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Improved Mumford-Shah Functional for Coupled Edge-Preserving Regularization and Image Segmentation


An improved Mumford-Shah functional for coupled edge-preserving regularization and image segmentation is presented. A nonlinear smooth constraint function is introduced that can induce edge-preserving regularization thus also facilitate the coupled image segmentation. The formulation of the functional is considered from the level set perspective, so that explicit boundary contours and edge-preserving regularization are both addressed naturally. To reduce computational cost, a modified additive operator splitting (AOS) algorithm is developed to address diffusion equations defined on irregular domains and multi-initial scheme is used to speed up the convergence rate. Experimental results by our approach are provided and compared with that of Mumford-Shah functional and other edge-preserving approach, and the results show the effectiveness of the proposed method.


  1. 1.

    Mumford D, Shah J: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics 1989, 42: 577–685. 10.1002/cpa.3160420503

    MathSciNet  Article  Google Scholar 

  2. 2.

    Geman S, Geman D: Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 1984, PAMI-6(6):721–741.

    Article  Google Scholar 

  3. 3.

    Shah J: A common framework for curve evolution, segmentation and anisotropic diffusion. Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '96), June 1996, San Francisco, Calif, USA 136–142.

    Google Scholar 

  4. 4.

    Teboul S, Féraud LB, Aubert G, Barlaud M: Variational approach for edge-preserving regularization using coupled PDE's. IEEE Transactions on Image Processing 1998, 7(3):387–397. 10.1109/83.661189

    Article  Google Scholar 

  5. 5.

    Chan TF, Vese LA: Active contours without edges. IEEE Transactions on Image Processing 2001, 10(2):266–277. 10.1109/83.902291

    Article  Google Scholar 

  6. 6.

    Vese LA, Chan TF: A multiphase level set framework for image segmentation using the Mumford and Shah model. International Journal of Computer Vision 2002, 50(3):271–293. 10.1023/A:1020874308076

    Article  Google Scholar 

  7. 7.

    Sonia F, Philippe M: Segmentation d'images par contours actifs sur le modèle de Mumford-Shah. Isabelle Bloch, Encadrant: Najib Gadi, Janvier–Avril 2004

    Google Scholar 

  8. 8.

    Tsai A, Yezzi A Jr., Willsky AS: Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Transactions on Image Processing 2001, 10(8):1169–1186. 10.1109/83.935033

    Article  Google Scholar 

  9. 9.

    Maso GD, Morel JM, Solimini S: A variational method in image segmentation: existence and approximation results. Acta Matematica 1992, 168: 89–151. 10.1007/BF02392977

    MathSciNet  Article  Google Scholar 

  10. 10.

    Ambrosio L, Tortorelli VM:Approximation of functionals depending on jumps by elliptic functionals via-convergence. Communications on Pure and Applied Mathematics 1990, 43: 999–1036. 10.1002/cpa.3160430805

    MathSciNet  Article  Google Scholar 

  11. 11.

    Richardson TJ: Scale independent piecewise smooth segmentation of images via variational methods, Ph.D. dissertation. Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Mass, USA; 1989.

    Google Scholar 

  12. 12.

    Weickert J: A review of nonlinear diffusion filtering. Scale-Space Theory in Computer Vision, July 1997, Utrecht, The Netherlands, Lecture Notes in Computer Science 1252: 3–28.

    Google Scholar 

  13. 13.

    Sethian JA: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Material Science. Cambridge University Press, Cambridge, UK; 1999.

    Google Scholar 

  14. 14.

    Weickert J, Zuiderveld KJ, ter Haar Romeny BM, Niessen WJ: Parallel implementations of AOS schemes: a fast way of nonlinear diffusion filtering. Proceedings of IEEE International Conference on Image Processing, October 1997, Santa Barbara, Calif, USA 3: 396–399.

    Article  Google Scholar 

  15. 15.

    Choi J, Kim G, Park P, et al.: Efficient PDE-based segmentation algorithms and their application to CT-scan images. Journal of the Korean Institute of Plant Engineering 2003, 1–17.

    Google Scholar 

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Correspondence to Zhang Hongmei.

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Hongmei, Z., Mingxi, W. Improved Mumford-Shah Functional for Coupled Edge-Preserving Regularization and Image Segmentation. EURASIP J. Adv. Signal Process. 2006, 037129 (2006).

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  • Information Technology
  • Computational Cost
  • Convergence Rate
  • Quantum Information
  • Image Segmentation