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基于加权总广义变差的Mumford-Shah模型

张文娟 冯象初 王旭东

张文娟, 冯象初, 王旭东. 基于加权总广义变差的Mumford-Shah模型. 自动化学报, 2012, 38(12): 1913-1922. doi: 10.3724/SP.J.1004.2012.01913
引用本文: 张文娟, 冯象初, 王旭东. 基于加权总广义变差的Mumford-Shah模型. 自动化学报, 2012, 38(12): 1913-1922. doi: 10.3724/SP.J.1004.2012.01913
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. doi: 10.3724/SP.J.1004.2012.01913
Citation: 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. doi: 10.3724/SP.J.1004.2012.01913

基于加权总广义变差的Mumford-Shah模型

doi: 10.3724/SP.J.1004.2012.01913
详细信息
    通讯作者:

    张文娟

Mumford-Shah Model Based on Weighted Total Generalized Variation

  • 摘要: 给出了加权总广义变差(Total generalized variation, TGV)的定义. 利用图像的2阶加权TGV半范作为正则项, 利用水平集函数的2阶加权TGV半范近似边界长度, 提出了基于加权TGV的Mumford-Shah模型. 对未知函数分别利用交替Split-Bregman方法、Fenchel对偶方法及FISTA (Fast iterative shrinkage-thresholding algorithm)给出数值计算模型. 仿真实验结果表明, 利用图像的2阶加权TGV半范的去噪效果优于常用的梯度模2范数和加权TV (Total variation)半范正则化; 利用水平集函数的2阶加权TGV半范近似边界长度的边缘检测效果优于传统的TV半范和加权TV半范约束.
  • [1] Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 1989, 42(5): 577-685[2] Li Xiao-Mao, Zhu Lin-Lin, Tang Yan-Dong. Boundary detection using open spline curve based on Mumford-Shah model. Acta Automatica Sinica, 2009, 35(2): 132-136(李小毛, 朱琳琳, 唐延东.基于Mumford-Shah模型和开样条曲线的边界检测. 自动化学报, 2009, 35(2): 132-136)[3] Li C M, Huang R, Ding Z H, Gatenby J C, Metaxas D N, Gore J C. A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI. IEEE Transactions on Image Processing, 2011, 20(7): 2007-2016[4] Vese L A, Chan T F. A multiphase level set framework for image segmentation using the Mumford and Shah model. International Journal of Computer Vision, 2002, 50(3): 271-293[5] Khare M, Srivastava R K. Level set method for segmentation of medical images without reinitialization. Journal of Medical Imaging and Health Informatics, 2012, 2(2): 158-167[6] Wang L L, Shi Y Y, Tai X C. Robust edge detection using Mumford-Shah model and binary level set method. In: Proceedings of the 3rd International Conference on Scale Space and Variational Methods in Computer Vision. Berlin, Heidelberg: Springer-Verlag, 2011. 203-301[7] Rudin L I, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D, 1992, 60(1-4): 259-268[8] He Jin-Rong. An Iteratively Reweighted Norm Algorithm for Image Restoration Based on Total Variation Regularization [Master dissertation], Wuhan University of Technology, China, 2009 (何进荣.基于加权范数迭代算法的总变差正则化图像复原方法[硕士学位论文],武汉理工大学, 中国, 2009)[9] Zhang J, Wei Z H. A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising. Applied Mathematical Modelling, 2011, 35(5): 2516-2528[10] Chan T F, Esedoglu S, Park F E. A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In: Proceedings of the 17th IEEE International Conference on Image. Hong Kong: IEEE, 2010. 4137-4140[11] Chan T, Marquina A, Mulet P. High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 2000, 22(2): 503-516[12] Póschl C, Scherzer O. Characterization of minimizers of convex regularization functionals. Contemporary Mathematics, 2008, 451: 219-248[13] Bredies K, Kunisch K, Pock T. Total generalized variation. SIAM Journal on Imaging Sciences, 2010,3(3): 492-526[14] Simon S. Operator splittings, bregman methods and frame shrinkage in image processing. International Journal of Computer Vision, 2011, 92(3): 265-280[15] Carter J. Dual Methods for Total Variation Based Image Restoration [Ph.D. dissertation], University of California, Los Angeles, USA, 2001[16] Beck A, Teboulle M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2009, 2(1): 183-202[17] 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 (王旭东, 冯象初, 霍雷刚. 去除乘性噪声的重加权各向异性全变差模型.自动化学报, 2012, 38(3): 444-451)[18] Han Yu, Wang Wei-Wei, Feng Xiang-Chu. Iteratively reweighted method based nonrigid image registration. Acta Automatica Sinica, 2011, 37(9): 1059-1066 (韩雨, 王卫卫, 冯象初. 基于迭代重加权的非刚性图像配准. 自动化学报,2011, 37(9): 1059-1066)[19] Chambolle A. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 2004, 20(1-2): 89-97[20] Canny J F. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1986, 8(6): 679-698[21] Smith S M, Brady J M. SUSAN——a new approach to low level image processing. International Journal of Computer Vision, 1997, 23(1): 45-78
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出版历程
  • 收稿日期:  2012-05-15
  • 修回日期:  2012-08-31
  • 刊出日期:  2012-12-20

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