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全变差图像恢复的自适应步长梯度投影算法

张本鑫 朱志斌

张本鑫, 朱志斌. 全变差图像恢复的自适应步长梯度投影算法. 自动化学报, 2016, 42(9): 1347-1355. doi: 10.16383/j.aas.2016.c150146
引用本文: 张本鑫, 朱志斌. 全变差图像恢复的自适应步长梯度投影算法. 自动化学报, 2016, 42(9): 1347-1355. doi: 10.16383/j.aas.2016.c150146
ZHANG Ben-Xin, ZHU Zhi-Bin. Gradient Projection Algorithm for Total Variation Image Restoration by Adaptive Steplength Selection Rules. ACTA AUTOMATICA SINICA, 2016, 42(9): 1347-1355. doi: 10.16383/j.aas.2016.c150146
Citation: ZHANG Ben-Xin, ZHU Zhi-Bin. Gradient Projection Algorithm for Total Variation Image Restoration by Adaptive Steplength Selection Rules. ACTA AUTOMATICA SINICA, 2016, 42(9): 1347-1355. doi: 10.16383/j.aas.2016.c150146

全变差图像恢复的自适应步长梯度投影算法

doi: 10.16383/j.aas.2016.c150146
基金项目: 

国家自然科学基金 11361018

广西自动检测技术与仪器重点实验室基金 YQ16112

桂林市科技攻关项目 20140127-2

广西自动检测技术与仪器重点实验室基金 YQ15112

广西省自然科学基金 2014GXNSFFA118001

广西和桂林电子科技大学研究生教育创新计划项目 YJCXB201502

广西高校科研一般项目 KY2016YB167

国家自然科学基金 11461015

详细信息
    作者简介:

    张本鑫桂林电子科技大学电子工程与自动化学院博士研究生.主要研究方向为最优化方法和变分法在图像处理中的应用.E-mail:zbx913@163.com

    通讯作者:

    朱志斌桂林电子科技大学数学与计算科学学院教授.2004年获西安交通大学理学博士学位.主要研究方向为最优化方法及其在图像处理和反问题中的应用.本文通信作者.E-mail:optimization_zhu@163.com

Gradient Projection Algorithm for Total Variation Image Restoration by Adaptive Steplength Selection Rules

Funds: 

National Natural Science Foundation of China 11361018

Guangxi Key Laboratory of Automatic Detecting Technology and Instruments YQ16112

Guilin Science and Technology Project 20140127-2

Guangxi Key Laboratory of Automatic Detecting Technology and Instruments YQ15112

Natural Science Foundation of Guangxi Province 2014GXNSFFA118001

Innovation Project of Guangxi and Guilin University of Electronic Technology Graduate Education YJCXB201502

Guangxi Education Scientific Research Program KY2016YB167

National Natural Science Foundation of China 11461015

More Information
    Author Bio:

    Ph. D. candidate at the School of Electronic Engineering and Automation, Guilin University of Electronic Technology. His research interest covers optimization algorithm, variational method and their applications in image processing

    Corresponding author: ZHU Zhi-Bin Professor at the School of Mathematics and Computing Science, Guilin University of Electronic Technology. He received his Ph. D. degree in applied mathematics from Xi0an Jiaotong University in 2004. His research interest covers optimization and their applications in image processing and its inverse problem
  • 摘要: 针对图像去噪问题,本文基于全变差对偶公式提出一个新的梯度投影算法.算法采用改进的非单调线搜索和自适应BB(Barzilai-Borwein)步长,有效地改善了Chambolle梯度投影算法收敛慢的缺点.数值结果表明新算法优于一些已有的梯度投影算法.
  • 图  1  Boat(1 024×1 024)去噪结果

    Fig.  1  Denoising results of Boat (1 024×1 024)

    图  2  Boat(1 024×1 024)梯度投影的相对范数vs. CPU时间(秒)

    Fig.  2  Relative norm of projected gradient vs. CPU time (s) of Boat (1 024×1 024)

    图  3  Cameraman( $Tol\, =\, 10^{-4}$ )和Barbara( $Tol\, =\, 10^{-6})$ MGPSSABB的去噪结果

    Fig.  3  Denoising results of MGPSSABB of Cameraman ( $Tol\, =\, 10^{-4}$ ) and Barbara ( $Tol\, =\, 10^{-6}$ )

    图  4  四个算法的对偶间隙vs. CPU时间(秒) (Tol=10−6)

    Fig.  4  Duality gap vs. CPU time(s) for four algorithms (Tol=10−6)

    表  1  数值结果

    Table  1  Numerical results

    Image Algorithm Niter Time(s) RMSE
    Shape(128×128) Chambolle 737 2.06 0.0739
    Chambolle1 592 1.86 0.0738
    MChambolle 196 0.81 0.0412
    GPSSABB 182 0.70 0.0414
    MGPSSABB 130 0.61 0.0414
    Lena(256×256) Chambolle 352 5.79 0.0829
    Chambolle1 272 4.73 0.0827
    MChambolle 129 3.62 0.0221
    GPSSABB 152 4.13 0.0214
    MGPSSABB 85 2.34 0.0214
    Peppers(512×512) Chambolle 389 35.80 0.0525
    Chambolle1 298 26.27 0.0520
    MChambolle 124 19.77 0.0520
    GPSSABB 124 19.67 0.0079
    MGPSSABB 92 15.79 0.0079
    Boat(1024×1024) Chambolle 388 141.80 0.0078
    Chambolle1 300 117.05 0.0078
    MChambolle 140 101.66 0.0078
    GPSSABB 156 107.90 0.0078
    MGPSSABB 97 67.16 0.0078
    下载: 导出CSV

    表  2  Cameraman (256 × 256)迭代步数和CPU时间(秒)

    Table  2  Number of iterations (Iter) and CPU time (s) of Cameraman (256 × 256)

    Algorithms Tol=10−2 Tol=10−3 Tol=10−4 Tol=10−6
    Iter Time(s) Iter Time(s) Iter Time(s) Iter Time(s)
    Chambolle 27 0.45 163 3.18 822 16.04 14625 271.89
    GPCL 32 0.59 114 1.98 549 10.19 10116 187.28
    GPLS 18 0.53 132 5.38 607 26.22 12100 550.32
    GPBBM 20 0.56 128 3.24 596 16.99 11032 295.31
    GPBBM(3) 20 0.58 72 1.78 292 7.53 3186 79.15
    SQPBBM 14 0.50 43 1.95 167 7.66 2532 105.18
    GPBBsafe 16 0.33 47 1.05 165 4.23 2398 73.68
    GPBBNM 16 0.34 48 1.06 162 3.42 2974 62.19
    GPABB 16 0.45 47 1.23 179 4.68 2276 49.45
    GPSSABB 13 0.41 48 1.36 146 4.31 1372 38.50
    MGPSSABB 13 0.37 47 1.33 129 3.67 678 18.90
    下载: 导出CSV

    表  3  Barbara (512 × 512)迭代步数和CPU时间(秒)

    Table  3  Number of iterations (Iter) and CPU time (s) of Barbara (512 × 512)

    Algorithms Tol=10−2 Tol=10−3 Tol=10−4 Tol=10−6
    Iter Time(s) Iter Time(s) Iter Time(s) Iter Time(s)
    Chambolle 27 3.04 131 14.27 536 60.47 8583 939.73
    GPCL 24 2.78 79 9.31 331 37.10 5854 641.63
    GPLS 36 6.61 90 20.90 319 77.84 5532 1316.20
    GPBBM 20 3.03 84 13.32 340 52.10 6096 872.62
    GPBBM(3) 20 2.89 74 11.31 229 34.94 2588 365.88
    SQPBBM 14 2.75 33 7.71 106 23.20 1530 325.25
    GPBBsafe 15 2.09 40 5.76 118 18.75 1494 290.38
    GPBBNM 15 2.42 41 5.55 116 15.69 1467 194.15
    GPABB 14 2.43 35 5.91 125 21.23 1123 180.51
    GPSSABB 12 2.03 34 5.97 104 19.25 878 159.95
    MGPSSABB 14 2.39 37 6.26 90 15.55 431 86.31
    下载: 导出CSV
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出版历程
  • 收稿日期:  2015-03-24
  • 录用日期:  2016-04-28
  • 刊出日期:  2016-09-01

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