Skip to main content
Log in

A Fast Approximation of the Bilateral Filter Using a Signal Processing Approach

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

The bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges. It has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics, and fast versions have been proposed. Unfortunately, little is known about the accuracy of such accelerations. In this paper, we propose a new signal-processing analysis of the bilateral filter which complements the recent studies that analyzed it as a PDE or as a robust statistical estimator. The key to our analysis is to express the filter in a higher-dimensional space where the signal intensity is added to the original domain dimensions. Importantly, this signal-processing perspective allows us to develop a novel bilateral filtering acceleration using downsampling in space and intensity. This affords a principled expression of accuracy in terms of bandwidth and sampling. The bilateral filter can be expressed as linear convolutions in this augmented space followed by two simple nonlinearities. This allows us to derive criteria for downsampling the key operations and achieving important acceleration of the bilateral filter. We show that, for the same running time, our method is more accurate than previous acceleration techniques. Typically, we are able to process a 2 megapixel image using our acceleration technique in less than a second, and have the result be visually similar to the exact computation that takes several tens of minutes. The acceleration is most effective with large spatial kernels. Furthermore, this approach extends naturally to color images and cross bilateral filtering.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /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.

Similar content being viewed by others

References

  • Adalsteinsson, D., & Sethian, J. A. (1995). A fast level set method for propagating interfaces. Journal of Computational Physics, 118, 269–277.

    Article  MATH  MathSciNet  Google Scholar 

  • Aurich, V., & Weule, J. (1995). Non-linear Gaussian filters performing edge preserving diffusion. In Proceedings of the DAGM symposium.

  • Bae, S., Paris, S., & Durand, F. (2006). Two-scale tone management for photographic look. ACM Transactions on Graphics, 250(3), 637–645. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Barash, D. (2002). A fundamental relationship between bilateral filtering, adaptive smoothing and the nonlinear diffusion equation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(6), 844.

    Article  Google Scholar 

  • Barash, D., Schlick, T., Israeli, M., & Kimmel, R. (2003). Multiplicative operator splittings in non-linear diffusion: from spatial splitting to multiplicative timesteps. Journal of Mathematical Imaging and Vision, 19, 33–48.

    Article  MATH  MathSciNet  Google Scholar 

  • Bennett, E. P., & McMillan, L. (2005). Video enhancement using per-pixel virtual exposures. ACM Transactions on Graphics, 24, 845–852. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Black, M. J., Sapiro, G., Marimont, D. H., & Heeger, D. (1998). Robust anisotropic diffusion. IEEE Transactions on Image Processing, 7(3), 421–432.

    Article  Google Scholar 

  • Blinn, J. F. (1996). Fun with premultiplied alpha. IEEE Computer Graphics and Applications, 16(5), 86–89.

    Article  Google Scholar 

  • Buades, A., Coll, B., & Morel, J.-M. (2005). Neighborhood filters and PDE’s (Technical Report 2005-04). CMLA.

  • Chen, J., Paris, S., & Durand, F. (2007). Real-time edge-aware image processing with the bilateral grid. ACM Transactions on Graphics 26(3). Proceedings of the ACM SIGGRAPH conference.

  • Choudhury, P., & Tumblin, J. E. (2003). The trilateral filter for high contrast images and meshes. In Proceedings of the Eurographics symposium on rendering.

  • Durand, F., & Dorsey, J. (2002). Fast bilateral filtering for the display of high-dynamic-range images. ACM Transactions on Graphics, 21(3), 257–266. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Eisemann, E., & Durand, F. (2004). Flash photography enhancement via intrinsic relighting. ACM Transactions on Graphics, 23(3), 673–678. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Elad, M. (2002). On the bilateral filter and ways to improve it. IEEE Transactions on Image Processing, 11(10), 1141–1151.

    Article  MathSciNet  Google Scholar 

  • Elad, M. (2005). Retinex by two bilateral filters. In Proceedings of the scale-space conference.

  • Felsberg, M., Forssén, P.-E., & Scharr, H. (2006). Channel smoothing: Efficient robust smoothing of low-level signal features. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(2), 209–222.

    Article  Google Scholar 

  • Fleishman, S., Drori, I., & Cohen-Or, D. (2003). Bilateral mesh denoising. ACM Transactions on Graphics, 22(3), 950–953. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. M., & Stahel, W. A. (1986). Robust statistics—the approach based on influence functions. New York: Wiley–Interscience. ISBN 0-471-73577-9.

    MATH  Google Scholar 

  • Huber, P. J. (1981). Robust statistics. Probability and statistics. New York: Wiley–Interscience.

    Google Scholar 

  • Jones, T. R., Durand, F., & Desbrun, M. (2003). Non-iterative, feature-preserving mesh smoothing. ACM Transactions on Graphics, 22(3), 943–949. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Koenderink, J. J., & van Doorn, A. J. (1999). The structure of locally orderless images. International Journal of Computer Vision, 31(2–3), 159–168.

    Article  Google Scholar 

  • Liu, C., Freeman, T., Szeliski, R., & Kang, S. (2006). Noise estimation from a single image. In Proceedings of the computer vision and pattern recognition conference. New York: IEEE Press.

    Google Scholar 

  • Margulis, D. (2005). Photoshop LAB color: The canyon conundrum and other adventures in the most powerful colorspace. Berkeley: Peachpit. ISBN: 0321356780.

    Google Scholar 

  • Mrázek, P., Weickert, J., & Bruhn, A. (2006). Geometric properties from incomplete data. In On robust estimation and smoothing with spatial and tonal kernels. Berlin: Springer.

    Google Scholar 

  • Oh, B. M., Chen, M., Dorsey, J., & Durand, F. (2001) Image-based modeling and photo editing. In Proceedings of the ACM SIGGRAPH conference. New York: ACM.

    Google Scholar 

  • Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79, 12–49.

    Article  MATH  MathSciNet  Google Scholar 

  • Paris, S., & Durand, F. (2006). A fast approximation of the bilateral filter using a signal processing approach. In Proceedings of the European conference on computer vision.

  • Paris, S., Briceño, H., & Sillion, F. (2004). Capture of hair geometry from multiple images. ACM Transactions on Graphics, 23(3), 712–719. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Petschnigg, G., Agrawala, M., Hoppe, H., Szeliski, R., Cohen, M., & Toyama, K. (2004). Digital photography with flash and no-flash image pairs. ACM Transactions on Graphics, 23(3), 664–672. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Pham, T. Q. (2006). Spatiotonal adaptivity in super-resolution of undersampled image sequences. Ph.D. thesis, Delft University of Technology.

  • Pham, T. Q., & van Vliet, L. J. (2005). Separable bilateral filtering for fast video preprocessing. In International conference on multimedia and expo. New York: IEEE Press.

    Google Scholar 

  • Porter, T., & Duff, T. (1984). Compositing digital images. Computer Graphics, 18(3), 253–259.

    Article  Google Scholar 

  • Sand, P., & Teller, S. (2006). Particle video: Long-range motion estimation using point trajectories. In Proceedings of the computer vision and pattern recognition conference.

  • Shannon, C. E. (1949). Communication in the presence of noise. Proceedings of the Institute of Radio Engineers, 37(1), 10–21.

    MathSciNet  Google Scholar 

  • Smith, S. (2002). Digital signal processing. London: Newnes. ISBN: 075067444X.

    Google Scholar 

  • Smith, S. M., & Brady, J. M. (1997). SUSAN—a new approach to low level image processing. International Journal of Computer Vision, 23(1), 45–78.

    Article  Google Scholar 

  • Sochen, N., Kimmel, R., & Malladi, R. (1998). A general framework for low level vision. IEEE Transactions in Image Processing, 7, 310–318.

    Article  MATH  MathSciNet  Google Scholar 

  • Sochen, N., Kimmel, R., & Bruckstein, A. M. (2001). Diffusions and confusions in signal and image processing. Journal of Mathematical Imaging and Vision, 14(3), 237–244.

    Article  MathSciNet  Google Scholar 

  • Tomasi, C., & Manduchi, R. (1998). Bilateral filtering for gray and color images. In Proceedings of the international conference on computer vision (pp. 839–846). New York: IEEE Press.

    Google Scholar 

  • van de Weijer, J., & van den Boomgaard, R. (2001). Local mode filtering. In Proceedings of the conference on computer vision and pattern recognition.

  • Weickert, J., ter Haar Romeny, B. M., & Viergever, M. A. (1998). Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing, 7, 398–410.

    Article  Google Scholar 

  • Weiss, B. (2006). Fast median and bilateral filtering. ACM Transactions on Graphics, 25(3), 519–526. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Willis, P. J. (2006). Projective alpha colour. Computer Graphics Forum, 25(3), 557–566. Proceedings of the Eurographics conference.

    Article  MathSciNet  Google Scholar 

  • Winnemöller, H., Olsen, S. C., & Gooch, B. (2006). Real-time video abstraction. ACM Transactions on Graphics, 25(3), 1221–1226. Proceedings of the ACM SIGGRAPH conference.

    Article  Google Scholar 

  • Wong, W. C. K., Chung, A. C. S., & Yu, S. C. H. (2004). Trilateral filtering for biomedical images. In Proceedings of the international symposium on biomedical imaging. New York: IEEE Press.

    Google Scholar 

  • Xiao, J., Cheng, H., Sawhney, H., Rao, C., & Isnardi, M. (2006). Bilateral filtering-based optical flow estimation with occlusion detection. In Proceedings of the European conference on computer vision.

  • Yaroslavsky, L. P. (1985). Digital picture processing. An introduction. Berlin: Springer.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sylvain Paris.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paris, S., Durand, F. A Fast Approximation of the Bilateral Filter Using a Signal Processing Approach. Int J Comput Vis 81, 24–52 (2009). https://doi.org/10.1007/s11263-007-0110-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-007-0110-8

Keywords

Navigation