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Some Nonlocal Filters Formulation Using Functional Rearrangements

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Scale Space and Variational Methods in Computer Vision (SSVM 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9087))

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Abstract

We present an exact reformulation of a broad class of nonlocal filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements.

Independently of the image spatial dimension, these filters are expressed as integral operators defined in a one-dimensional space, corresponding to the level sets measures.

We provide some insight into the properties of this new formulation and show some numerical demonstrations to illustrate them.

The authors are partially supported by the Spanish DGI Project MTM2013-43671-P.

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References

  1. Adams, A., Baek, J., Davis, M.A.: Fast high-dimensional filtering using the permutohedral lattice. Computer Graphics Forum 29(2), 753–762 (2010)

    Article  Google Scholar 

  2. Álvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. ii. Siam J Numer Anal 29(3), 845–866 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Álvarez, L., Mazorra, L.: Signal and image restoration using shock filters and anisotropic diffusion. Siam J Numer Anal 31(2), 590–605 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barash, D.: Fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE T Pattern Anal 24(6), 844–847 (2002)

    Article  Google Scholar 

  5. Barash, D., Comaniciu, D.: A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift. Image Vision Comput 22(1), 73–81 (2004)

    Article  Google Scholar 

  6. Brainweb. http://brainweb.bic.mni.mcgill.ca/brainweb,Vol.t1_icbm_normal_1mm_pn9_rf20_resampled_brain.nii.gz

  7. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model Sim 4(2), 490–530 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Buades, A., Coll, B., Morel, J.M.: Neighborhood filters and pde’s. Numer Math 105(1), 1–34 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Buades, A., Coll, B., Morel, J.M.: Image denoising methods. a new nonlocal principle. Siam Rev 52(1), 113–147 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chang, C.I., Du, Y., Wang, J., Guo, S.M., Thouin, P.: Survey and comparative analysis of entropy and relative entropy thresholding techniques. In: IEE Proceedings-Vision, Image and Signal Processing, vol. 153, pp. 837–850. IET (2006)

    Google Scholar 

  11. Chaudhury, K.N., Sage, D., Unser, M.: Fast O(1) bilateral filtering using trigonometric range kernels. TIP 2011 (2011)

    Google Scholar 

  12. Durand, F., Dorsey, J.: Fast bilateral filtering for the display of high-dynamic-range images. ACM Siggraph 21, 257–266 (2002)

    Google Scholar 

  13. Eisemann, E., Durand, F.: Flash photography enhancement via intrinsic relighting. Siggraph 23(3), 673–678 (2004)

    Article  Google Scholar 

  14. Elad, M.: On the origin of the bilateral filter and ways to improve it. IEEE T Image Process 11(10), 1141–1151 (2002)

    Article  MathSciNet  Google Scholar 

  15. Freesurfer. http://freesurfer.net

  16. FMRIB Software Library. http://fsl.fmrib.ox.ac.uk/fsl/fslwiki

  17. Galiano, G., Velasco, J.: On a non-local spectrogram for denoising one-dimensional signals. Appl Math Comput 244, 859–869 (2014)

    Article  MathSciNet  Google Scholar 

  18. Galiano, G., Velasco, J.: Neighborhood filters and the decreasing rearrangement. J Math Imaging Vision (2014). doi:10.1007/s10851-014-0522-3

  19. Galiano, G., Velasco, J.: On a fast bilateral filtering formulation using functional rearrangements. arXiv:1406.7128v1 (2015) (submitted)

  20. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model Sim 7(3), 1005–1028 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge, U.P. (1964)

    Google Scholar 

  22. Kass, M., Solomon, J.: Smoothed local histogram filters. ACM TOG 29(4), 100:1–100:10 (2010)

    Article  Google Scholar 

  23. Lieb, E.H., Loss, M.: Analysis, vol. 4. American Mathematical Soc. (2001)

    Google Scholar 

  24. Nath, S., Agarwal, S., Kazmi, Q.A.: Image histogram segmentation by multi-level thresholding using hill climbing algorithm. Int J Comput Appl 35(1) (2011)

    Google Scholar 

  25. Paris, S., Durand, F.: A fast approximation of the bilateral filter using a signal processing approach. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3954, pp. 568–580. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  26. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE T Pattern Anal 12(7), 629–639 (1990)

    Article  Google Scholar 

  27. Petschnigg, G., Agrawala, M., Hoppe, H., Szeliski, R., Cohen, M., Toyama, K.: Digital photography with flash and no-flash image pairs. ACM Siggraph 23(3), 664–672 (2004)

    Article  Google Scholar 

  28. Peyré, G.: Image processing with nonlocal spectral bases. Multiscale Model Sim 7(2), 703–730 (2008)

    Article  MATH  Google Scholar 

  29. Porikli, F.: Constant time O(1) bilateral filtering. CVPR 2008 (2008)

    Google Scholar 

  30. Rakotoson, J.M.: Réarrangement Relatif: Un instrument d’estimations dans les problčmes aux limites, vol 64. Springer (2008)

    Google Scholar 

  31. Ram, I., Elad, M., Cohen, I.: Image processing using smooth ordering of its patches. IEEE T Image Process 22(7), 2764–2774 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. Singer, A., Shkolnisky, Y., Nadler, B.: Diffusion interpretation of nonlocal neighborhood filters for signal denoising. SIAM J Imaging Sci 2(1), 118–139 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Smith, S.M., Brady, J.M.: Susan. a new approach to low level image processing. Int J Comput Vision 23(1), 45–78 (1997)

    Article  Google Scholar 

  35. Statistical Parametric Mapping (SPM). http://www.fil.ion.ucl.ac.uk/spm

  36. Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Sixth International Conference on Computer Vision, pp. 839–846. IEEE (1998)

    Google Scholar 

  37. Weiss, B.: Fast median and bilateral filtering. ACM Siggraph 25, 519–526 (2006)

    Article  Google Scholar 

  38. Yang, Q., Tan, K.H., Ahuja, N.: Real-time O(1) bilateral filtering. CVPR 2009 (2009)

    Google Scholar 

  39. Yang, Q., Ahuja, N., Tan, K.H.: Constant time median and bilateral filtering. Int J Comput Vis (2014). doi:10.1007/s11263-014-0764-y

  40. Yaroslavsky, L.P.: Digital picture processing. An introduction. Springer Verlag, Heidelberg (1985)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Universidad de Oviedo, Oviedo, Spain

    Gonzalo Galiano & Julián Velasco

Authors
  1. Gonzalo Galiano
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  2. Julián Velasco
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Corresponding author

Correspondence to Gonzalo Galiano .

Editor information

Editors and Affiliations

  1. University of Bordeaux, Talence, France

    Jean-François Aujol

  2. ENS Cachan, Cachan, France

    Mila Nikolova

  3. University of Bordeaux, Talence, France

    Nicolas Papadakis

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Galiano, G., Velasco, J. (2015). Some Nonlocal Filters Formulation Using Functional Rearrangements. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_14

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  • DOI: https://doi.org/10.1007/978-3-319-18461-6_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18460-9

  • Online ISBN: 978-3-319-18461-6

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