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Adaptive Second-Order Total Variation: An Approach Aware of Slope Discontinuities

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Scale Space and Variational Methods in Computer Vision (SSVM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 7893))

Abstract

Total variation (TV) regularization, originally introduced by Rudin, Osher and Fatemi in the context of image denoising, has become widely used in the field of inverse problems. Two major directions of modifications of the original approach were proposed later on. The first concerns adaptive variants of TV regularization, the second focuses on higher-order TV models. In the present paper, we combine the ideas of both directions by proposing adaptive second-order TV models, including one anisotropic model. Experiments demonstrate that introducing adaptivity results in an improvement of the reconstruction error.

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References

  1. Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification. In: Vision, Modeling, and Visualization Proceedings, pp. 293–300 (2006)

    Google Scholar 

  2. Bredies, K., Kunisch, K., Pock, T.: Total Generalized Variation. SIAM J. Imaging Sciences 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Q., Montesinos, P., Sun, Q.S., Heng, P.A., Xia, D.S.: Adaptive total variation denoising based on difference curvature. Image Vision Comput. 28(3), 298–306 (2010)

    Article  Google Scholar 

  4. Dong, Y., Hintermüller, M.: Multi-scale total variation with automated regularization parameter selection for color image restoration. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 271–281. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  5. Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proc. ISPRS Conf. on Fast Processing of Photogrammetric Data, pp. 281–305 (1987)

    Google Scholar 

  6. Frick, K., Marnitz, P., Munk, A.: Statistical multiresolution estimation for variational imaging: With an application in Poisson-biophotonics. J. Math. Imaging Vis, 1–18 (2012)

    Google Scholar 

  7. Grasmair, M.: Locally adaptive total variation regularization. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 331–342. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Grasmair, M., Lenzen, F.: Anisotropic Total Variation Filtering. Appl. Math. Optim. 62(3), 323–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu, Y., Jacob, M.: Higher degree total variation (HDTV) regularization for image recovery. IEEE Trans. Image Processing 21, 2559–2571 (2012)

    Article  MathSciNet  Google Scholar 

  10. Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lefkimmiatis, S., Bourquard, A., Unser, M.: Hessian-based norm regularization for image restoration with biomedical applications. IEEE Transactions on Image Processing 21(3), 983–995 (2012)

    Article  MathSciNet  Google Scholar 

  12. Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Optim. Appl. (2012) (online first )

    Google Scholar 

  13. Lysaker, M., Tai, X.-C.: Iterative image restoration combining total variation minimization and a second-order functional. Int. J. Comp. Vis. 66, 5–18 (2006)

    Article  Google Scholar 

  14. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1-4), 259–268 (1992)

    Article  MATH  Google Scholar 

  15. Scherzer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60, 1–27 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer (2009)

    Google Scholar 

  17. Setzer, S., Steidl, G., Teuber, T.: Infimal convolution regularizations with discrete l1-type functionals. Comm. Math. Sci. 9, 797–872 (2011)

    MathSciNet  Google Scholar 

  18. Steidl, G., Teuber, T.: Anisotropic smoothing using double orientations. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 477–489. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

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Lenzen, F., Becker, F., Lellmann, J. (2013). Adaptive Second-Order Total Variation: An Approach Aware of Slope Discontinuities. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2013. Lecture Notes in Computer Science, vol 7893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38267-3_6

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  • DOI: https://doi.org/10.1007/978-3-642-38267-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38266-6

  • Online ISBN: 978-3-642-38267-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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