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Convex Relaxation of a Class of Vertex Penalizing Functionals

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Abstract

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.

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References

  1. Ambrosio, L., Masnou, S.: A direct variational approach to a problem arising in image reconstruction. Interfaces Free Bound. 5, 63–81 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon/Oxford University Press, New York (2000)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)

    Google Scholar 

  5. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  7. Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  9. Chambolle, A., Cremers, D., Pock, T.: A convex approach for computing minimal partitions. Technical Report 649, CMAP, Ecole Polytechnique, France (2008)

  10. Chan, T.F., Shen, J.: Nontexture inpainting by curvature driven diffusion (cdd). J. Vis. Commun. Image Represent. 12, 436–449 (2001)

    Article  Google Scholar 

  11. Chan, T.F., Kang, S.-H., Shen, J.: Euler’s elastica and curvature based inpaintings. SIAM J. Appl. Math. 63, 564–594 (2002)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. DiBenedetto, E.: Real Analysis. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser, Boston (2002)

    Book  MATH  Google Scholar 

  14. Droske, M., Rumpf, M.: A level set formulation for Willmore flow. Interfaces Free Bound. 6(3), 361–378 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  20. Goldluecke, B., Cremers, D.: Introducing total curvature for image processing. In: IEEE International Conference on Computer Vision (ICCV) (2011)

    Google Scholar 

  21. Gol’shtein, E.G., Tret’yakov, N.V.: Modified Lagrangians in convex programming and their generalizations. Math. Program. Stud. 10, 86–97 (1979)

    Article  MathSciNet  Google Scholar 

  22. Grzhibovskis, R., Heintz, A.: A convolution-thresholding approximation of generalized curvature flows. SIAM J. Numer. Anal. 42, 2652–2670 (2004)

    Article  MathSciNet  Google Scholar 

  23. He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for total variation image restoration. Technical report, Nanjing University, China (2010)

  24. Kanizsa, G.: Organization in Vision. Praeger, New York (1979)

    Google Scholar 

  25. Lellmann, J., Schnörr, C.: Continuous multiclass labeling approaches and algorithms. SIAM J. Imaging Sci. 4(4), 1049–1096 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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. Mumford, D.: Elastica and computer vision. In: Algebraic Geometry and Its Applications, pp. 491–506 (1994)

    Chapter  Google Scholar 

  28. Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nitzberg, M., Mumford, D., Shiota, T.: Filtering, segmentation, and depth. In: Lecture Notes in Comp. Sci., vol. 662 (1993)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  32. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  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)]

    Article  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Weickert, J.: Anisotropic diffusion image processing. Kaiserslautern (1996)

  40. Willmore, T.J.: Riemannian Geometry. Clarendon, Oxford (1993)

    MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010, Graz, Austria

    Kristian Bredies

  2. Institute for Computer Graphics and Vision, Graz University of Technology, Inffeldgasse 16, 8010, Graz, Austria

    Thomas Pock

  3. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, 10012, NY, USA

    Benedikt Wirth

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  1. Kristian Bredies
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  2. Thomas Pock
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  3. Benedikt Wirth
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Correspondence to Kristian Bredies.

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Bredies, K., Pock, T. & Wirth, B. Convex Relaxation of a Class of Vertex Penalizing Functionals. J Math Imaging Vis 47, 278–302 (2013). https://doi.org/10.1007/s10851-012-0347-x

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