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A variational formulation for physical noised image segmentation

  • Qiong Lou
  • Jia-lin Peng
  • De-xing KongEmail author
Article
  • 97 Downloads

Abstract

Image segmentation is a hot topic in image science. In this paper we present a new variational segmentation model based on the theory of Mumford-Shah model. The aim of our model is to divide noised image, according to a certain criterion, into homogeneous and smooth regions that should correspond to structural units in the scene or objects of interest. The proposed region-based model uses total variation as a regularization term, and different fidelity term can be used for image segmentation in the cases of physical noise, such as Gaussian, Poisson and multiplicative speckle noise. Our model consists of five weighted terms, two of them are responsible for image denoising based on fidelity term and total variation term, the others assure that the three conditions of adherence to the data, smoothing, and discontinuity detection are met at once. We also develop a primal-dual hybrid gradient algorithm for our model. Numerical results on various synthetic and real images are provided to compare our method with others, these results show that our proposed model and algorithms are effective.

Keywords

image segmentation variational method image denoising primal-dual hybrid gradient algorithm non-Gaussian noise 

MR Subject Classification

65K10 68U10 49M30 

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References

  1. [1]
    L Ambrosio, N Fusco, D Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, 2000.zbMATHGoogle Scholar
  2. [2]
    X Cai, R Chan, T Zeng. Image segmentation by convex approximation of the Mumford-Shah model, 2012, preprint.Google Scholar
  3. [3]
    V Caselles, R Kimmel, G Sapiro. Geodesic active contours, Int J Comput Vision, 1997, 22: 61–79.CrossRefzbMATHGoogle Scholar
  4. [4]
    T Chan, S Esedoglu, F Park, A Yip. Recent developments in total variation image restoration, In: Mathematical Models in Computer Vision, Springer Verlag, 2005.Google Scholar
  5. [5]
    T F Chan, L A Vese. Active contours without edges, IEEE Trans Image Process, 2001, 10: 266–277.CrossRefzbMATHGoogle Scholar
  6. [6]
    D Cremers, M Rousson, R Deriche. A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape, Int J Comput Vision, 2007, 72: 195–215.CrossRefGoogle Scholar
  7. [7]
    D Gabay, B Mercier. A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput Math Appl, 1976, 2: 17–40.CrossRefzbMATHGoogle Scholar
  8. [8]
    T Goldstein, S Osher. The split Bregman method for L1-regularized problems, SIA M J Imaging Sci, 2009, 2: 323–343.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    M Kass, A Witkin, D Terzopoulos. Snakes: active contour models, Int J Comput Vision, 1988, 1: 321–331.CrossRefGoogle Scholar
  10. [10]
    D Krishnan Q V Pham, A M Yip. A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmention problems, Adv Comput Math, 2009, 31: 237–266.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    H Lantéri, C Theys. Restoration of astrophysical images: the case of Poisson data with additive Gaussian noise, EURASIP J Adv Signal Process, 2005, 15: 2500–2513.CrossRefGoogle Scholar
  12. [12]
    C Li, C Xu, C Gui, M D Fox. Level set evolution without re-initialization: a new variational formulation, In: Proc of IEEE Conference on Computer Vision and Pattern Recognition, 2005, 1: 430–436.Google Scholar
  13. [13]
    M Mueller, K Segl, H Kaufmann. Edge-and region-based segmentation technique for the extraction of large, man-made objects in high-resolution satellite imagery, Pattern Recogn, 2004, 37: 1619–1628.CrossRefGoogle Scholar
  14. [14]
    D Mumford, J Shah. Optimal approximations by piecewise smooth functions and associated variatioanl problems, Comm Pure Appl Math, 1989, 42: 577–685.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    S Osher, R Fedkiw. Level Set Methods and Dynamic Implicit Surfaces, A MS Vol 153, Springer, 2003.CrossRefzbMATHGoogle Scholar
  16. [16]
    J L Peng, F F Dong, D X Kong. Recent advances of variational model in medical imaging and applications to computer aided surgery, Appl Math J Chinese Univ Ser B, 2012, 27: 379–411.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    T Pock, D Cremers, A Chambolle, H Bischof. A convex relaxation approach for computing minimal partitions, In: Proc of CVPR, 2009: 810–817.Google Scholar
  18. [18]
    P Rodríguez. Total variation regularization algorithms for images corrupted with different noise models: a review, J Electrical Comput Eng, vol 2013, 2013, Article ID 217021.Google Scholar
  19. [19]
    L Rudin, S Osher, E Fatemi. Nonlinear total variation based noise removal algorithms, Phys D, 1992, 60: 259–268.CrossRefzbMATHGoogle Scholar
  20. [20]
    A Savitzky, M J Golay. Smoothing and differentiation of data by simplified least squares procedures, Anal Chem, 1964, 36(8): 1627–1639.CrossRefGoogle Scholar
  21. [21]
    A Sawatzky, D Tenbrinck, X Jiang, M Burger. A variational framework for region-based segmentation incorporating physical noise models, J Math Imaging Vision, 2012: 1–31.Google Scholar
  22. [22]
    D Strong, T Chan. Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 2003, 19(6): S165.Google Scholar
  23. [23]
    Y Vardi, L A Shepp, L Kaufman. A statistical model for positron emission tomography, J Amer Statist Assoc, 1985, 80: 8–20.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    L A Vese, T F Chan. A multiphase level set framework for image segmentation using the Mumford and Shah model, Int J Comput Vision, 2002, 50: 271–293.CrossRefzbMATHGoogle Scholar
  25. [25]
    A Vitti. The Mumford-Shah variational model for image segmentation: an overview of the theory, implementation and use, ISPRS J Photogramm, 2012, 69: 50–64.CrossRefGoogle Scholar
  26. [26]
    M A Wani, B G Batchelor. Edge-region-based segmentation of range images, IEEE T Pattern Anal, 1994, 16: 314–319.CrossRefGoogle Scholar
  27. [27]
    M N Wernick, J N Aarsvold. Emission Tomography: The Fundamentals of PET and SPECT, Access Online via Elsevier, 2004.Google Scholar
  28. [28]
    X Z hang, M Burger, S Osher. Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIA M J Imaging Sci, 2010, 3: 253–276.CrossRefGoogle Scholar
  29. [29]
    Y Zhang, B J Matuszewski, L K Shark, C J Moore. Medical image segmentation using new hybrid level-set method, In: Bio Medical Visualization, 2008, 71–76.Google Scholar
  30. [30]
    M Zhu, T Chan. An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CA M Report 08-34, 2008.Google Scholar
  31. [31]
    M Zhu, S J Wright, T F Chan. Duality-based algorithms for total variation image restoration, Comput Optim Appl, 2010, 47: 377–400.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Center of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.The School of Computer Science and TechnologyHuaqiao UniversityXiamenChina
  3. 3.Department of MathematicsZhejiang UniversityHangzhouChina

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