贝叶斯框架下图像恢复及相关技术的研究
|
摘要
作为图像处理和计算机视觉的基础,图像恢复能够解决图像质量的退化问题。在许多应用领域,如遥感、医学成像和军事监察等,图像退化都是一个普遍存在且亟需解决的问题。因此,图像恢复技术受到广泛研究和应用。
本论文研究的重点是贝叶斯框架下的图像恢复问题,包括了参数估计、图像降噪和图像去模糊等。对贝叶斯框架下常见的图像恢复方法进行了深入的研究,针对其中存在的不足之处,如无法找到后验概率闭合形式的解,参数估计缺少良好的准则,最大后验概率属于点估计等,引入了分层最大似然估计方法和变分贝叶斯方法,并通过实验进行了分析验证。
本文的研究成果与创新主要包括:
1、针对实证分析法通常难以找到闭合形式解的问题,提出了一种分层最大似然估计法。实验结果表明,将该方法应用于图像恢复,可以加快图像恢复问题的求解速度,并且获得了比较理想的恢复结果。
2、针对Morozov离差原理、L曲线、广义交叉验证和期望最大化等经典参数估计算法的不足,提出了一种新的贝叶斯参数估计算法,该算法利用分层最大似然估计法,可以在图像恢复的过程中,同步地对多个模型参数进行估计。
3、提出了一种基于扩散技术的图像降噪算法,引入梯度算子作为原始图像的先验模型,并采用变分贝叶斯方法估计原始图像。实验结果表明,该算法克服了传统经验贝叶斯方法的不足。
4、提出了一种基于空间投影和混合模型的彩色图像降噪算法。在YCbCr空间下,该算法采用混合模型作为原始图像的先验模型,并基于分层最大似然估计方法估计原始图像。实验结果表明,该算法可使降噪后图像的信噪比平均提高1dB~3dB。
5、研究了噪声的分布特征,提出用拉普拉斯模分布模型刻画噪声,总变分模型描述原始图像的先验特征,并采用迭代重加权最小二乘法解决该了引入总变分模型和拉普拉斯模型带来的L1-优化问题。实验结果显示,拉普拉斯分布模型可以更真实地反映噪声的分布情况,迭代重加权最小二乘法的引入有效地提高了算法的时间性能。
6、将稀疏表示与分层最大似然估法相结合,提出了一种了稀疏图像去模糊算法。该算法引入Donoho等人的稀疏特征模型刻画系数向量的稀疏性,并采用总变分模型描述原始图像的先验特征,最后利用分层最大似然估计法推导原始图像。实验结果显示,该算法为稀疏表示技术用于图像恢复提供一通用的算法框架。
7、提出了一种基于调和模型图像去模糊算法,该算法利用调和模型刻画原始图像的分布特征,并通过分层最大似然估计法推导原始图像。实验结果表明,与基于最大后验概率的算法相比,该算法的时间性能和恢复结果更优。
8、提出了一种贝叶斯多信道图像盲去模糊算法,算法采用交叉关系模型和平滑模型组成的混合模型来描述点扩散函数的先验特征,同时还引入总变分模型刻画原始图像的先验特征,并基于分层最大似然估计算法推导原始图像和点扩散函数。实验结果表明,相比单信道的图像去模糊算法,多信道图像去模糊算法的恢复结果更优。与同类的多信道算法相比,该算法的速度更快,恢复效果更优。
As the foundation of image processing and computer vision, image restoration can solve the problem of image degradation. In many applications, such as remote sensing, medical imaging, and military reconnaissance, image degradation remainds a common and urgent problem to be solved. Thus, image restoration has always been followed with interest and studied earnestly.
This thesis focuses on the image restoration in Bayesian framework, which mainly contains research on model parameter estimation, image denoising, and image deblurring. By studying classical algorithm in Bayesian framework, several problems are found, such as the lack of a close-formed solution for posterior probability, the Maximum a Posteriori (MAP) method falling into the point estimation, and the lack of good principles for parameter estimation. This thesis proposes a hierarchical maximum likelihood method and introduces the variational Bayesian method in response to these problems. Both methods are experimentally analyzed and verified.
These are the major achievements and innovations:
1. Due to the difficulty of using the evidence analysis method in achieving close-formed solutions, we propose a novel hierarchical maximum likelihood method. The experiments demonstrate that the proposed method can obtain better results with satisfactory speeds when used in image restoration.
2. A novel Bayesian parameter estimation algorithm is proposed to overcome the drawbacks in Morozov discrepancy principle, L-curve method, generalized cross validation, and expectation maximum method. The proposed algorithm uses the hierarchical maximum likelihood method to estimate multiple model parameters together with image restoration.
3. We propose a diffusion-based image denoising algorithm that introduces the gradient operator as the prior model of the original image and utilizes the variational Bayesian method to estimate the original image. Results show that the proposed algorithm overcomes the shortcomings of the classical empirical Bayesian method.
4. We propose a novel color image denoising algorithm based on space projection and hybrid model. In the YCbCr space, the proposed algorithm uses the hybrid model to characterize the original image, and the hierarchical maximum likelihood framework is used to estimate the original image. Experiments indicate that the SNR values of denoised images are improved by 1dB to 2dB on average.
5. By studying the distribution properties of the noise, we introduce the Laplace distribution model to characterize the noises. We also propose the total variation to model the original image. To solve the problem of L1-optimization difficulty caused by the Laplace model and the total variation, we incorporate the iteratively reweighted norm method into our algorithm. Experiments clearly show that the Laplace model gives a true reflection of the noise natures, and the iteratively reweighted norm truly speeds up the proposed algorithm.
6. By combining the sparse representation with the hierarchical maximum likelihood method, a sparse representation-based image deblurring method is proposed. This algorithm introduces Donoho’s sparse model to characterize the sparse coefficient vector, and the total variation model is used to characterize the original image, which is estimated using the hierarchical maximum likelihood method. We theoretically and experimentally prove that the proposed algorithm supplies a unified framework for sparse image restoration.
7. We propose a novel harmonic image deblurring algorithm that uses the harmonic model to characterize the original image, and the hierarchical maximum likelihood method to estimate it. Experiments demonstrate that the proposed algorithm achieves better results with faster speed compared to the MAP-based algorithms.
8. We propose a novel Bayesian multichannel image deblurring algorithm that utilizes a hybrid model composed of the cross relation model and the smoothing model to characterize point spread function. The proposed algorithm also utilizes total variation to model the original image. Point spread function, together with original image, is estimated alternatively using the hierarchical maximum likelihood method. Results show the competitive performance of the proposed algorithm in vision effects and speed compared to other algorithms.
本论文研究的重点是贝叶斯框架下的图像恢复问题,包括了参数估计、图像降噪和图像去模糊等。对贝叶斯框架下常见的图像恢复方法进行了深入的研究,针对其中存在的不足之处,如无法找到后验概率闭合形式的解,参数估计缺少良好的准则,最大后验概率属于点估计等,引入了分层最大似然估计方法和变分贝叶斯方法,并通过实验进行了分析验证。
本文的研究成果与创新主要包括:
1、针对实证分析法通常难以找到闭合形式解的问题,提出了一种分层最大似然估计法。实验结果表明,将该方法应用于图像恢复,可以加快图像恢复问题的求解速度,并且获得了比较理想的恢复结果。
2、针对Morozov离差原理、L曲线、广义交叉验证和期望最大化等经典参数估计算法的不足,提出了一种新的贝叶斯参数估计算法,该算法利用分层最大似然估计法,可以在图像恢复的过程中,同步地对多个模型参数进行估计。
3、提出了一种基于扩散技术的图像降噪算法,引入梯度算子作为原始图像的先验模型,并采用变分贝叶斯方法估计原始图像。实验结果表明,该算法克服了传统经验贝叶斯方法的不足。
4、提出了一种基于空间投影和混合模型的彩色图像降噪算法。在YCbCr空间下,该算法采用混合模型作为原始图像的先验模型,并基于分层最大似然估计方法估计原始图像。实验结果表明,该算法可使降噪后图像的信噪比平均提高1dB~3dB。
5、研究了噪声的分布特征,提出用拉普拉斯模分布模型刻画噪声,总变分模型描述原始图像的先验特征,并采用迭代重加权最小二乘法解决该了引入总变分模型和拉普拉斯模型带来的L1-优化问题。实验结果显示,拉普拉斯分布模型可以更真实地反映噪声的分布情况,迭代重加权最小二乘法的引入有效地提高了算法的时间性能。
6、将稀疏表示与分层最大似然估法相结合,提出了一种了稀疏图像去模糊算法。该算法引入Donoho等人的稀疏特征模型刻画系数向量的稀疏性,并采用总变分模型描述原始图像的先验特征,最后利用分层最大似然估计法推导原始图像。实验结果显示,该算法为稀疏表示技术用于图像恢复提供一通用的算法框架。
7、提出了一种基于调和模型图像去模糊算法,该算法利用调和模型刻画原始图像的分布特征,并通过分层最大似然估计法推导原始图像。实验结果表明,与基于最大后验概率的算法相比,该算法的时间性能和恢复结果更优。
8、提出了一种贝叶斯多信道图像盲去模糊算法,算法采用交叉关系模型和平滑模型组成的混合模型来描述点扩散函数的先验特征,同时还引入总变分模型刻画原始图像的先验特征,并基于分层最大似然估计算法推导原始图像和点扩散函数。实验结果表明,相比单信道的图像去模糊算法,多信道图像去模糊算法的恢复结果更优。与同类的多信道算法相比,该算法的速度更快,恢复效果更优。
As the foundation of image processing and computer vision, image restoration can solve the problem of image degradation. In many applications, such as remote sensing, medical imaging, and military reconnaissance, image degradation remainds a common and urgent problem to be solved. Thus, image restoration has always been followed with interest and studied earnestly.
This thesis focuses on the image restoration in Bayesian framework, which mainly contains research on model parameter estimation, image denoising, and image deblurring. By studying classical algorithm in Bayesian framework, several problems are found, such as the lack of a close-formed solution for posterior probability, the Maximum a Posteriori (MAP) method falling into the point estimation, and the lack of good principles for parameter estimation. This thesis proposes a hierarchical maximum likelihood method and introduces the variational Bayesian method in response to these problems. Both methods are experimentally analyzed and verified.
These are the major achievements and innovations:
1. Due to the difficulty of using the evidence analysis method in achieving close-formed solutions, we propose a novel hierarchical maximum likelihood method. The experiments demonstrate that the proposed method can obtain better results with satisfactory speeds when used in image restoration.
2. A novel Bayesian parameter estimation algorithm is proposed to overcome the drawbacks in Morozov discrepancy principle, L-curve method, generalized cross validation, and expectation maximum method. The proposed algorithm uses the hierarchical maximum likelihood method to estimate multiple model parameters together with image restoration.
3. We propose a diffusion-based image denoising algorithm that introduces the gradient operator as the prior model of the original image and utilizes the variational Bayesian method to estimate the original image. Results show that the proposed algorithm overcomes the shortcomings of the classical empirical Bayesian method.
4. We propose a novel color image denoising algorithm based on space projection and hybrid model. In the YCbCr space, the proposed algorithm uses the hybrid model to characterize the original image, and the hierarchical maximum likelihood framework is used to estimate the original image. Experiments indicate that the SNR values of denoised images are improved by 1dB to 2dB on average.
5. By studying the distribution properties of the noise, we introduce the Laplace distribution model to characterize the noises. We also propose the total variation to model the original image. To solve the problem of L1-optimization difficulty caused by the Laplace model and the total variation, we incorporate the iteratively reweighted norm method into our algorithm. Experiments clearly show that the Laplace model gives a true reflection of the noise natures, and the iteratively reweighted norm truly speeds up the proposed algorithm.
6. By combining the sparse representation with the hierarchical maximum likelihood method, a sparse representation-based image deblurring method is proposed. This algorithm introduces Donoho’s sparse model to characterize the sparse coefficient vector, and the total variation model is used to characterize the original image, which is estimated using the hierarchical maximum likelihood method. We theoretically and experimentally prove that the proposed algorithm supplies a unified framework for sparse image restoration.
7. We propose a novel harmonic image deblurring algorithm that uses the harmonic model to characterize the original image, and the hierarchical maximum likelihood method to estimate it. Experiments demonstrate that the proposed algorithm achieves better results with faster speed compared to the MAP-based algorithms.
8. We propose a novel Bayesian multichannel image deblurring algorithm that utilizes a hybrid model composed of the cross relation model and the smoothing model to characterize point spread function. The proposed algorithm also utilizes total variation to model the original image. Point spread function, together with original image, is estimated alternatively using the hierarchical maximum likelihood method. Results show the competitive performance of the proposed algorithm in vision effects and speed compared to other algorithms.
引文
[1] Raj A., Thakur K. Fast and stable Bayesian image expansion using sparse edge priors [J]. IEEE Transactions on Image Processing, 2007, 16(4): 1073-1084.
[2] Dobigeon N., Hero A. O., Tournerest J. Hierarchical Bayesian sparse image reconstruction with application to MRFM [J]. IEEE Transactions on Signal Processing, 2009, 18(9): 2059-2070.
[3] Guerrero-Colon J. A., Mancera L., Portilla J. Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids [J]. IEEE Transactions on Image Processing, 2008, 17(1): 27-41.
[4] Figueiredo M. A. T., Bioucas-Dias J. M., Nowak R. D. Majorization-minimization algorithms for wavelet-based image restoration [J]. IEEE Transactions on Image Processing. 2007, 16(12): 2980-2091.
[5] Joshi N., Matusik W., Adelson E. H., et al. Personal photo enhancement using example images [J]. ACM Transactions on Graphics, 2010, 29(2), article 12: 1-15.
[6] Duijster A., Scheunders P., De Backer S. Wavelet-based EM algorithm for multispectral-image restoration [J]. IEEE Transactions on Geoscience and Remote Sensing, 2009, 47(11): 3892-3898.
[7] Goossens B., Pizurica A. Image denoising using mixtures of projected Gaussian scale mixtures [J]. IEEE Transactions on Image Processing, 2009, 18(8): 1689-1702.
[8] Rivera M., Ocegueda O., Marroquin J. L. Entropy-controlled quadratic Markov measure field models for efficient image segmentation [J]. IEEE Transactions on Image Processing, 2007, 16(12): 3047-3057.
[9] Zhu D., Razaz M., Fisher M. An adaptive algorithm for image restoration using combined penalty functions [J]. Pattern Recognition Letters, 2006, 27(12): 1336-1341.
[10] Zhu D., Razaz M., Lee R. Adaptive penalty likelihood for reconstruction of multidimensional confocal microscopy images [J]. Computerized Medical Imaging Graphics, 2005, 29(5): 319-331.
[11] Mairal J., Elad M., Sapiro G. Sparse representation for color image restoration [J]. IEEE Transactions on Image Processing, 2008, 17(1): 53-69.
[12] Darbon J., Sigelle M. Image restoration with discrete constrained total variation part I: fast and exact optimization [J]. Journal of Mathematical Imaging and Vision, 2006, 26(3): 261-276.
[13] Darbon J., Sigelle M. Image restoration with discrete constrained total variation part II:levelable functions, convex priors and non-convex cases [J]. Journal of Mathematical Imaging and Vision, 2006, 26(3): 277-291.
[14] Heimer A., Cohen I. Multichannel seismic deconvolution using Markov-Bernoulli random-field modeling [J], IEEE Transactions on Geoscience and Remote Sensing, 2009, 47(7): 2047-2058.
[15] Mondal P. P., Vicidomini G., Diaspro A. Markov random field aided Bayesian approach for image reconstruction in confocal microscopy [J]. Journal of Applied Physics, 2007, 102(4): 044701-1-044701-9.
[16] Fevotte C., Godsill S. J. Sparse linear regression in unions of bases via Bayesian variable selection [J]. IEEE Signal Processing Letters, 2006, 13(7): 441-444.
[17] El Naqa I., Kawrakow I., Fippel M., et al. A comparison of Monte Carlo dose calculation denoising techniques [J]. Physics in Medicine and Biology, 2005, 50(5): 909-922.
[18] Gibbs A. L. Convergence in the Wasserstein metric for Markov chain Monte Carlo algorithms with applications to image restoration [J]. Stochastic Models, 2004, 20(4): 473-492.
[19] Rudin L. I., Osher S., Fatemi E. Nonlinear total variation based noise removal algorithms [J]. Physica D: Nonlinear Phenomena, 1992, 60(1-4): 259-268.
[20] Bertalmio M., Caselles V., Rouge B., et al. TV based image restoration with local constraints [J]. Journal of Scientific Computing, 2003, 19(1-3): 95-122.
[21] Bioucas-Dias J. M., Figueiredo M. A. T., Oliveira J. P. Adaptive total-variation image deconvolution: a majorization-minimization approach [A]. Proceedings of the 14th European Signal Processing Conference [C]. EURASIP, 2006.
[22] Babacan S. D., Molina R., Katsaggelos A. K. Parameter estimation in TV image restoration using variational distribution approximation [J]. IEEE Transactions on Image Processing, 2008, 17(3): 326-339.
[23] Perona P., Malik J. Scale-space and edge detection using anisotropic diffusion [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(7): 629-639.
[24] Weickert J. Multiscale texture enhancement [A]. Proceedings of the 6th International Conference on Computer Analysis of Images and Patterns [C]. Lecture Notes in Computer Science, 1995, 970: 230-237.
[25] Gilboa G., Sochen N., Zeevi Y. Y. Forward-and-backward diffusion processes for adaptive image enhancement and denoising [J], IEEE Transactions on Image Processing, 2002, 11(7): 689–703.
[26] You Y. L., Kaveh M. Fouth-order partial differential equations for noise removal [J]. IEEE Transactions on Image Processing, 2000, 9(10): 1723-1730.
[27] Guarnieri G., Marsi S., Ramponi G. Fast bilateral filter for edge-preserving smoothing [J]. Electronics Letters, 2006, 42(7): 396-397.
[28] Donoho D. L., Johnstone I. M. Ideal spatial adaptation by wavelet shrinkage [J]. Biometrika, 1994, 81(3): 425-455.
[29] Donoho D. L., Johnstone I. M. Adapting to unknown smoothness via wavelet shrinkage [J]. Journal of American Statistical Association, 1995, 90(432):1200-1224.
[30] Nason G. P. Wavelet shrinkage using cross validation [J]. Journal of the Royal Statistical Society, Series B, 1996, 58(2): 463-479.
[31] Ruggeri F., Vidakovic B. A Bayesian decision theoretic approach to the choice of thresholding parameter [J]. Statistica Sinica, 1999, 9(1), 183-197.
[32] Chang S. G., Yu B., Vetterli M. Adaptive wavelet thresholding for image denoising and compression [J]. IEEE Trasactions on Image Processing, 2000, 9(9): 1532-1546.
[33] Gao H. Y. Wavelet shrinkage denoising using the nonnegative garrote [J]. Journal of Computational and Graphical Statistics, 1998, 7(4): 469-488.
[34] Bruce A. G., Gao H. Y. WaveShrink: shrinkage functions and thresholds [A]. Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing III [C]. SPIE, 1995, 2569: 270-281.
[35] Simoncelli E. P., Adelson E. H. Noise removal via Bayesian wavelet coring [A]. Proceedings of the IEEE International Conference on Image Processing [C]. IEEE Signal Processing Society, 1996: 379-382.
[36] Chen P., Suter D. Shift-invariant wavelet denoising using interscale dependency [A]. International Conference on Image Processing [C]. IEEE Computer Society, 2004: 1005-1008.
[37] Chipman H., Kolaczyk E., McCulloch R. Adaptive Bayesian wavelet shrinkage [J]. Journal of the American Statistical Association, 1997, 92(440): 1413-1421.
[38] Solbo S., Eltoft T. Homomorphic wavelet-based statistical despeckling of SAR images [J]. IEEE Transactions on Geoscience and Remote Sensing, 2004, 42(4): 711-721.
[39] Achim A., Tsakalides P., Bezarianos A. SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling [J]. IEEE Transactions on Geoscience and Remote Sensing, 2003, 41(8):1773-1784.
[40] Crouse M. S., Baraniuk R. G. Contextual hidden Markov models for wavelet-domain signal processing [A]. Proceedings of the 31th Asilomar Conference on Signals,Systems, and Computers [C]. IEEE Computer Society, 1997, 1: 95-100.
[41] Fan G. L., Xia X. G. Improved hidden Markov models in the wavelet domain [J]. IEEE Transactions on Signal Processing, 2001, 49(1):115-120.
[42] Romberg J. K., Choi H., Baraniuk R. G. Bayesian tree-structured image modeling using wavelet-domain hidden Markov models [J]. IEEE Transactions on Image Processing, 2001, 10(7): 1056-1068.
[43] Malfait M., Roose D. Wavelet-based image denoising using a Markov random field a priori model [J]. IEEE Transactions on Image Processing, 1997, 6(4): 549-565.
[44] Jansen M., Bultheel A. Empirical Bayes approach to improve wavelet thresholding for image noise reduction [J]. Journal of the American Statistical Association, 2001, 96(454): 629-639.
[45] Pizurica A., Philips W., Lemahieu I., et al. A joint inter-and intrascale statistical model for Bayesian wavelet based image denoisin [J]. IEEE Transactions on Image Processing, 2002, 11(5): 545-557.
[46] Gabor D. Information theory in electron microscopy [J]. Laboratory Investigation, 1965, 14(6): 801-807.
[47] Jain A. K. Partial differential equations and finite-difference methods in image processing, part I: image representation [J]. Journal of optimaizaiotn theory and applications, 1977, 23(48): 65-91.
[48] Chan T. F., Wong C. K. Total variation blind deconvolution [J]. IEEE Transactions on Image Processing, 1998, 7(3): 370-375.
[49] You Y. L., Kaveh M. Blind image restoration by anisotropic regularization [J]. IEEE Transactions on Image Processing, 1999, 8(3): 396-407.
[50] Alliney S. A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing, 1997, 45(4): 913-917.
[51] Nilkolova M. Minimizers of cost-functions involving nonsmooth data fidelity terms [J]. SIAM Journal on Numerical Analysis, 2002(40): 965-994.
[52] Chan T. F., Marquina A., Mulet P. High-order total variation-based image restoration [J]. SIAM Journal on Scientific Computing, 2000, 22(2): 503–516.
[53] Money J. H., Kang S. H. Total variation minimizing blind deconvolution with shock filter reference [J]. Image and Vision Computing, 2008, 26(2): 302-314.
[54] Banham M. R., Katsaggelos A. K. Spatially adptive wavelet-based multiscale image restoration [J]. IEEE Transactions on Image Proeessing, 1996, 5(4): 619-634.
[55] Regularization E., Belge M., Kilmer M., et al. Wavelet domain image restoration with adaptiv edge-preserving regularization [J]. IEEE Transactions on Image Proeessing, 2000, 9(4): 597-608.
[56] Figueiredo M. A. T, Nowak R. D. An EM algorithm for wavelet-based image restoration [J]. IEEE Transactions on Image Proeessing, 2003, 12(8): 906-916.
[57] Bioucas-Dias J. M. Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors [J]. IEEE Transactions on Image Proeessing, 2006, 15(4): 937-951.
[58] Vonesch C., Unser M. A fast multilevel algorithm for wavelet-regularized image restoration [J]. IEEE Transactions on Image Processing, 2009, 18(3): 509-523.
[59] Candes E. J., Donoho D. L. Curvelets, multiresolution representation and scaling laws [A]. Proceedings of SPIE Conference on Wavelet Application in Signal and Image Preeessing VIII [C]. SPIE, 2000, 4119: l-12.
[60] Donoho D. L., Duncan M. R. Digital curvelet transform: strategy, implentation and expenments [A]. Proceedings of SPIE Conference on Wavelet Application in Signal and Image Preeessing VIII [C]. SPIE, 2000, 4056: 12-30.
[61] Candes E. J. Ridgelets: theory and applications [D]. Department of Statisties, Stanford University, USA, 1998.
[62] Candes E. J. Monoseale ridgelets for the representation of images with Edges [R]. Stanford, CA, USA: Department of Statisties, Stanford University, 1999.
[63] Le Pennec E., Mallat S. Sparse geometric image representation with bandelets [J]. IEEE Transactions on Image Proeessing, 2005, 14(4): 423-438.
[64] Le Pennec E., Mallat S. Bandelet image approximation and compression [J]. Multiscale Modeling and Simulation, 2005, 4(3): 992-1039.
[65] Do M. N., Vetterli M. Beyond Wavelets [M]. New York: Academic Press, 2003.
[66] Do M. N., Vetterli M. Framing pyramids [J]. IEEE Transactions on Signal Processing, 2003, 51(9): 2329~2342.
[67] Patel V. M., Easley G. R., Healy D. M. Shearlet-based deconvolution [J]. IEEE Transactions on Image Processing, 2009, 18(12): 2673-2685.
[68] Starck J. L., Neuyen M. K., Murtagh F. Wavelets and curvelets for image deconvolution: a combined approach [J]. Signal Processing, 2003, 83(10): 2279-2283.
[69] Chantas G., Galatsanos N. P., Molina R., et al. Variational Bayesian image restoration with a product of spatially weighted total variation image priors [J]. IEEE Transactions on Image Processing, 2010, 19(2): 351-362.
[70] Geman S., Geman D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1984, 6(6): 721-741.
[71] Molina R., Katsaggelos A. K., Abad J., et al. A Bayesian approach to blind deconvolution based on Dirichlet distributions [A]. Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing [C]. IEEE Signal Processing Society, 1997, 4: 2809-2812.
[72] Babacan S. D., Molina R., Katsaggelos A. K. Variational Bayesian blind deconvolution using a total variation prior [J]. IEEE Transactions on Image Processing, 2009, 18(1): 12-26.
[73] Vega M., Mateos J., Molina R., et al. Super resolution of multispectral images using l1 image models and interband correlations [A]. Proceedings of the IEEE International Workshop on Machine Learing for Signal Processing [C]. IEEE Signal Processing Society, 2009: 1-6.
[74] Tzikas D. G., Likas A. C., Galatsanos N. P. Variational Bayesian sparse kernel-based blind image deconvolution with student’s-t priors [J]. IEEE Transactions on Image Processing, 2009, 18(4): 753-764.
[75] Molina R., Katsaggelos A. K., Matcos J. Bayesian and regularization methods for hyperparameter estimation in image restoration [J]. IEEE Transactions on Image Proeessing, 1999, 8(2): 231-246.
[76] Kullback S. Information Theory and Statistics [M]. New York: Dover, 1959.
[77] Kilmer M. E., O’Leary D. P. Choosing regularization parameters in iterative methods for ill-Posed problems [J]. SIAM Journal on Matrix Analysis and Applications, 2001, 22(4): 1204-1221.
[78] Calvetti D., Reichel L., Zhang Q. Applied and Computational Control, Signals and Circuits [M]. Boston: Birkhauser, 1999: 313-367.
[79] Nguyen N., Milanfar P., Golub G. Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement [J]. IEEE Transactions on Image Processing, 2001, 10(9): 1299-1308.
[80] Hansen P. C. Analysis of discrete ill-posed problems by means of the L-curve [J]. SIAM Review, 1992, 34(4): 561-580.
[81] Dempster A. P., Laird N. M., Rubin D. B. Maximum likelihood from incomplete data via the EM algorithm [J]. Journal of the Royal Statistical Society, Series B, 1977, 39(1): 1-38.
[82] Raiffa H., Schlaifer R. Applied Statistical Decision Theory [M]. Boston: MIT Press, 1961.
[83] Portilla J, Simoncelli E. Image restration using Gaussian scale mixtures in the wavelet domain [A]. Proceedings of the IEEE Conference on Image Processing [C]. IEEE Signal Processing Society, 2003, 2: 965-968.
[84] Bioucas-Dias J. M., Figueiredo, M. A. T., Oliveira J. P. Total variation-based image deconvolution: a majorization-minimization approach [A]. Proceedings of the IEEE Conference on Acoustics, Speech and Signal Processing [C]. IEEE Signal Processing Society, 2006, 2: 861–864.
[85]王大凯,侯榆青,彭进业.图像处理的偏微分方程方法[M].北京:科学出版社, 2008.
[86] Catte F., Lions P., Morel J., et al. Image selective smoothing and edge-detection by nonlinear diffusion [J]. SIAM Joumal on Numerical Analysis, 1992, 29(1): 182-193.
[87]林宙辰,石青云.一个能去噪和保持真实感的各向异性扩散方程[J].计算机学报, 1999, 22(11): 1133-1137.
[88]林宙辰,石青云. An anisotropic diffusion PDE for noise reduction and thin edge preservation [A]. Proceedings of the International Conference on Image Analysis and Processing [C]. IEEE Computer Society, 1999: 102-107.
[89] Bajla I., Hollander I., Witkovsky V. A performance evaluation method for geometry-driven diffusion filters [J]. Journal of Electrical Engineering, 2003, 54(l-2): 3-12.
[90] Kacur J., Mikula K. Slow and fast diffusion effects in image processing [J]. Computing and Visualization in Science, 2001, 3(4):185-195.
[91] Gilboa G., Sochen N., Zeevi Y. Y. Forward-and-backward diffusion processes for adaptive image enhancement and denoising [J]. IEEE Transactions on Image Processing, 2002, 11(7): 689-703.
[92] Gilboa G., Sochen N., Zeevi Y. Y. Image enhancement and denoising by complex diffusion processes [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004, 25(8): 1020-1036.
[93] Koenderink J. J. The structure of imgaes [J]. Biological Cybernetics, 1984, 50(5): 363-370.
[94] Chan T. F., Shen J. Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[M]. Philadelphia: SIAM, 2005.
[95] Sendur L., Selesnick I. W. Bivariate shrinkage functions for wavelet-Based denoising exploiting interscale dependency [J]. IEEE Transactions on Signal Processing, 2002, 50(11): 2744-2756.
[96] Zhang M., Gunturk B. K. Multiresolution bilateral filtering for image denoising [J]. IEEE Transactions on Image Processing, 2008, 17(12): 2324-2333.
[97] Easley G. R., Labate D., Colonna F. Shearlet-based total variation diffusion for denoising [J]. Transactions on Image Processing, 2009, 18(2):260-268.
[98] Katsaggelos A. K., Lay K. T., Galatsanos N. P. A general framework for frequency domain multi-channel signal processing [J]. IEEE Transactions on Image Processing, 1993, 2(3): 417–420.
[99] Wang Y. L., Yang J. F., Yin W. T., et al. A new alternating minimization algorithm for total variation image reconstruction [J]. SIAM Journal on Imaging Sciences, 2008, 1(3): 248–272.
[100] Tonelli L. The calculus of variations [J]. Bulletin of the American Mathematical Society, 1925, 31(3-4): 163-172.
[101] Cesari L. On the representation of surfaces [J]. American Journal of Mathematics, 1950, 72(2): 335-346.
[102] Ambrosio L. Metric space valued functions of bounded variation [J]. Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, Serie 4, 1990, 17(3): 439-478.
[103] Ambrosioa L., Miranda L., Manigliac S., et al. BV functions in abstract Wiener spaces [J]. Journal of Functional Analysis, 2010, 258(3): 785-813.
[104] Candes E. J., Donoho D. L. Curvelets: a surprisingly effective non-adaptive representation for objects with edges [R]. Stanford, CA, USA: Department of Statistics, Stanford University, 1999.
[105] Mumford D., Shah J. Optimal approximations by piecewise smooth functions and associated variational problems [J]. Communications on Pure and Applied Mathematics, 1989, 42(5): 577-685.
[106] Ambrosio L. Regularity of solutions of a degenerate elliptic variational problem [J]. Manuscripta Mathematica, 1990, 68(1): 309-326.
[107] Ambrosio L., Tortorelli V. M. Approximation of functional depending on jumps by elliptic functional via gamma-convergence [J]. Communication on Pure and Applied Mathematics [J]. 2006, 43(8): 999-1036.
[108] Strong D. M., Blomgren P., Chan T. F. Spatially adaptive local feature-driven total variation minimizing image restoration [A]. Proceedings of SPIE Conference onStatistical and Stochastic Methods in Image Processing II [C]. SPIE, 1997, 3167: 222-233.
[109] Blomgren P., Chan T. F. Color TV: total variation methods for restoration of vector-valued images [J]. IEEE Transactions on Image Processing, 1998, 7(3): 304-309.
[110] Shen J. H. On the foundations of vision modeling: I. Weber’s law and Weberized TV restoration [J]. Physica D: Nonlinear Phenomena 2003, 175(3-4): 241-251.
[111] Vogel C. R., Oman M. E. Iterative methods for total variation denoising [J]. SIAM Journal on Seientific Computing, 1996, 17(1): 227-238.
[112] Chan T. F., Mulet P. On the convergence of the lagged diffusivity fixed point method in total variation image restoration [J]. SIAM Journal on Numerical Analysis, 1999, 36(2): 354-367.
[113] Chan R H, Chan T. F., Zhou H. M. Continuation method for total variation denoising Problem [R]. Los Angeles, CA, USA: University of Califonia, 1995.
[114] Chan T. F., Golub G. H., Mulet P. A nonlinear primal dual method for total variation-based image restoration [R]. Los Angeles, CA, USA: University of Califonia, 1995.
[115] Pizurica A., Philips W. Estimating the probability of the presence of a signal of interest in multiresolution single- and multiband image denoising [J]. IEEE Transactions on Image Processing, 2006, 15(3): 654-665.
[116] Ben-Shahar O., Zucher S. W. Hue fields and color curvatures: a perceptual organization approach to color image denoising [A]. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition [C]. IEEE Computer Society, 2003: 713-720.
[117] Lian N. X., Zagorodnov V., Tan Y. P. Edge-preserving image denoising via optimal color space projection [J]. IEEE Transactions on Image Process, 2006, 15(9): 2575-2587.
[118] Farsiu S., Elad M., Milafar P. Multiframe demosaicing and super-resolution of color images [J]. IEEE Transactions on Image Processing, 2006, 15(1): 141-159.
[119] Scales J. A., Gersztenkorn A. Robust methods in inverse theory [J]. Inverse Problems, 1998, 4(4): 1071–1091.
[120] Foi A., Katkovnik V., Egiazarian K. Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images [J]. IEEE Transactions on Image Processing, 2007, 16(5): 1395-1411.
[121] Bresson X., Chan T. F. Fast dual minimization of the vectorial total variation norm and applications to color image processing [J]. Inverse Problems and Imaging, 2008, 2(4): 455-484.
[122] Mallat S., Zhang Z. Matching pursuits with time-frequency dictionaries [J]. IEEE Transaction on Signal Process, 1993, 41(12): 3397-3415.
[123] Neff R., Zakhor A. Very low bit-rate video coding based on matching pursuits [J]. IEEE Transaction on Circuits and Systems for Video Technology, 1997, 7(1): 158–171.
[124] Qian S., Chen D. Signal representation using adaptive normalized Gaussian functions [J]. Signal Processing, 1994, 36(1): 1-11.
[125] Tropp J. A., Gilbert A. C. Signal recovery from random measurements via orthogonal matching pursuit [J]. IEEE Transactions on Information Theory, 2007, 53(12): 4655-4666.
[126] Chen S. S., Donoho D. L., Saunders M. A. Atomic decomposition by basis pursuit [J]. SIAM Journal on Scientific Computing, 1999, 20(1): 33-61.
[127] Jaggi S., Karl W. C., Mallat S., et al. Feature extraction through high-resolution pursuit [J]. Applied and Computational Harmonic Analysis, 1998, 5(4): 428-449.
[128] Sardy S., Bruce A. G., Tseng P. Block coordinate relaxation methods for nonparametric wavelet denoising [J]. Journal of Computational and Graphical Statistics, 2000, 9(2): 361-379.
[129] Tropp J. A. Greed is good: algorithmic results for sparse approximation [J].IEEE Transactions on Information Theory, 2004, 50(10): 2231-2242.
[130] Coifman R. R., Wickerhauser M. V. Entropy-based algorithms for best basis selection [J]. IEEE Transactions on Information Theory, 1992, 38(2): 713-718.
[131] Wipf D. P., Rao B. D. Sparse Bayesian learning for basis selection [J]. IEEE Transactions on Signal Processing, 2004, 52 (8): 2153-2164.
[132] Donho D. L., Huo X. Combined image representation using edgelets and wavelets [A]. Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII [C]. SPIE, 1999, 3813: 468-476.
[133]尹忠科,邵君, Vandergheynst P.利用FFT实现基于MP的信号稀疏分解[J].电子与信息学报, 2006, 28(4): 614-618.
[134] Kreutz K., Murray J. F. Dictionary learning algorithms for sparse representation [J]. Neural Computation, 2003, 15(2): 349-396.
[135] Aharon M., Elad M., Bruckstein A. M. The K-SVD: an algorithm for designing ofovercomplete dictionaries for sparse representation [J]. IEEE Transactions on Signal Processing, 2006, 54(11): 4311-4322.
[136] Mairal J., Elad M., Sapiro G. Sparse representation for color image restoration [J]. IEEE Transactions on Image Processing, 2008, 17(1): 53-69.
[137] Bronstein M. M., Bronstein A. M., Zibulevsky M., et al. Blind deconvolution of images using optimal sparse representations [J]. IEEE Transactions on Image Processing, 2005, 14(6): 726-736.
[138] Gutierrez J., Ferri F. J., Malo J. Regularization operators for natural images based on nonlinear perception models [J]. IEEE Transactions on Image Processing, 2006, 15(1): 189-200.
[139] Bioucas-Dias J. M., Figueiredo M. A. T. A new TwIST: two-step iterative shrinkage / thresholding algorithms for image restoration [J]. IEEE Transactions on Image Processing, 2007, 16(12): 2992-3004.
[140] Kingsbury N. Complex wavelets for shift Invariant analysis and filtering of signals [J]. Applied and Computational Harmonic Analysis, 2001, 10(3): 234-253.
[141] Selesnick I. W. The double-density dual-tree DWT [J]. IEEE Transactions on Signal Processing, 2004, 52(5): 1304-1314.
[142] Yin W. T., Osher S., Goldfarb D., et al. Bregman iterative algorithms for L1-minimization with applications to compressed sensing [J]. SIAM Journal on Imaging Sciences, 2008, 1(1): 143-168.
[143] Tikhonov A. N., Goncharsky A. V., Stepono V. V., et al. Numerical Methods for the Solutions of Ill-Posed Problems [M]. Dordrecht: Kluwer Academic Publishers, 1995.
[144] Paik J. K., Katsaggelos A. K. Image restoration using a modified Hopfield network. [J]. IEEE Transactions on Image Process, 1992, 1(1): 49-63.
[145]梁立孚.变分原理及其应用[M].第二版.哈尔滨:哈尔滨工程大学出版社: 2007.
[146] Geman S., Geman D., Abend K., et al. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of Images [J]. Journal of Applied Statistics, 1993, 20(5-6): 25-62.
[147] Likas A., Galatsanos N. P. A variational approach for Bayesian blind image deconvolution [J]. IEEE Transactions on Signal Processing, 2004, 52(8): 2222-2233.
[148] Levin A., Fergus R., Durand F., et al. Image and depth from a conventional camera with a coded aperture [J]. ACM Transactions on Graphics, 2007, 26(3), articale 70: 1-9.
[149] Yang J. F., Zhang Y., Yin W. T. An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise [J]. SIAM Journal on ScientificComputing, 2009, 31(4): 2842-2865.
[150] Harikumar G., Bresler Y. Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms [J]. IEEE Transactions on Image Processing, 1999, 8(2): 202–219.
[151] Giannakis G., Heath R. Blind identification of multichannel FIR blurs and perfect image restoration [J] IEEE Transactions on Image Processing, 2000, 9(11): 1877–1896.
[152] Pai H. T., Bovik A. Exact multichannel blind image restoration [J]. IEEE Signal Processing Letter, 1997, 4(8): 217–220.
[153] Pillai S., Liang B. Blind image deconvolution using a robust GCD approach [J]. IEEE Transactions on Image Processing, 1999, 8(2): 295–301.
[154] Chaudhuri S., Rajagopalan A. N. A recursive algorithm for maximum likelihood-based identification of blur from multiple observations [J]. IEEE Transactions on Image Processing, 1998, 7(7): 1075-1079.
[155] Panci G., Campisi P., Colonnese S., et al. Multichannel blind image deconvolution using the bussgang algorithm: spatial and multiresolution approaches [J]. IEEE Transactions on Image Processing, 2003, 12(11): 1324–1337.
[156] Sroubek F., Flusser J. Multichannel blind iterative image restoration [J]. IEEE Transactions on Image Processing, 2003, 12(9): 1094–1106.
[157] Sroubek F., Flusser J. Multichannel blind deconvolution of spatially misaligned images [J]. IEEE Transactions on Image Processing, 2005, 14(7): 874-883.
[158] Welk M., Nagy J. G. Variational deconvolution of multi-channel images with inequality constraints [A]. Proceedings of the 3rd Iberian Conference on Pattern Recognition and Image Analysis [C]. Lecture Notes in Computer Science, 2007, 4477: 386-393.
[159] Tong L., Perreau S. Multichannel blind identification: from subspace to maximum likelihood methods [J]. Proceedings of the IEEE, 1998, 86(10): 1951-1968.
[160] Chantas G. K., Calatsanos N. P., Likas A. C. Bayesian restoration using a new nonstationary edge-preserving image prior [J]. IEEE Transactions on Image Processing, 2006, 15(10): 2987-2997.
[161] Souidene W., Beghdadi A., Abed-Meraim K., et al. Regularized MRE method for blind multichannel image deconvolution [A]. Proceedings of the 38th Asilomar Conference on Signals, Systems and Computers [C]. IEEE Computer Society, 2004, 2: 2233-2237.
[162] Yan J. F., Yin W. T., Zhang Y., et al. A Fast algorithm for edge-preserving variational multichannel image restoration [J]. SIAM Journal on Imaging Sciences, 2009, 2(2): 569-592.
[2] Dobigeon N., Hero A. O., Tournerest J. Hierarchical Bayesian sparse image reconstruction with application to MRFM [J]. IEEE Transactions on Signal Processing, 2009, 18(9): 2059-2070.
[3] Guerrero-Colon J. A., Mancera L., Portilla J. Image restoration using space-variant Gaussian scale mixtures in overcomplete pyramids [J]. IEEE Transactions on Image Processing, 2008, 17(1): 27-41.
[4] Figueiredo M. A. T., Bioucas-Dias J. M., Nowak R. D. Majorization-minimization algorithms for wavelet-based image restoration [J]. IEEE Transactions on Image Processing. 2007, 16(12): 2980-2091.
[5] Joshi N., Matusik W., Adelson E. H., et al. Personal photo enhancement using example images [J]. ACM Transactions on Graphics, 2010, 29(2), article 12: 1-15.
[6] Duijster A., Scheunders P., De Backer S. Wavelet-based EM algorithm for multispectral-image restoration [J]. IEEE Transactions on Geoscience and Remote Sensing, 2009, 47(11): 3892-3898.
[7] Goossens B., Pizurica A. Image denoising using mixtures of projected Gaussian scale mixtures [J]. IEEE Transactions on Image Processing, 2009, 18(8): 1689-1702.
[8] Rivera M., Ocegueda O., Marroquin J. L. Entropy-controlled quadratic Markov measure field models for efficient image segmentation [J]. IEEE Transactions on Image Processing, 2007, 16(12): 3047-3057.
[9] Zhu D., Razaz M., Fisher M. An adaptive algorithm for image restoration using combined penalty functions [J]. Pattern Recognition Letters, 2006, 27(12): 1336-1341.
[10] Zhu D., Razaz M., Lee R. Adaptive penalty likelihood for reconstruction of multidimensional confocal microscopy images [J]. Computerized Medical Imaging Graphics, 2005, 29(5): 319-331.
[11] Mairal J., Elad M., Sapiro G. Sparse representation for color image restoration [J]. IEEE Transactions on Image Processing, 2008, 17(1): 53-69.
[12] Darbon J., Sigelle M. Image restoration with discrete constrained total variation part I: fast and exact optimization [J]. Journal of Mathematical Imaging and Vision, 2006, 26(3): 261-276.
[13] Darbon J., Sigelle M. Image restoration with discrete constrained total variation part II:levelable functions, convex priors and non-convex cases [J]. Journal of Mathematical Imaging and Vision, 2006, 26(3): 277-291.
[14] Heimer A., Cohen I. Multichannel seismic deconvolution using Markov-Bernoulli random-field modeling [J], IEEE Transactions on Geoscience and Remote Sensing, 2009, 47(7): 2047-2058.
[15] Mondal P. P., Vicidomini G., Diaspro A. Markov random field aided Bayesian approach for image reconstruction in confocal microscopy [J]. Journal of Applied Physics, 2007, 102(4): 044701-1-044701-9.
[16] Fevotte C., Godsill S. J. Sparse linear regression in unions of bases via Bayesian variable selection [J]. IEEE Signal Processing Letters, 2006, 13(7): 441-444.
[17] El Naqa I., Kawrakow I., Fippel M., et al. A comparison of Monte Carlo dose calculation denoising techniques [J]. Physics in Medicine and Biology, 2005, 50(5): 909-922.
[18] Gibbs A. L. Convergence in the Wasserstein metric for Markov chain Monte Carlo algorithms with applications to image restoration [J]. Stochastic Models, 2004, 20(4): 473-492.
[19] Rudin L. I., Osher S., Fatemi E. Nonlinear total variation based noise removal algorithms [J]. Physica D: Nonlinear Phenomena, 1992, 60(1-4): 259-268.
[20] Bertalmio M., Caselles V., Rouge B., et al. TV based image restoration with local constraints [J]. Journal of Scientific Computing, 2003, 19(1-3): 95-122.
[21] Bioucas-Dias J. M., Figueiredo M. A. T., Oliveira J. P. Adaptive total-variation image deconvolution: a majorization-minimization approach [A]. Proceedings of the 14th European Signal Processing Conference [C]. EURASIP, 2006.
[22] Babacan S. D., Molina R., Katsaggelos A. K. Parameter estimation in TV image restoration using variational distribution approximation [J]. IEEE Transactions on Image Processing, 2008, 17(3): 326-339.
[23] Perona P., Malik J. Scale-space and edge detection using anisotropic diffusion [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(7): 629-639.
[24] Weickert J. Multiscale texture enhancement [A]. Proceedings of the 6th International Conference on Computer Analysis of Images and Patterns [C]. Lecture Notes in Computer Science, 1995, 970: 230-237.
[25] Gilboa G., Sochen N., Zeevi Y. Y. Forward-and-backward diffusion processes for adaptive image enhancement and denoising [J], IEEE Transactions on Image Processing, 2002, 11(7): 689–703.
[26] You Y. L., Kaveh M. Fouth-order partial differential equations for noise removal [J]. IEEE Transactions on Image Processing, 2000, 9(10): 1723-1730.
[27] Guarnieri G., Marsi S., Ramponi G. Fast bilateral filter for edge-preserving smoothing [J]. Electronics Letters, 2006, 42(7): 396-397.
[28] Donoho D. L., Johnstone I. M. Ideal spatial adaptation by wavelet shrinkage [J]. Biometrika, 1994, 81(3): 425-455.
[29] Donoho D. L., Johnstone I. M. Adapting to unknown smoothness via wavelet shrinkage [J]. Journal of American Statistical Association, 1995, 90(432):1200-1224.
[30] Nason G. P. Wavelet shrinkage using cross validation [J]. Journal of the Royal Statistical Society, Series B, 1996, 58(2): 463-479.
[31] Ruggeri F., Vidakovic B. A Bayesian decision theoretic approach to the choice of thresholding parameter [J]. Statistica Sinica, 1999, 9(1), 183-197.
[32] Chang S. G., Yu B., Vetterli M. Adaptive wavelet thresholding for image denoising and compression [J]. IEEE Trasactions on Image Processing, 2000, 9(9): 1532-1546.
[33] Gao H. Y. Wavelet shrinkage denoising using the nonnegative garrote [J]. Journal of Computational and Graphical Statistics, 1998, 7(4): 469-488.
[34] Bruce A. G., Gao H. Y. WaveShrink: shrinkage functions and thresholds [A]. Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing III [C]. SPIE, 1995, 2569: 270-281.
[35] Simoncelli E. P., Adelson E. H. Noise removal via Bayesian wavelet coring [A]. Proceedings of the IEEE International Conference on Image Processing [C]. IEEE Signal Processing Society, 1996: 379-382.
[36] Chen P., Suter D. Shift-invariant wavelet denoising using interscale dependency [A]. International Conference on Image Processing [C]. IEEE Computer Society, 2004: 1005-1008.
[37] Chipman H., Kolaczyk E., McCulloch R. Adaptive Bayesian wavelet shrinkage [J]. Journal of the American Statistical Association, 1997, 92(440): 1413-1421.
[38] Solbo S., Eltoft T. Homomorphic wavelet-based statistical despeckling of SAR images [J]. IEEE Transactions on Geoscience and Remote Sensing, 2004, 42(4): 711-721.
[39] Achim A., Tsakalides P., Bezarianos A. SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling [J]. IEEE Transactions on Geoscience and Remote Sensing, 2003, 41(8):1773-1784.
[40] Crouse M. S., Baraniuk R. G. Contextual hidden Markov models for wavelet-domain signal processing [A]. Proceedings of the 31th Asilomar Conference on Signals,Systems, and Computers [C]. IEEE Computer Society, 1997, 1: 95-100.
[41] Fan G. L., Xia X. G. Improved hidden Markov models in the wavelet domain [J]. IEEE Transactions on Signal Processing, 2001, 49(1):115-120.
[42] Romberg J. K., Choi H., Baraniuk R. G. Bayesian tree-structured image modeling using wavelet-domain hidden Markov models [J]. IEEE Transactions on Image Processing, 2001, 10(7): 1056-1068.
[43] Malfait M., Roose D. Wavelet-based image denoising using a Markov random field a priori model [J]. IEEE Transactions on Image Processing, 1997, 6(4): 549-565.
[44] Jansen M., Bultheel A. Empirical Bayes approach to improve wavelet thresholding for image noise reduction [J]. Journal of the American Statistical Association, 2001, 96(454): 629-639.
[45] Pizurica A., Philips W., Lemahieu I., et al. A joint inter-and intrascale statistical model for Bayesian wavelet based image denoisin [J]. IEEE Transactions on Image Processing, 2002, 11(5): 545-557.
[46] Gabor D. Information theory in electron microscopy [J]. Laboratory Investigation, 1965, 14(6): 801-807.
[47] Jain A. K. Partial differential equations and finite-difference methods in image processing, part I: image representation [J]. Journal of optimaizaiotn theory and applications, 1977, 23(48): 65-91.
[48] Chan T. F., Wong C. K. Total variation blind deconvolution [J]. IEEE Transactions on Image Processing, 1998, 7(3): 370-375.
[49] You Y. L., Kaveh M. Blind image restoration by anisotropic regularization [J]. IEEE Transactions on Image Processing, 1999, 8(3): 396-407.
[50] Alliney S. A property of the minimum vectors of a regularizing functional defined by means of the absolute norm [J]. IEEE Transactions on Signal Processing, 1997, 45(4): 913-917.
[51] Nilkolova M. Minimizers of cost-functions involving nonsmooth data fidelity terms [J]. SIAM Journal on Numerical Analysis, 2002(40): 965-994.
[52] Chan T. F., Marquina A., Mulet P. High-order total variation-based image restoration [J]. SIAM Journal on Scientific Computing, 2000, 22(2): 503–516.
[53] Money J. H., Kang S. H. Total variation minimizing blind deconvolution with shock filter reference [J]. Image and Vision Computing, 2008, 26(2): 302-314.
[54] Banham M. R., Katsaggelos A. K. Spatially adptive wavelet-based multiscale image restoration [J]. IEEE Transactions on Image Proeessing, 1996, 5(4): 619-634.
[55] Regularization E., Belge M., Kilmer M., et al. Wavelet domain image restoration with adaptiv edge-preserving regularization [J]. IEEE Transactions on Image Proeessing, 2000, 9(4): 597-608.
[56] Figueiredo M. A. T, Nowak R. D. An EM algorithm for wavelet-based image restoration [J]. IEEE Transactions on Image Proeessing, 2003, 12(8): 906-916.
[57] Bioucas-Dias J. M. Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors [J]. IEEE Transactions on Image Proeessing, 2006, 15(4): 937-951.
[58] Vonesch C., Unser M. A fast multilevel algorithm for wavelet-regularized image restoration [J]. IEEE Transactions on Image Processing, 2009, 18(3): 509-523.
[59] Candes E. J., Donoho D. L. Curvelets, multiresolution representation and scaling laws [A]. Proceedings of SPIE Conference on Wavelet Application in Signal and Image Preeessing VIII [C]. SPIE, 2000, 4119: l-12.
[60] Donoho D. L., Duncan M. R. Digital curvelet transform: strategy, implentation and expenments [A]. Proceedings of SPIE Conference on Wavelet Application in Signal and Image Preeessing VIII [C]. SPIE, 2000, 4056: 12-30.
[61] Candes E. J. Ridgelets: theory and applications [D]. Department of Statisties, Stanford University, USA, 1998.
[62] Candes E. J. Monoseale ridgelets for the representation of images with Edges [R]. Stanford, CA, USA: Department of Statisties, Stanford University, 1999.
[63] Le Pennec E., Mallat S. Sparse geometric image representation with bandelets [J]. IEEE Transactions on Image Proeessing, 2005, 14(4): 423-438.
[64] Le Pennec E., Mallat S. Bandelet image approximation and compression [J]. Multiscale Modeling and Simulation, 2005, 4(3): 992-1039.
[65] Do M. N., Vetterli M. Beyond Wavelets [M]. New York: Academic Press, 2003.
[66] Do M. N., Vetterli M. Framing pyramids [J]. IEEE Transactions on Signal Processing, 2003, 51(9): 2329~2342.
[67] Patel V. M., Easley G. R., Healy D. M. Shearlet-based deconvolution [J]. IEEE Transactions on Image Processing, 2009, 18(12): 2673-2685.
[68] Starck J. L., Neuyen M. K., Murtagh F. Wavelets and curvelets for image deconvolution: a combined approach [J]. Signal Processing, 2003, 83(10): 2279-2283.
[69] Chantas G., Galatsanos N. P., Molina R., et al. Variational Bayesian image restoration with a product of spatially weighted total variation image priors [J]. IEEE Transactions on Image Processing, 2010, 19(2): 351-362.
[70] Geman S., Geman D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1984, 6(6): 721-741.
[71] Molina R., Katsaggelos A. K., Abad J., et al. A Bayesian approach to blind deconvolution based on Dirichlet distributions [A]. Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing [C]. IEEE Signal Processing Society, 1997, 4: 2809-2812.
[72] Babacan S. D., Molina R., Katsaggelos A. K. Variational Bayesian blind deconvolution using a total variation prior [J]. IEEE Transactions on Image Processing, 2009, 18(1): 12-26.
[73] Vega M., Mateos J., Molina R., et al. Super resolution of multispectral images using l1 image models and interband correlations [A]. Proceedings of the IEEE International Workshop on Machine Learing for Signal Processing [C]. IEEE Signal Processing Society, 2009: 1-6.
[74] Tzikas D. G., Likas A. C., Galatsanos N. P. Variational Bayesian sparse kernel-based blind image deconvolution with student’s-t priors [J]. IEEE Transactions on Image Processing, 2009, 18(4): 753-764.
[75] Molina R., Katsaggelos A. K., Matcos J. Bayesian and regularization methods for hyperparameter estimation in image restoration [J]. IEEE Transactions on Image Proeessing, 1999, 8(2): 231-246.
[76] Kullback S. Information Theory and Statistics [M]. New York: Dover, 1959.
[77] Kilmer M. E., O’Leary D. P. Choosing regularization parameters in iterative methods for ill-Posed problems [J]. SIAM Journal on Matrix Analysis and Applications, 2001, 22(4): 1204-1221.
[78] Calvetti D., Reichel L., Zhang Q. Applied and Computational Control, Signals and Circuits [M]. Boston: Birkhauser, 1999: 313-367.
[79] Nguyen N., Milanfar P., Golub G. Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement [J]. IEEE Transactions on Image Processing, 2001, 10(9): 1299-1308.
[80] Hansen P. C. Analysis of discrete ill-posed problems by means of the L-curve [J]. SIAM Review, 1992, 34(4): 561-580.
[81] Dempster A. P., Laird N. M., Rubin D. B. Maximum likelihood from incomplete data via the EM algorithm [J]. Journal of the Royal Statistical Society, Series B, 1977, 39(1): 1-38.
[82] Raiffa H., Schlaifer R. Applied Statistical Decision Theory [M]. Boston: MIT Press, 1961.
[83] Portilla J, Simoncelli E. Image restration using Gaussian scale mixtures in the wavelet domain [A]. Proceedings of the IEEE Conference on Image Processing [C]. IEEE Signal Processing Society, 2003, 2: 965-968.
[84] Bioucas-Dias J. M., Figueiredo, M. A. T., Oliveira J. P. Total variation-based image deconvolution: a majorization-minimization approach [A]. Proceedings of the IEEE Conference on Acoustics, Speech and Signal Processing [C]. IEEE Signal Processing Society, 2006, 2: 861–864.
[85]王大凯,侯榆青,彭进业.图像处理的偏微分方程方法[M].北京:科学出版社, 2008.
[86] Catte F., Lions P., Morel J., et al. Image selective smoothing and edge-detection by nonlinear diffusion [J]. SIAM Joumal on Numerical Analysis, 1992, 29(1): 182-193.
[87]林宙辰,石青云.一个能去噪和保持真实感的各向异性扩散方程[J].计算机学报, 1999, 22(11): 1133-1137.
[88]林宙辰,石青云. An anisotropic diffusion PDE for noise reduction and thin edge preservation [A]. Proceedings of the International Conference on Image Analysis and Processing [C]. IEEE Computer Society, 1999: 102-107.
[89] Bajla I., Hollander I., Witkovsky V. A performance evaluation method for geometry-driven diffusion filters [J]. Journal of Electrical Engineering, 2003, 54(l-2): 3-12.
[90] Kacur J., Mikula K. Slow and fast diffusion effects in image processing [J]. Computing and Visualization in Science, 2001, 3(4):185-195.
[91] Gilboa G., Sochen N., Zeevi Y. Y. Forward-and-backward diffusion processes for adaptive image enhancement and denoising [J]. IEEE Transactions on Image Processing, 2002, 11(7): 689-703.
[92] Gilboa G., Sochen N., Zeevi Y. Y. Image enhancement and denoising by complex diffusion processes [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004, 25(8): 1020-1036.
[93] Koenderink J. J. The structure of imgaes [J]. Biological Cybernetics, 1984, 50(5): 363-370.
[94] Chan T. F., Shen J. Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods[M]. Philadelphia: SIAM, 2005.
[95] Sendur L., Selesnick I. W. Bivariate shrinkage functions for wavelet-Based denoising exploiting interscale dependency [J]. IEEE Transactions on Signal Processing, 2002, 50(11): 2744-2756.
[96] Zhang M., Gunturk B. K. Multiresolution bilateral filtering for image denoising [J]. IEEE Transactions on Image Processing, 2008, 17(12): 2324-2333.
[97] Easley G. R., Labate D., Colonna F. Shearlet-based total variation diffusion for denoising [J]. Transactions on Image Processing, 2009, 18(2):260-268.
[98] Katsaggelos A. K., Lay K. T., Galatsanos N. P. A general framework for frequency domain multi-channel signal processing [J]. IEEE Transactions on Image Processing, 1993, 2(3): 417–420.
[99] Wang Y. L., Yang J. F., Yin W. T., et al. A new alternating minimization algorithm for total variation image reconstruction [J]. SIAM Journal on Imaging Sciences, 2008, 1(3): 248–272.
[100] Tonelli L. The calculus of variations [J]. Bulletin of the American Mathematical Society, 1925, 31(3-4): 163-172.
[101] Cesari L. On the representation of surfaces [J]. American Journal of Mathematics, 1950, 72(2): 335-346.
[102] Ambrosio L. Metric space valued functions of bounded variation [J]. Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, Serie 4, 1990, 17(3): 439-478.
[103] Ambrosioa L., Miranda L., Manigliac S., et al. BV functions in abstract Wiener spaces [J]. Journal of Functional Analysis, 2010, 258(3): 785-813.
[104] Candes E. J., Donoho D. L. Curvelets: a surprisingly effective non-adaptive representation for objects with edges [R]. Stanford, CA, USA: Department of Statistics, Stanford University, 1999.
[105] Mumford D., Shah J. Optimal approximations by piecewise smooth functions and associated variational problems [J]. Communications on Pure and Applied Mathematics, 1989, 42(5): 577-685.
[106] Ambrosio L. Regularity of solutions of a degenerate elliptic variational problem [J]. Manuscripta Mathematica, 1990, 68(1): 309-326.
[107] Ambrosio L., Tortorelli V. M. Approximation of functional depending on jumps by elliptic functional via gamma-convergence [J]. Communication on Pure and Applied Mathematics [J]. 2006, 43(8): 999-1036.
[108] Strong D. M., Blomgren P., Chan T. F. Spatially adaptive local feature-driven total variation minimizing image restoration [A]. Proceedings of SPIE Conference onStatistical and Stochastic Methods in Image Processing II [C]. SPIE, 1997, 3167: 222-233.
[109] Blomgren P., Chan T. F. Color TV: total variation methods for restoration of vector-valued images [J]. IEEE Transactions on Image Processing, 1998, 7(3): 304-309.
[110] Shen J. H. On the foundations of vision modeling: I. Weber’s law and Weberized TV restoration [J]. Physica D: Nonlinear Phenomena 2003, 175(3-4): 241-251.
[111] Vogel C. R., Oman M. E. Iterative methods for total variation denoising [J]. SIAM Journal on Seientific Computing, 1996, 17(1): 227-238.
[112] Chan T. F., Mulet P. On the convergence of the lagged diffusivity fixed point method in total variation image restoration [J]. SIAM Journal on Numerical Analysis, 1999, 36(2): 354-367.
[113] Chan R H, Chan T. F., Zhou H. M. Continuation method for total variation denoising Problem [R]. Los Angeles, CA, USA: University of Califonia, 1995.
[114] Chan T. F., Golub G. H., Mulet P. A nonlinear primal dual method for total variation-based image restoration [R]. Los Angeles, CA, USA: University of Califonia, 1995.
[115] Pizurica A., Philips W. Estimating the probability of the presence of a signal of interest in multiresolution single- and multiband image denoising [J]. IEEE Transactions on Image Processing, 2006, 15(3): 654-665.
[116] Ben-Shahar O., Zucher S. W. Hue fields and color curvatures: a perceptual organization approach to color image denoising [A]. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition [C]. IEEE Computer Society, 2003: 713-720.
[117] Lian N. X., Zagorodnov V., Tan Y. P. Edge-preserving image denoising via optimal color space projection [J]. IEEE Transactions on Image Process, 2006, 15(9): 2575-2587.
[118] Farsiu S., Elad M., Milafar P. Multiframe demosaicing and super-resolution of color images [J]. IEEE Transactions on Image Processing, 2006, 15(1): 141-159.
[119] Scales J. A., Gersztenkorn A. Robust methods in inverse theory [J]. Inverse Problems, 1998, 4(4): 1071–1091.
[120] Foi A., Katkovnik V., Egiazarian K. Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images [J]. IEEE Transactions on Image Processing, 2007, 16(5): 1395-1411.
[121] Bresson X., Chan T. F. Fast dual minimization of the vectorial total variation norm and applications to color image processing [J]. Inverse Problems and Imaging, 2008, 2(4): 455-484.
[122] Mallat S., Zhang Z. Matching pursuits with time-frequency dictionaries [J]. IEEE Transaction on Signal Process, 1993, 41(12): 3397-3415.
[123] Neff R., Zakhor A. Very low bit-rate video coding based on matching pursuits [J]. IEEE Transaction on Circuits and Systems for Video Technology, 1997, 7(1): 158–171.
[124] Qian S., Chen D. Signal representation using adaptive normalized Gaussian functions [J]. Signal Processing, 1994, 36(1): 1-11.
[125] Tropp J. A., Gilbert A. C. Signal recovery from random measurements via orthogonal matching pursuit [J]. IEEE Transactions on Information Theory, 2007, 53(12): 4655-4666.
[126] Chen S. S., Donoho D. L., Saunders M. A. Atomic decomposition by basis pursuit [J]. SIAM Journal on Scientific Computing, 1999, 20(1): 33-61.
[127] Jaggi S., Karl W. C., Mallat S., et al. Feature extraction through high-resolution pursuit [J]. Applied and Computational Harmonic Analysis, 1998, 5(4): 428-449.
[128] Sardy S., Bruce A. G., Tseng P. Block coordinate relaxation methods for nonparametric wavelet denoising [J]. Journal of Computational and Graphical Statistics, 2000, 9(2): 361-379.
[129] Tropp J. A. Greed is good: algorithmic results for sparse approximation [J].IEEE Transactions on Information Theory, 2004, 50(10): 2231-2242.
[130] Coifman R. R., Wickerhauser M. V. Entropy-based algorithms for best basis selection [J]. IEEE Transactions on Information Theory, 1992, 38(2): 713-718.
[131] Wipf D. P., Rao B. D. Sparse Bayesian learning for basis selection [J]. IEEE Transactions on Signal Processing, 2004, 52 (8): 2153-2164.
[132] Donho D. L., Huo X. Combined image representation using edgelets and wavelets [A]. Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII [C]. SPIE, 1999, 3813: 468-476.
[133]尹忠科,邵君, Vandergheynst P.利用FFT实现基于MP的信号稀疏分解[J].电子与信息学报, 2006, 28(4): 614-618.
[134] Kreutz K., Murray J. F. Dictionary learning algorithms for sparse representation [J]. Neural Computation, 2003, 15(2): 349-396.
[135] Aharon M., Elad M., Bruckstein A. M. The K-SVD: an algorithm for designing ofovercomplete dictionaries for sparse representation [J]. IEEE Transactions on Signal Processing, 2006, 54(11): 4311-4322.
[136] Mairal J., Elad M., Sapiro G. Sparse representation for color image restoration [J]. IEEE Transactions on Image Processing, 2008, 17(1): 53-69.
[137] Bronstein M. M., Bronstein A. M., Zibulevsky M., et al. Blind deconvolution of images using optimal sparse representations [J]. IEEE Transactions on Image Processing, 2005, 14(6): 726-736.
[138] Gutierrez J., Ferri F. J., Malo J. Regularization operators for natural images based on nonlinear perception models [J]. IEEE Transactions on Image Processing, 2006, 15(1): 189-200.
[139] Bioucas-Dias J. M., Figueiredo M. A. T. A new TwIST: two-step iterative shrinkage / thresholding algorithms for image restoration [J]. IEEE Transactions on Image Processing, 2007, 16(12): 2992-3004.
[140] Kingsbury N. Complex wavelets for shift Invariant analysis and filtering of signals [J]. Applied and Computational Harmonic Analysis, 2001, 10(3): 234-253.
[141] Selesnick I. W. The double-density dual-tree DWT [J]. IEEE Transactions on Signal Processing, 2004, 52(5): 1304-1314.
[142] Yin W. T., Osher S., Goldfarb D., et al. Bregman iterative algorithms for L1-minimization with applications to compressed sensing [J]. SIAM Journal on Imaging Sciences, 2008, 1(1): 143-168.
[143] Tikhonov A. N., Goncharsky A. V., Stepono V. V., et al. Numerical Methods for the Solutions of Ill-Posed Problems [M]. Dordrecht: Kluwer Academic Publishers, 1995.
[144] Paik J. K., Katsaggelos A. K. Image restoration using a modified Hopfield network. [J]. IEEE Transactions on Image Process, 1992, 1(1): 49-63.
[145]梁立孚.变分原理及其应用[M].第二版.哈尔滨:哈尔滨工程大学出版社: 2007.
[146] Geman S., Geman D., Abend K., et al. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of Images [J]. Journal of Applied Statistics, 1993, 20(5-6): 25-62.
[147] Likas A., Galatsanos N. P. A variational approach for Bayesian blind image deconvolution [J]. IEEE Transactions on Signal Processing, 2004, 52(8): 2222-2233.
[148] Levin A., Fergus R., Durand F., et al. Image and depth from a conventional camera with a coded aperture [J]. ACM Transactions on Graphics, 2007, 26(3), articale 70: 1-9.
[149] Yang J. F., Zhang Y., Yin W. T. An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise [J]. SIAM Journal on ScientificComputing, 2009, 31(4): 2842-2865.
[150] Harikumar G., Bresler Y. Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms [J]. IEEE Transactions on Image Processing, 1999, 8(2): 202–219.
[151] Giannakis G., Heath R. Blind identification of multichannel FIR blurs and perfect image restoration [J] IEEE Transactions on Image Processing, 2000, 9(11): 1877–1896.
[152] Pai H. T., Bovik A. Exact multichannel blind image restoration [J]. IEEE Signal Processing Letter, 1997, 4(8): 217–220.
[153] Pillai S., Liang B. Blind image deconvolution using a robust GCD approach [J]. IEEE Transactions on Image Processing, 1999, 8(2): 295–301.
[154] Chaudhuri S., Rajagopalan A. N. A recursive algorithm for maximum likelihood-based identification of blur from multiple observations [J]. IEEE Transactions on Image Processing, 1998, 7(7): 1075-1079.
[155] Panci G., Campisi P., Colonnese S., et al. Multichannel blind image deconvolution using the bussgang algorithm: spatial and multiresolution approaches [J]. IEEE Transactions on Image Processing, 2003, 12(11): 1324–1337.
[156] Sroubek F., Flusser J. Multichannel blind iterative image restoration [J]. IEEE Transactions on Image Processing, 2003, 12(9): 1094–1106.
[157] Sroubek F., Flusser J. Multichannel blind deconvolution of spatially misaligned images [J]. IEEE Transactions on Image Processing, 2005, 14(7): 874-883.
[158] Welk M., Nagy J. G. Variational deconvolution of multi-channel images with inequality constraints [A]. Proceedings of the 3rd Iberian Conference on Pattern Recognition and Image Analysis [C]. Lecture Notes in Computer Science, 2007, 4477: 386-393.
[159] Tong L., Perreau S. Multichannel blind identification: from subspace to maximum likelihood methods [J]. Proceedings of the IEEE, 1998, 86(10): 1951-1968.
[160] Chantas G. K., Calatsanos N. P., Likas A. C. Bayesian restoration using a new nonstationary edge-preserving image prior [J]. IEEE Transactions on Image Processing, 2006, 15(10): 2987-2997.
[161] Souidene W., Beghdadi A., Abed-Meraim K., et al. Regularized MRE method for blind multichannel image deconvolution [A]. Proceedings of the 38th Asilomar Conference on Signals, Systems and Computers [C]. IEEE Computer Society, 2004, 2: 2233-2237.
[162] Yan J. F., Yin W. T., Zhang Y., et al. A Fast algorithm for edge-preserving variational multichannel image restoration [J]. SIAM Journal on Imaging Sciences, 2009, 2(2): 569-592.