservation of the important image features. To achieve a good performance in this respect, a denoising algorithm has to adapt to image discontinuities. The wavelet representation naturally facilitates the construction of such spatially adaptive algorithms. In particular, Donoho and Johnstone [1] developed a very simple denoising procedure consisting in thresholding the noisy wavelet coefficients (shrinkage). Indeed, the wavelet transform is good at energy compaction, small coefficients are more likely due to noise, and large coefficients due to important signal features (such as edges). Wavelet shrinkage is, thus, effective for signals with sparse wavelet representations. In this context, the key issue is the threshold value selection. From an asymptotically minimax analysis, Donoho and Johostone have proposed to apply the universal threshold in the VisuShrink method. Subsequently, they suggested to choose
A
Manuscript received July 8, 2007; revised October 31, 2007. This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 60573027 and Grant No. 10601068. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Philippe Salembier. F.-X. Yan and L.-Z. Cheng are with the Department of Mathematics and System, School of Sciences, National University of Defense Technology, Changsha 410073, China (e-mail: [email protected]; [email protected]). S.-L. Peng is with the Institute of Automation, Chinese Academy of Sciences, Beijing 100080, China (e-mail: [email protected]). Digital Object Identifier 10.1109/LSP.2007.914790
the optimal threshold by minimizing Stein’s unbiased risk estimator (SURE) [2] and called the corresponding denosing approach SureShrink [3]. The SURE principle was also used by Pesquet et al. to develop sophisticated multivariate estimates for multicomponent image denoising [4]–[6]. Recently, by exploiting the Stein’s principle, Luisier et al. introduced a new interscale SURE-based approach to orthonormal wavelet image denoising that does not need any prior statistical modelization of the wavelet coefficients [7]. Instead of postulating a statistical model for the wavelet coefficients, they directly parameterized the denoising process as a sum of elementary nonlinear processes with unknown weights. The key point of their techniques is to take advantage of Stein’s MSE estimate, which depends on the noisy image alone, not on the clean one. Using this approach, they designed an image denoising algorithm that takes into account interscale dependencies, but discards intrascale correlations. In order to compensate for feature misalignment, they developed a rigorous procedure based on the relative group delay between the scaling and wavelet filters-group delay compensation. Their denoising results demonstrated that, for most of the images, the interscale SURE-based approach is competitive with the best techniques available that consider nonredundant orthonormal transforms, but with the noteworthy exception for images that have substantial high-frequency contents such as the Barbara image. The main reason for that is that the interscale correlations may be too weak for such images, which indicates that an efficient denoising process may require intrascale information as well. In this letter, we will create an orthonormal wavelet thresholding which considering both the intrascale and interscale dependencies of the wavelet coefficients. Specifically, we define a local spatial predictor to capture the intrascale correlations and propose the intrascale and interscale wavelet thresholding within the SURE-based approach. Experiments results show that our algorithm is superior to the interscale SURE-based denoising approach, and especially efficient for those images that have substantial textures such as the Barbara image. II. SURE-BASED INTRASCALE AND INTERSCALE THRESHOLDING In this letter, the denoising of an image corrupted by additive will be considered, i.e., white Gaussian noise with variance
(1) where are independent and identically distributed and independent of . The (i.i.d.) as normal
1070-9908/$25.00 © 2008 IEEE
140
goal is to obtain an estimate of mizes the mean-squared error (MSE)
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
which mini-
(2) An orthogonal wavelet transformation of the noisy input yields an equivalent additive white noise model in the wavelet domain. In each wavelet subband at a given scale (there are three subbands in each scale) and orientation, we have (3) where are noisy wavelet coefficients, are noise-free are i.i.d. normal random variables, wavelet coefficients, is the which are statistically independent from , and number of coefficients in a subband. Here, we will only consider orthonormal wavelet transform, so the MSE in the space domain defined in (2) is a weighted sum of the MSE of each individual wavelet subband. Different from Luisier’s bivariate denoising function which only considers interscale dependencies, in this letter, we choose to estimate each by the following pointwise denoising function which takes into account both interscale and intrascale dependencies: (4) where is a (weakly) differentiable function from to is the interscale predictor of , and is the intrascale predictor of . In order to compensate for feature misalignment, we also use the group delay compensation procedure [7] which is based on the relative group delay between the scaling and wavelet filters to obtain the interscale predictor . Then, we introduce a local spatial activity indicator [9] (LSAI) as an intrascale predictor as follows:
(5)
In practice, we only have access to the noisy signal and not to the original signal . In (6), we, thus, need to remove the explicit dependence on . Since has no influence in the minimization process, we do not need to estimate it. The remaining problematic term . However, for Gaussian adis only ditive noise, Stein’s formula applied to our multivariate model in terms of oballows to express served data only. Recall that the randomness of only results from the Gaussian white noise , because no statistical model is assumed on the noise-free data ; and here, we only consider an orthonormal transform—which transforms Gaussian white noise into Gaussian white noise, that is, any wavelet coefficient other than is independent of . As defined in (5), the intrascale predictor is a weighted sum of the magnitude of the coefficients in the neighborhood of . Thus, the interscale predictor (which is built out of the lowpass subband at the same scale as ) and the intrascale predictor are both independent of . Consequently, according to Stein’s theorem [2], we have
(7)
is an unbiased estimator of the MSE defined in (6), i.e.,
(8)
and where is expectation operator. Note that variables are both independent of ; it is easy to proof (8), and the concrete proof procedure is similar with [7, proof of Theorem 1]. and the LSAI into the To suitably integrate the parent pointwise denoising function, we use them as discriminators between high SNR wavelet coefficients and low SNR wavelet coefficients and choose to build a linearly parameterized denoising function of the following form:
Here, represents the locally averaged magnitude of the coof a fixed size efficients in a relatively small square window , and to be derived with respect to its center component (but is not included). Our goal is to find a function that minimizes the MSE of each wavelet subband (9)
(6)
In (9), the function depends linearly on parameters, , and —degree of freedom—which we will determine exactly by minimizing . For the sake of convenience, we put the paand together into a vector rameters , and then for ; at the same time, we define the weight variables
YAN et al.: NEW INTERSCALE AND INTRASCALE ORTHONORMAL WAVELET THRESHOLDING
141
TABLE I COMPARISON OF SEVERAL DENOISING ALGORITHMS
as for and for and for . If we introduce (9) into the estimate of the MSE given in (8), according to (7), perform differentiations over the parameters , we have for all
(10) It is equivalent to
(11)
These equations can be summarized in matrix form as , where and are is a matrix of vectors of size 6 1, and size 6 6. This linear system is solved for by (12) Formulation (9) is a general form of denoising function. Now, and the decision we should choose suitable basis functions function . As in Luisier’s work, we choose the basis functions as derivatives of Gaussian (DOG) as follows:
(13) , and these basis functions can guarantee that where the denoising function in (9) satisfies differentiability, antisymmetry, and linear behavior of large coefficients [7]. Then, we choose the decision function in (9) as
(14)
Substituting the formulations (14) and (13) into the formulation (9), we can obtain our interscale and intrascale-dependent denoising function. Then, the procedure of our proposed intrascale and interscale SURE-based orthonormal wavelet denoising can be summed up as follows. 1) Perform a level discrete wavelet transform (DWT) to the noisy data and obtain noisy wavelet . subimages 2) For each wavelet subimages, compute the interscale preusing group delay compensation and dictor compute the intrascale predictor using (5). , and 3) Solve the linear system to obtain the parameters according to (14) and (12). estimates of the noise-free high4) Compute using (4) and (14). pass subbands 5) Reconstruct the denoised image by applying the inverse discrete wavelet transform (IDWT) on the processed highpass wavelet subimages to obtain an esof the noise-free data timate .
III. EXPERIMENTAL RESULTS We tested our algorithm on “Barbara,” which represent those images that have substantial textures, and “Boat,” which represent those images that have less textures, to make a comparison with some other related algorithms [7]–[9]. To compare with Luisier’s interscale SURE-based method, we also use a critically sampled orthornormal wavelet basis with eight vanishing moments (sym8) over five decomposition stages. For the tested Barbara image, when integrating the intrascale dependencies, the window size 5 5 yielded maximum PSNR. For those tested imaged with less texture, the window size 3 3 yielded maximum PSNR. Experimental results in Table I show that the resulting SURE-based method which consider the interscale and intrascale dependencies always yields an improved PSNR as compared to the interscale version [7], and also, superior to some selected techniques reflecting the state of the art in orthonormal wavelet denoising [8], [9]. When compared to the interscale SURE-based approach [7], db, and for textural images, the PSNR gains are up to for those images with less texture, the PSNR gains are about db. Even though these PSNR gains may seem marginal, the differences can be seen visually. The visual improvement
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IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
Fig. 1. (a) Part of the noisy Barbara image, PSNR = 22:11 dB. (b) Denoised result using the interscale SURE-based approach, PSNR = 27:96 dB. (c) Denoised result using our interscale and intrascale-dependent thresholding function (9): PSNR = 28:88 dB.
mainly consists of better suppressing noise in uniform areas as can be seen from Fig. 1. IV. CONCLUSION In this letter, improvements to the original interscale SURE orthonormal wavelet image denoising method introduced in [7] were proposed. In order to efficiently integrate the intrascale correlations within the SURE-based approach, we introduce a LSAI to describe the interscale dependencies. Experimental results show that the proposed algorithm gives better denoising performance, especially for images that have substantial highfrequency contents. For the next step, we will work on a more effective integration of the intrascale and interscale correlations within the SURE-based approach. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive suggestions that improved the presentation of this letter. The authors also would like to thank F. Luisier, T. Blu, M. Unser, and their Biomedical Imaging Group (BIG) for generously sharing the source codes for the interscale SUREbased denoising.
REFERENCES [1] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, 1994. [2] C. Stein, “Estimation of the mean of a multivariate normal distribution,” Ann. Statist., vol. 9, pp. 1135–1151, 1981. [3] D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Amer. Statist. Assoc., vol. 90, no. 432, pp. 1200–1224, Dec. 1995. [4] A. Benazza-Benyahia and J.-C. Pesquet, “An extended SURE approach for multicomponent image denoising,” in Proc. ICASSP’2004, Montreal, QC, Canada, May 2004. [5] C. Chaux, A. Benazza-Benyahia, and J. C. Pesquet, “Using Stein’s principle for multichannel image denoising,” in Proc. IEEE EURASIP Int. Symp. Control, Communications, and Signal Processing (ISCCSP 2006), Marrakech, Morocco, Mar. 2006. [6] A. Benazza-Benyahia and J. C. Pesquet, “Building robust wavelet estimators for multicomponent image using Stein’s principle,” IEEE Trans. Image Process., vol. 14, no. 11, pp. 1814–1830, Nov. 2005. [7] F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: Interscale ortonormal wavelet thresholding,” IEEE Trans. Image Process., vol. 16, no. 3, pp. 593–606, Mar. 2007. [8] L. Sendur and I. W. Selesnick, “Bivariate shrinkage with local variance estimation,” IEEE Signal Process. Lett., vol. 9, no. 12, pp. 438–441, Dec. 2002. [9] A. Pizurica and W. Philips, “Estimating the probability of the presence of a signal of interest in multiresolution single—and multiband image denoising,” IEEE Trans. Image Process., vol. 15, no. 3, pp. 645–665, Mar. 2006.
