Journal of Mathematical Imaging and Vision

, Volume 47, Issue 3, pp 278–302 | Cite as

Convex Relaxation of a Class of Vertex Penalizing Functionals

  • Kristian BrediesEmail author
  • Thomas Pock
  • Benedikt Wirth


We investigate a class of variational problems that incorporate in some sense curvature information of the level lines. The functionals we consider incorporate metrics defined on the orientations of pairs of line segments that meet in the vertices of the level lines. We discuss two particular instances: One instance that minimizes the total number of vertices of the level lines and another instance that minimizes the total sum of the absolute exterior angles between the line segments. In case of smooth level lines, the latter corresponds to the total absolute curvature. We show that these problems can be solved approximately by means of a tractable convex relaxation in higher dimensions. In our numerical experiments we present preliminary results for image segmentation, image denoising and image inpainting.


Variational methods Convex relaxation Higher order penalties Roto-translation space Vertex counting regularization Total curvature regularization Binary image segmentation Image denoising Image inpainting 



We thank Antonin Chambolle and Stefan Heber for fruitful discussions. The first and the third author acknowledge support from the special research grant SFB F32 “Mathematical Optimization and Applications in Biomedical Sciences” of the Austrian Science Fund (FWF) and the second author acknowledges support from the Austrian Science Fund (FWF) under the grant P22492-N23.


  1. 1.
    Ambrosio, L., Masnou, S.: A direct variational approach to a problem arising in image reconstruction. Interfaces Free Bound. 5, 63–81 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon/Oxford University Press, New York (2000) zbMATHGoogle Scholar
  3. 3.
    Ballester, C., Bertalmio, M., Caselles, V., Sapiro, G., Verdera, J.: Filling-in by joint interpolation of vector fields and gray levels. IEEE Trans. Image Process. 10, 1200–1211 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987) Google Scholar
  5. 5.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cai, X., Gu, G., He, B., Yuan, X.: A relaxed customized proximal point algorithm for separable convex programming. Technical report, Department of Mathematics, Hong Kong Baptist University, China (2011) Google Scholar
  7. 7.
    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2010) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chambolle, A., Cremers, D., Pock, T.: A convex approach for computing minimal partitions. Technical Report 649, CMAP, Ecole Polytechnique, France (2008) Google Scholar
  10. 10.
    Chan, T.F., Shen, J.: Nontexture inpainting by curvature driven diffusion (cdd). J. Vis. Commun. Image Represent. 12, 436–449 (2001) CrossRefGoogle Scholar
  11. 11.
    Chan, T.F., Kang, S.-H., Shen, J.: Euler’s elastica and curvature based inpaintings. SIAM J. Appl. Math. 63, 564–594 (2002) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24(3), 307–326 (2006) MathSciNetCrossRefGoogle Scholar
  13. 13.
    DiBenedetto, E.: Real Analysis. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser, Boston (2002) zbMATHCrossRefGoogle Scholar
  14. 14.
    Droske, M., Rumpf, M.: A level set formulation for Willmore flow. Interfaces Free Bound. 6(3), 361–378 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3, Ser. A), 293–318 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    El-Zehiry, N., Grady, L.: Fast global optimization of curvature. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR2010), pp. 3257–3264 (2010) Google Scholar
  17. 17.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992) zbMATHGoogle Scholar
  18. 18.
    Franken, M., Rumpf, M., Wirth, B.: A phase field based PDE constraint optimization approach to time discrete Willmore flow. Int. J. Numer. Anal. Model. (2011) Google Scholar
  19. 19.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6(6), 721–741 (1984) CrossRefGoogle Scholar
  20. 20.
    Goldluecke, B., Cremers, D.: Introducing total curvature for image processing. In: IEEE International Conference on Computer Vision (ICCV) (2011) Google Scholar
  21. 21.
    Gol’shtein, E.G., Tret’yakov, N.V.: Modified Lagrangians in convex programming and their generalizations. Math. Program. Stud. 10, 86–97 (1979) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Grzhibovskis, R., Heintz, A.: A convolution-thresholding approximation of generalized curvature flows. SIAM J. Numer. Anal. 42, 2652–2670 (2004) MathSciNetCrossRefGoogle Scholar
  23. 23.
    He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for total variation image restoration. Technical report, Nanjing University, China (2010) Google Scholar
  24. 24.
    Kanizsa, G.: Organization in Vision. Praeger, New York (1979) Google Scholar
  25. 25.
    Lellmann, J., Schnörr, C.: Continuous multiclass labeling approaches and algorithms. SIAM J. Imaging Sci. 4(4), 1049–1096 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Masnou, S., Morel, J.-M.: Level-lines based disocclusion. In: Proceedings of 5th IEEE International Conference on Image Processing (ICIP), pp. 259–263 (1998) Google Scholar
  27. 27.
    Mumford, D.: Elastica and computer vision. In: Algebraic Geometry and Its Applications, pp. 491–506 (1994) CrossRefGoogle Scholar
  28. 28.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Nitzberg, M., Mumford, D., Shiota, T.: Filtering, segmentation, and depth. In: Lecture Notes in Comp. Sci., vol. 662 (1993) Google Scholar
  30. 30.
    Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms. In: International Conference of Computer Vision (ICCV 2011), pp. 1762–1769 (2011) CrossRefGoogle Scholar
  31. 31.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: Global solutions of variational models with convex regularization. SIAM J. Imaging Sci. 3, 1122 (2010). doi: 10.1137/090757617 MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Rudin, L., Osher, S.J., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992). [Also in Experimental Mathematics: Computational Issues in Nonlinear Science (Proc. Los Alamos Conf. 1991)] zbMATHCrossRefGoogle Scholar
  34. 34.
    Schoenemann, T., Cremers, D.: Introducing curvature into globally optimal image segmentation: minimum ratio cycles on product graphs. In: International Conference on Computer Vision (ICCV2007), Rio de Janeiro, Brazil, October 2007 Google Scholar
  35. 35.
    Schoenemann, T., Masnou, S., Cremers, D.: The elastic ratio: introducing curvature into ratio-based image segmentation. IEEE Trans. Image Process. 20(9), 2565–2581 (2011) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schoenemann, T., Kahl, F., Masnou, S., Cremers, D.: A linear framework for region-based image segmentation and inpainting involving curvature penalization. Int. J. Comput. Vis. 99(1), 53–68 (2012). doi: 10.1007/s11263-012-0518-7 MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Strekalovskiy, E., Cremers, D.: Total variation for cyclic structures: convex relaxation and efficient minimization. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR2011), pp. 1905–1911, June 2011 Google Scholar
  38. 38.
    Tai, X.-C., Hahn, J., Chung, G.J.: A fast algorithm for Euler’s elastica model using augmented Lagrangian method. SIAM J. Imaging Sci. 4(1), 313–344 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Weickert, J.: Anisotropic diffusion image processing. Kaiserslautern (1996) Google Scholar
  40. 40.
    Willmore, T.J.: Riemannian Geometry. Clarendon, Oxford (1993) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  2. 2.Institute for Computer Graphics and VisionGraz University of TechnologyGrazAustria
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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