Advertisement

Journal of Mathematical Imaging and Vision

, Volume 61, Issue 1, pp 1–20 | Cite as

Pointwise Besov Space Smoothing of Images

  • Gregery T. Buzzard
  • Antonin Chambolle
  • Jonathan D. Cohen
  • Stacey E. Levine
  • Bradley J. LucierEmail author
Article
  • 159 Downloads

Abstract

We formulate various variational problems in which the smoothness of functions is measured using Besov space semi-norms. Equivalent Besov space semi-norms can be defined in terms of moduli of smoothness or sequence norms of coefficients in appropriate wavelet expansions. Wavelet-based semi-norms have been used before in variational problems, but existing algorithms do not preserve edges, and many result in blocky artifacts. Here, we devise algorithms using moduli of smoothness for the \(B^1_\infty (L_1(I))\) Besov space semi-norm. We choose that particular space because it is closely related both to the space of functions of bounded variation, \({\text {BV}}(I)\), that is used in Rudin–Osher–Fatemi image smoothing, and to the \(B^1_1(L_1(I))\) Besov space, which is associated with wavelet shrinkage algorithms. It contains all functions in \({\text {BV}}(I)\), which include functions with discontinuities along smooth curves, as well as “fractal-like” rough regions; examples are given in an appendix. Furthermore, it prefers affine regions to staircases, potentially making it a desirable regularizer for recovering piecewise affine data. While our motivations and computational examples come from image processing, we make no claim that our methods “beat” the best current algorithms. The novelty in this work is a new algorithm that incorporates a translation-invariant Besov regularizer that does not depend on wavelets, thus improving on earlier results. Furthermore, the algorithm naturally exposes a range of scales that depends on the image data, noise level, and the smoothing parameter. We also analyze the norms of smooth, textured, and random Gaussian noise data in \(B^1_\infty (L_1(I))\), \(B^1_1(L_1(I))\), \({\text {BV}}(I)\) and \(L^2(I)\) and their dual spaces. Numerical results demonstrate properties of solutions obtained from this moduli of smoothness-based regularizer.

Keywords

Image smoothing Besov spaces Variational image smoothing Norms of image features—noise smooth regions textures—in smoothness spaces and duals of smoothness spaces 

Notes

Acknowledgements

The authors would like to thank Kristian Bredies for providing the penguin images in Fig. 1 and for confirming the \({\mathrm{TGV}}\) experiments related to Fig. 4.

References

  1. 1.
    Anzellotti, G., Giaquinta, M.: BV functions and traces. Rend. Sem. Mat. Univ. Padova 60, 1–21 (1979) (1978). http://www.numdam.org/item?id=RSMUP_1978__60__1_0
  2. 2.
    Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005).  https://doi.org/10.1007/s11263-005-4948-3 CrossRefzbMATHGoogle Scholar
  3. 3.
    Aujol, J.F., Aubert, G., Blanc-Fraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005).  https://doi.org/10.1007/s10851-005-4783-8 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010).  https://doi.org/10.1137/090769521 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition: 5th International Workshop, EMMCVPR 2005, St. Augustine, FL, USA, November 9–11, 2005: Proceedings, LNCS 3757. Springer, pp. 136–152 (2005)Google Scholar
  6. 6.
    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 (2011).  https://doi.org/10.1007/s10851-010-0251-1 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chambolle, A., DeVore, R.A., Lee, N.Y., Lucier, B.J.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998).  https://doi.org/10.1109/83.661182 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chambolle, A., Levine, S.E., Lucier, B.J.: An upwind finite-difference method for total variation-based image smoothing. SIAM J. Imaging Sci. 4(1), 277–299 (2011).  https://doi.org/10.1137/090752754 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cramér, H.: Mathematical Methods of Statistics. Princeton Mathematical Series, vol. 9. Princeton University Press, Princeton (1946)zbMATHGoogle Scholar
  10. 10.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)Google Scholar
  11. 11.
    DeVore, R.A., Lucier, B.J.: Fast wavelet techniques for near-optimal image processing. In: Military Communications Conference, 1992. MILCOM’92, Conference Record. Communications—Fusing Command, Control and Intelligence, vol. 3. IEEE, pp. 1129–1135 (1992).  https://doi.org/10.1109/MILCOM.1992.244110
  12. 12.
    DeVore, R.A., Popov, V.A.: Interpolation of Besov spaces. Trans. Am. Math. Soc. 305(1), 397–414 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    DeVore, R.A., Jawerth, B., Lucier, B.J.: Image compression through wavelet transform coding. IEEE Trans. Inf. Theory 38(2, part 2), 719–746 (1992).  https://doi.org/10.1109/18.119733 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ditzian, Z., Ivanov, K.G.: Minimal number of significant directional moduli of smoothness. Anal. Math. 19(1), 13–27 (1993).  https://doi.org/10.1007/BF01904036 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995). http://links.jstor.org/sici?sici=0162-1459(199512)90:432<1200:ATUSVW>2.0.CO;2-K&origin=MSN
  16. 16.
    Haddad, A., Meyer, Y.: Variational methods in image processing. In: Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, vol. 446, American Mathematical Society, Providence, RI, pp. 273–295, (2007).  https://doi.org/10.1090/conm/446/08636
  17. 17.
    Papafitsoros, K., Schönlieb, C.: A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis. 48(2), 308–338 (2014).  https://doi.org/10.1007/s10851-013-0445-4 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ridders, C.: A new algorithm for computing a single root of a real continuous function. IEEE Trans. Circuits Syst. 26(11), 979–980 (1979).  https://doi.org/10.1109/TCS.1979.1084580 CrossRefzbMATHGoogle Scholar
  19. 19.
    Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). (reprint of the 1970 original, Princeton Paperbacks)Google Scholar
  20. 20.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(14), 259–268 (1992).  https://doi.org/10.1016/0167-2789(92)90242-F MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sauer, K., Bouman, C.: Bayesian estimation of transmission tomograms using segmentation based optimization. IEEE Trans. Nuclear Sci. 39(4), 1144–1152 (1992).  https://doi.org/10.1109/23.159774 CrossRefGoogle Scholar
  22. 22.
    Schneider, C.: Trace operators in Besov and Triebel–Lizorkin spaces. Z. Anal. Anwend. 29(3), 275–302 (2010).  https://doi.org/10.4171/ZAA/1409 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang, Z.: SSIM index with automatic downsampling, version 1.0. (2009). https://www.mathworks.com/matlabcentral/answers/uploaded_files/29995/ssim.m. Accessed 27 July 2017
  24. 24.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004).  https://doi.org/10.1109/TIP.2003.819861 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.CMAP, École Polytechnique, CNRSPalaiseau CedexFrance
  3. 3.Department of Mathematics and Computer ScienceDuquesne UniversityPittsburghUSA
  4. 4.Google Inc.Mountain ViewUSA

Personalised recommendations