区间约束的全变差图像复原和自动参数估计

何川; 胡昌华; 张伟; 师彪

doi:10.3724/SP.J.1004.2014.01804

区间约束的全变差图像复原和自动参数估计

doi: 10.3724/SP.J.1004.2014.01804

1.
西安高新技术研究所西安 710025

计量
- 文章访问数: 1779
- HTML全文浏览量: 66
- PDF下载量: 993
- 被引次数: 0
出版历程
- 收稿日期: 2013-08-29
- 修回日期: 2014-03-24
- 刊出日期: 2014-08-20

Box-constrained Total-variation Image Restoration with Automatic Parameter Estimation

1.
Hi-Tech Institute of Xi'an, Xi'an 710025, China

Funds:

Supported by National Natural Science Foundation of China (61174030, 61104223, 61203189, 61374120, 61203007), the Postdoc-toral Science Foundation of China (2012M512147), and National Sci-ence Fund for Distinguished Young Scholars of China (61025014)

摘要

摘要: 因为数字图像的像素仅能取得给定动态范围内的有限值，像素值的区间约束在图像复原中引起广泛关注. 该文研究了带有正则化参数自动估计的区间约束全变差图像复原问题. 通过变量分裂并引入多组辅助变量，区间约束的全变差最小化问题被分解为一系列更易求解的子问题. 随后，交替方向法被用以求解相关的子问题. 根据Morozov偏差准则，在每步迭代中，正则化参数以闭合形式实现自适应更新. 图像复原实验表明，当较高比例的图像像素值位于给定动态范围的边界时，所提方法可以获得更为精确的复原结果.
- 全变差 /
- 区间约束 /
- 变量分裂 /
- 交替方向法
Abstract: The box constraints in image restoration have been arousing great attention, since the pixels of a digital image can attain only a finite number of values in a given dynamic range. This paper studies the box-constrained total-variation (TV) image restoration problem with automatic regularization parameter estimation. By adopting the variable splitting technique and introducing some auxiliary variables, the box-constrained TV minimization problem is decomposed into a sequence of subproblems which are easier to solve. Then the alternating direction method (ADM) is adopted to solve the related subproblems. By means of Morozov0s discrepancy principle, the regularization parameter can be updated adaptively in a closed form in each iteration. Image restoration experiments indicate that with our strategies, more accurate solutions are achieved, especially for image with high percentage of pixel values lying on the boundary of the given dynamic range.
- Total variation (TV) /
- box constraints /
- variable splitting /
- alternating direction method (ADM)

HTML全文

参考文献(29)

[1]	Campisi P, Egiazarian K. Blind Image Deconvolution-Theory and Applications. Boca Raton: CRC Press, 2007. 1-32
[2]	[2] Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D, 1992, 60(1-4): 259-268
[3]	[3] Wang Xu-Dong, Feng Xiang-Chu, Huo Lei-Gang. Iteratively reweighted anisotropic-TV based multiplicative noise removal model. Acta Automatica Sinica, 2012, 38(3): 444-451 (in Chinese)
[4]	[4] Kim D, Sra S, Dhillon I S. Tackling box-constrained optimization via a new projected quasi-Newton approach. SIAM Journal on Scientific Computing, 2010, 32(6): 3548-3563
[5]	[5] Morini B, Pocelli M, Chan R H. A reduced Newton method for constrained linear least-squares problems. Journal of Computational and Applied Mathematics, 2010, 233(9): 2200-2212
[6]	[6] Becky A, Teboulle M. Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Transactions on Image Processing, 2009, 18(11): 2419-2434
[7]	[7] Chan R H, Ma J. A multiplicative iterative algorithm for box-constrained penalized likelihood image restoration. IEEE Transactions on Image Processing, 2012, 21(7): 3168-3181
[8]	[8] Chan R H, Tao M, Yuan X M. Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM Journal on Imaging Sciences, 2013, 6(1): 680-697
[9]	[9] Ma J. Multiplicative algorithms for maximum penalized likelihood inversion with non-negative constraints and generalized error distributions. Communications in Statistics-Theory and Methods, 2006, 35(5): 831-848
[10]	Ma J. Positively constrained multiplicative iterative algorithm for maximum penalized likelihood tomographic reconstruction. IEEE Transactions on Nuclear Science, 2010, 57(1): 181-192
[11]	Chan R H, Liang H X, Ma J. Positively constrained total variation penalized image restoration. Advances in Adaptive Data Analysis, 2011, 3(1-2): 187-201
[12]	Wen Y W, Yip A M. Adaptive parameter selection for total variation image deconvolution. Numerical Mathematics-Theory Methods and Applications, 2009, 2(4): 427-438
[13]	Ng M, Weiss P, Yuan X. Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM Journal on Scientific Computing, 2010, 32(5): 2710-2736
[14]	Afonso M V, Bioucas-Dias J M, Figueiredo M A T. An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Transactions on Image Processing, 2011, 20(3): 681-695
[15]	Wen Y W, Chan R H. Parameter selection for total-variation-based image restoration using discrepancy principle. IEEE Transactions on Image Processing, 2012, 21(4): 1770-1781
[16]	Blomgren P, Chan T F. Modular solvers for image restoration problems using the discrepancy principle. Numerical Linear Algebra with Applications, 2002, 9(5): 347-358
[17]	Liao H Y, Li F, Ng M K. Selection of regularization parameter in total variation image restoration. Journal of the Optical Society of America A-Optics Image Science and Vision, 2009, 26(11): 2311-2320
[18]	Engl H W, Grever W. Using the L-curve for determining optimal regularization parameters. Numerische Mathematik, 1994, 69(1): 561-580
[19]	An Yao-Zu, Lu Yao, Zhao Hong. An adaptive-regularized image super-resolution. Acta Automatica Sinica, 2012, 38(4): 601-608 (in Chinese)
[20]	Lin Y Z, Wohlberg B, Guo H B. UPRE method for total variation parameter selection. Signal Processing, 2010, 90(8): 2546-2551
[21]	Chambolle A. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 2004, 20(1-2): 89-97
[22]	He B S, Yuan X M. Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM Journal on Imaging Sciences, 2012, 5(1): 119-149
[23]	Papadakis N, Peyr G, Oudet D. Optimal transport with proximal splitting. SIAM Journal on Imaging Sciences, 2014, 7(1): 212-238
[24]	Zhang Wen-Juan, Feng Xiang-Chu, Wang Xu-Dong. Mumford-Shah model based on weighted total generalized variation. Acta Automatica Sinica, 2012, 38(12): 1913-1922 (in Chinese)
[25]	Raguet H, Fadili J, Peyr G. A generalized forward-backward splitting. SIAM Journal on Imaging Sciences, 2013, 6(3): 1199-1226
[26]	Xue Qian, Yang Cheng-Yi, Wang Hua-Xiang. Alternating direction method for salt-and-pepper denoising. Acta Automatica Sinica, 2013, 39(12): 2071-2076 (in Chinese)
[27]	Wu C L, Tai X C. Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM Journal on Imaging Sciences, 2010, 3(3): 300-339
[28]	Wang Y L, Yang J F, Yin W T, Zhang Y. A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 2008, 1(3): 248-272
[29]	Combettes P L, Pesquet J C. Fixed-point Algorithms for Inverse Problems in Science and EngineeringChapter 10: Proximal Splitting Methods in Signal Processing. New York: Springer, 2011. 185-212