泊松化图像复原的交替最小化算法

2014-10-27 08:44刘新武��
经济数学 2014年3期
关键词:图像复原

刘新武��

摘 要 为了快速地去除图像中的泊松噪声, 本文在传统的交替方向算法基础上, 结合松弛算法提出了一个改进的快速交替最小化算法. 与经典的数值算法相比,数值试验表明提出的新算法不但能有效地实现泊松化图像复原, 还能大幅度地提高数值计算的速率, 并显著地减少电脑的CPU运行时间.

关键词 图像复原;泊松噪声;全变差;交替最小化算法

中图分类号 TP391 文献标识码 A

Alternating Minimization Algorithm

for Poissonian Image Restoration

LIU Xinwu

(School of Mathematics and Computational Science, Hunan University of Science

and Technology, Xiangtan, Hunan 411201,China)

Abstract To quickly remove Poisson noise, based on the traditional alternating direction method, this paper combined the relaxation method and proposed an improved alternating minimization algorithm. Compared with the classical numerical algorithm, numerical simulations demonstrate that the proposed strategy not only removes Poisson noise efficiently, but improves the speed of calculation substantially and reduces the computer CPU time noticeably.

Key words image restoration; Poisson noise; total variation; alternating minimization algorithm

1 引 言

图像在形成、传输和存储过程中, 不可避免地会受到噪声的影响. 譬如, 在天文成像[1]和电子显微镜成像[2,3]中, 获得的图像就往往会受到泊松噪声污染,并出现明显的降质现象,因此图像复原就显得尤为重要. 目前,图像复原技术已在天文学、医学、刑侦、军事以及金融学等领域得到了广泛的应用. 例如,在商业和金融行业中,一个新兴的融合信息科学、金融学和管理学的先进金融信息技术(如模式识别、人工智能等)已有效地应用于金融票据识别、金融票据影像处理及打印中,并成功地解决了一系列经济领域中的热点和难点问题.

4 结 论

本文研究了一个基于TV正则化模型的泊松化图像复原问题. 为了提高数值计算的速率, 本文结合传统的交替方向法和松弛算法, 提出了一个改进的交替最小化算法. 数值试验表明, 新算法在泊松去噪中具有显著的优越性和高效性.同时,该算法也必将在金融行业中的金融票据识别和票据影像处理中得到进一步的发展和应用.

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[7] M FIGUEIREDO, J BIOUCASDIAS. Restoration of Poissonian images using alternating direction optimization [J]. IEEE Transactions on Image Processing, 2010, 19(12): 3133-3145.

[8] S SETZER, G STEIDL, T TEUBER. Deblurring Poissonian images by split Bregman techniques [J]. Journal of Visual Communication and Image Representation, 2010, 21(3): 193-199.

[9] X LIU, L HUANG. Total bounded variationbased Poissonian images recovery by split Bregman iteration [J]. Mathematical Methods in the Applied Sciences, 2012, 35(5): 520-529.

[10]T F CHAN, P MULET. 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.

[11]C VOGEL, M OMAN. Iteration methods for total variation denoising [J]. SIAM Journal on Scientific Computing, 1996, 17(1): 227-238.

[12]A CHAMBOLLE. An algorithm for total variation minimization and application [J]. Journal of Mathematical Imaging and Vision, 2004, 20(1-2): 89-97.

[13]M K NG, L QI, Y YANG, Y HUANG. On semismooth Newtons methods for total variation minimization [J]. Journal of Mathematical Imaging and Vision, 2007, 27(3): 265-276.

[14]T GOLDSTEIN, S OSHER. The split Bregman algorithm for L1 regularized problems [J]. SIAM Journal on Imaging Sciences, 2009, 2(2): 323-343.

[15]J F CAI, S OSHER, Z SHEN. Split Bregman methods and frame based image restoration [J]. Multiscale Modeling & Simulation, 2009, 8(2): 337-369.

[16]X LIU, L HUANG. Split Bregman iteration algorithm for total bounded variation regularization based image deblurring [J]. Journal of Mathematical Analysis and Applications, 2010, 372(2): 486-495.

[17]W YIN, S OSHER, D GOLDFARB, J DARBON. Bregman iterative algorithms for L1minimization with applications to compressed sensing [J]. SIAM Journal on Imaging Sciences, 2008, 1(1): 143-168.

[18]RQ JIA, H ZHAO, W ZHAO. Relaxation methods for image denoising based on difference schemes [J]. Multiscale Modeling & Simulation, 2011, 9(1): 355-372.

[7] M FIGUEIREDO, J BIOUCASDIAS. Restoration of Poissonian images using alternating direction optimization [J]. IEEE Transactions on Image Processing, 2010, 19(12): 3133-3145.

[8] S SETZER, G STEIDL, T TEUBER. Deblurring Poissonian images by split Bregman techniques [J]. Journal of Visual Communication and Image Representation, 2010, 21(3): 193-199.

[9] X LIU, L HUANG. Total bounded variationbased Poissonian images recovery by split Bregman iteration [J]. Mathematical Methods in the Applied Sciences, 2012, 35(5): 520-529.

[10]T F CHAN, P MULET. 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.

[11]C VOGEL, M OMAN. Iteration methods for total variation denoising [J]. SIAM Journal on Scientific Computing, 1996, 17(1): 227-238.

[12]A CHAMBOLLE. An algorithm for total variation minimization and application [J]. Journal of Mathematical Imaging and Vision, 2004, 20(1-2): 89-97.

[13]M K NG, L QI, Y YANG, Y HUANG. On semismooth Newtons methods for total variation minimization [J]. Journal of Mathematical Imaging and Vision, 2007, 27(3): 265-276.

[14]T GOLDSTEIN, S OSHER. The split Bregman algorithm for L1 regularized problems [J]. SIAM Journal on Imaging Sciences, 2009, 2(2): 323-343.

[15]J F CAI, S OSHER, Z SHEN. Split Bregman methods and frame based image restoration [J]. Multiscale Modeling & Simulation, 2009, 8(2): 337-369.

[16]X LIU, L HUANG. Split Bregman iteration algorithm for total bounded variation regularization based image deblurring [J]. Journal of Mathematical Analysis and Applications, 2010, 372(2): 486-495.

[17]W YIN, S OSHER, D GOLDFARB, J DARBON. Bregman iterative algorithms for L1minimization with applications to compressed sensing [J]. SIAM Journal on Imaging Sciences, 2008, 1(1): 143-168.

[18]RQ JIA, H ZHAO, W ZHAO. Relaxation methods for image denoising based on difference schemes [J]. Multiscale Modeling & Simulation, 2011, 9(1): 355-372.

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