Skip to main content
Springer Nature Link
Log in

Fast Kernel Smoothing by a Low-Rank Approximation of the Kernel Toeplitz Matrix

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Kernel smoothing methods, including the bilateral filter, are commonly used in data processing/modeling and edge-aware image smoothing. Due to their nonlinear nature, these filters require significant computational time. In this paper, we address this problem by studying a practical case in which the data to be processed are integers. The basic idea is to use eigendecomposition to approximate the kernel matrix which is a real symmetric Toeplitz matrix. This approximation leads to more efficient computation. We study the distribution of its eigenvalues and show that the upper bounds of the eigenvalues can be expressed analytically in terms of the Fourier transform of the kernel function. This result not only captures the relationship between the order of the low-rank approximation of the kernel matrix and the filtering quality, but also shows that among the three kernel functions considered in this work, the Gaussian kernel can be most efficiently approximated. We have applied the proposed fast algorithm to implement the bilateral filter. By taking advantage of a property of the Gaussian kernel, we have also proposed another algorithm with even faster speed. Experimental results show that the performance of the proposed algorithms is competitive with those state-of-the-art algorithms in terms of speed and quality.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Reviewer A pointed out this result.

  2. Suggested by Reviewer B.

  3. https://github.com/wosugi/compressive-bilateral-filter

  4. http://people.csail.mit.edu/jiawen/#code

  5. http://github.com/wosugi/compressive-bilateral-filter

  6. http://au.mathworks.com/matlabcentral/fileexchange/56158-fast-and-accurate-bilateral-filtering

  7. http://r0k.us/graphics/kodak/index.html

References

  1. Bishop, C.: Pattern Recognition and Machine Learning. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Böttcher, A., Grudsky, G.: Spectral Properties of Banded Toeplitz Matrices. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  3. Bottou, L., Chapelle, O., DeCoste, D., Weston, J.: Large-Scale Kernel Machines (Neural Information Processing). The MIT Press, Cambridge (2007)

    Google Scholar 

  4. Caraffa, L., Tarel, J.P., Charbonnier, P.: The guided bilateral filter: when the joint/cross bilateral filter becomes robust. IEEE Trans. Image Process. 24, 1199–1208 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chaudhury, K.N., Dabhade, S.D.: Fast and provably accurate bilateral filtering. IEEE Trans. Image Process. 25, 2519 (2016)

    Article  MathSciNet  Google Scholar 

  6. Chaudhury, K.N., Sage, D., Unser, M.: Fast o(1) bilateral filtering using trigonometric range kernels. IEEE Trans. Image Process. 20(12), 3376–3382 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dai, L., Yuan, M., Zhang, X.: Accelerate bilateral filter using hermite polynomials. Electron. Lett. 50(20), 1432–1434 (2014)

    Article  Google Scholar 

  8. Deng, G.: Fast compressive bilateral filter. Electron. Lett. 53(3), 150–152 (2017)

    Article  Google Scholar 

  9. Ferreira, P.J.S.: Localization of the eigenvalues of Toeplitz matrices using additive decomposition embedding in circulants and the Fourier transform. In: Proceedings of the IFAC 10th International Symposium on System Identification, pp. 271–276 (1994)

  10. Getreuer, P.: A survey of Gaussian convolution algorithms. Image Proces. Line 3, 286–310 (2015)

    Article  Google Scholar 

  11. Girosi, F.: Models of noise and robust estimates. Technical Report A.I. Memo No. 1287, Artificial Intelligence Laboratory, MIT (1991)

  12. Gray, R.M.: Toeplitz and circulant matrices: a review. Found. Trends Commun. Inf. Theory 2, 155–239 (2006)

    Article  MATH  Google Scholar 

  13. Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Stat. Comput. 12(1), 79–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Griebel, M., Wissel, D.: Fast approximation of the discrete Gauss transform in higher dimensions. J. Sci. Comput. 55(1), 149–172 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gunturk, B.K.: Fast bilateral filter with arbitrary range and domain kernels. IEEE Trans. Image Process. 20(9), 2690–2696 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer Series in Statistics. Springer, New York (2001)

    MATH  Google Scholar 

  17. Kailath, T., Sayed, A.H.: Fast Reliable Algorithms for Matrices with Structure. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  19. Ma, Z., He, K., Wei, Y., Sun, J., Wu, E.: Constant time weighted median filtering for stereo matching and beyond. In: Proceedings of the IEEE ICCV’13, pp. 49–56 (2013)

  20. Markovsky, I.: Low Rank Approximation: Algorithms, Implementation, Applications. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  21. Milanfar, P.: A tour of modern image filtering: new insights and methods, both practical and theoretical. IEEE Signal Process. Mag. 30(1), 106–128 (2013)

    Article  MathSciNet  Google Scholar 

  22. Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice Hall, Upper Saddle River (1989)

    MATH  Google Scholar 

  23. Paris, S., Durand, F.: A fast approximation of the bilateral filter using a signal processing approach. In: Proceedings of the European Conference on Computer Vision, pp. 568–580 (2006)

  24. Paris, S., Kornprobst, P., Tumblin, J., Durand, F.: Bilateral filtering: theory and applications. Found. Trends Comput. Graph. Vis. 4(1), 1–73 (2009)

    Article  MATH  Google Scholar 

  25. Porikli, F.: Constant time o(1) bilateral filtering. In: Proceedings of the Internationl Conference on Computer Vision Pattern Recognition, pp. 1–8 (2008)

  26. Rey, W.J.J.: Introduction to Robust and Quasi-Robust Statistical Methods. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  27. Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  28. Sugimoto, K., Breckon, T., Kamata, S.: Constant-time bilateral filter using spectral decomposition. In: Proceedings of the IEEE International Conference on Image Processing, pp. 3319–3323 (2016)

  29. Sugimoto, K., Kamata, S.I.: Compressive bilateral filtering. IEEE Trans. Image Process. 24(11), 3357–3369 (2015)

    Article  MathSciNet  Google Scholar 

  30. Sugimoto, K., Kamata, S.I.: Efficient constant-time Gaussian filtering with sliding dct/dst-5 and dual-domain error minimization. ITE Trans. Media Technol. Appl. 3, 12–21 (2015)

    Article  Google Scholar 

  31. Takeda, H., Farsiu, S., Milanfar, P.: Kernel regression for image processing and reconstruction. IEEE Trans. Image Process. 16(2), 349–366 (2007)

    Article  MathSciNet  Google Scholar 

  32. Tyrtyshnikov, E.: A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232, 1–43 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tyrtyshnikov, E., Zamarashkin, N.: Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl. 270, 15–27 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, C., Duraiswami, R., Davis, L.: Efficient kernel machines using the improved fast Gauss transform. In: Saul, L.K., Weiss, Y., Bottou, L. (eds.) Advances in Neural Information Processing Systems, vol. 17, pp. 1561–1568. MIT Press, Cambridge (2005)

    Google Scholar 

  35. Yoshizawa, S., Belyaev, A., Yokota, H.: Fast Gauss bilateral filtering. Comput. Graph. Forum 29(1), 60–74 (2010)

    Article  Google Scholar 

  36. Zhang, Q., Xu, L., Jia, J.: 100+ times faster weighted median filter (WMF). In: Proceedings of the IEEE CVPR’14, pp. 2830–2837 (2014)

  37. Zhang, Z.: Parameter estimation techniques: a tutorial with application to conic fitting. Image Vis. Comput. 15, 59–76 (1997)

    Article  Google Scholar 

Download references

Acknowledgements

We thank the two reviewers for providing pertinent and constructive comments which help us to improve the technical content of this paper.

Author information

Authors and Affiliations

  1. Department of Engineering, La Trobe University, Bundoora, VIC, 3086, Australia

    Guang Deng & Song Wang

  2. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, VIC, 3010, Australia

    Jonathan H. Manton

Authors
  1. Guang Deng
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jonathan H. Manton
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Song Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Guang Deng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, G., Manton, J.H. & Wang, S. Fast Kernel Smoothing by a Low-Rank Approximation of the Kernel Toeplitz Matrix. J Math Imaging Vis 60, 1181–1195 (2018). https://doi.org/10.1007/s10851-018-0804-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-018-0804-2

Keywords