Skip to main content
SpringerLink
Log in

Separable Anisotropic Diffusion

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

Abstract

Anisotropic diffusion has many applications in image processing, but the high computational cost usually requires accuracy trade-offs in order to grant its applicability in practical problems. This is specially true when dealing with 3D images, where anisotropic diffusion should be able to provide interesting results for many applications, but the usual implementation methods greatly scale in complexity with the additional dimension. Here we propose a separable implementation of the most general anisotropic diffusion formulation, based on Gaussian convolutions, whose favorable computational complexity scales linearly with the number of dimensions, without any assumptions about specific parameterizations. We also present variants that bend the Gaussian kernels for improved results when dealing with highly anisotropic curved or sharp structures. We test the accuracy, speed, stability, and scale-space properties of the proposed methods, and present some results (both synthetic and real) which show their advantages, including up to 60 times faster computation in 3D with respect to the explicit method, improved accuracy and stability, and min–max preservation.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  • Acton, S., & Landis, J. (1997). Multi-spectral anisotropic diffusion. International Journal of Remote Sensing, 18(13), 2877–2886.

    Article  Google Scholar 

  • Aubert, G., & Kornprobst, P. (2006). Mathematical problems in image processing: Partial differential equations and the calculus of variations, applied mathematical sciences (2nd ed., Vol. 147). Berlin: Springer.

    MATH  Google Scholar 

  • Barash, D. (2002). Fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(6), 844–847.

    Article  Google Scholar 

  • Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., et al. (1994). Templates for the solution of linear systems: Building blocks for iterative methods (2nd ed.). Philadelphia, PA: SIAM.

    Book  MATH  Google Scholar 

  • Benmansour, F., & Cohen, L. D. (2011). Tubular structure segmentation based on minimal path method and anisotropic enhancement. International Journal of Computer Vision, 92(2), 192–210.

    Article  Google Scholar 

  • Cabral, B., & Leedom, L. C. (1993) Imaging vector fields using line integral convolution. In Proceedings SIGGRAPH (pp. 263–270).

  • Carmona, R. A., & Zhong, S. (1998). Adaptive smoothing respecting feature directions. IEEE Transactions on Image Processing, 7(3), 353–8.

    Article  Google Scholar 

  • Despotovic, I., Goossens, B., & Wilfried, P. (2015). MRI segmentation of the human brain: Challenges, methods, and applications. Computational and Mathematical Methods in Medicine, 2015(e450), 341.

    Google Scholar 

  • Direkoglu, C., Dahyot, R., & Manzke, M. (2012). On using anisotropic diffusion for skeleton extraction. International Journal of Computer Vision, 100(2), 170–189.

    Article  MathSciNet  Google Scholar 

  • Dongarra, J., Lumsdaine, A., Pozo, R., & Remington, K. (1996). IML++ v.1.2 iterative method library. http://math.nist.gov/iml++.

  • Duarte-Carvajalino, J. M., Castillo, P. E., & Velez-Reyes, M. (2007). Comparative study of semi-implicit schemes for nonlinear diffusion in hyperspectral imagery. IEEE Transactions on Image Processing, 16(5), 1303–1314.

    Article  MathSciNet  Google Scholar 

  • Duarte-Carvajalino, J. M., Sapiro, G., Velez-Reyes, M., & Castillo, P. E. (2008). Multiscale representation and segmentation of hyperspectral imagery using geometric partial differential equations and algebraic multigrid methods. IEEE Transactions on Geoscience and Remote Sensing, 46(8), 2418–2434.

    Article  Google Scholar 

  • Florack, L. M. J., & Ter Haar Romeny, B. M. (1993). Cartesian differential invariants in scale-space. Journal of Mathematical Imaging and Vision, 3(4), 327–348.

    Article  Google Scholar 

  • Frangakis, A., & Hegerl, R. (2001). Noise reduction in electron tomographic reconstructions using nonlinear anisotropic diffusion. Journal of Structural Biology, 135(3), 239–250.

    Article  Google Scholar 

  • Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., & Seidel, H. (2008). Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision, 31(23), 255–269.

    MathSciNet  MATH  Google Scholar 

  • Gilboa, G., Sochen, N., & Zeevi, Y. Y. (2006). Estimation of optimal PDE-based denoising in the SNR sense. IEEE Transactions on Image Processing, 15(8), 2269–2280.

    Article  Google Scholar 

  • Grewenig, S., Weickert, J., & Bruhn, A. (2010). From box filtering to fast explicit diffusion. In: M. Goesele, S. Roth, A. Kuijper, B. Schiele, & K. Schindler (Eds.) Pattern Recognition, Springer, Berlin, Lecture Notes in Computer Science (Vol. 6376, pp. 533–542). Source code available from http://www.mia.uni-saarland.de/Research/SC_FED.shtml.

  • Hajiaboli, M. R. (2011). An anisotropic fourth-order diffusion filter for image noise removal. International Journal of Computer Vision, 92(2), 177–191.

    Article  MathSciNet  MATH  Google Scholar 

  • Hosssain, Z., & Möller, T. (2010). Edge aware anisotropic diffusion for 3D scalar data. IEEE Transactions on Visualization and Computer Graphics, 16(6), 1376–85.

    Article  Google Scholar 

  • Koenderink, J. (1984). The structure of images. Biological Cybernetics, 50(5), 363–370.

    Article  MathSciNet  MATH  Google Scholar 

  • Kopp, J. (2008). Efficient numerical diagonalization of Hermitian \(3\times 3\) matrices. International Journal of Modern Physics C, 19(5), 845–845.

    Article  MathSciNet  MATH  Google Scholar 

  • Krissian, K., & Aja-Fernandez, S. (2009). Noise-driven anisotropic diffusion filtering of MRI. IEEE Transactions on Image Processing, 18(10), 2265–2274.

    Article  MathSciNet  MATH  Google Scholar 

  • Lampert, C. H., & Wirjadi, O. (2006). An optimal nonorthogonal separation of the anisotropic gaussian convolution filter. IEEE Transactions on Image Processing, 15(11), 3501–3513.

    Article  MathSciNet  Google Scholar 

  • LeVeque, R. (2007). Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems. Philadelphia: SIAM.

    Book  MATH  Google Scholar 

  • Lindeberg, T. (1990). Scale-space for discrete signals. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(3), 234–254.

    Article  Google Scholar 

  • Lindeberg, T. (1994). Scale-space theory in computer vision. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Martín-Herrero, J. (2007). Anisotropic diffusion in the hypercube. IEEE Transactions on Geoscience and Remote Sensing, 45(5), 1386–1398.

    Article  Google Scholar 

  • Martín-Herrero, J., & Germain, C. (2007). Microstructure reconstruction of fibrous C/C composites from X-ray microtomography. Carbon, 45(6), 1242–1253.

    Article  Google Scholar 

  • Méndez-Rial, R., & Martín-Herrero, J. (2012). Efficiency of semi-implicit schemes for anisotropic diffusion in the hypercube. IEEE Transactions on Image Processing, 21(5), 2389–2398.

    Article  MathSciNet  MATH  Google Scholar 

  • Méndez-Rial, R., Calvino-Cancela, M., & Martín-Herrero, J. (2010). Accurate implementation of anisotropic diffusion in the hypercube. IEEE Geoscience and Remote Sensing Letters, 7(4), 870–874.

    Article  Google Scholar 

  • Méndez-Rial, R., Calvino-Cancela, M., & Martín-Herrero, J. (2011). Anisotropic inpainting of the hypercube. IEEE Geoscience and Remote Sensing Letters, 9(2), 214–218.

    Article  Google Scholar 

  • Osher, S., & Rudin, L. I. (1990). Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis, 27(4), 919–940.

    Article  MATH  Google Scholar 

  • Perona, P., & Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), 629–639.

    Article  Google Scholar 

  • Pham, T. Q., & Van Vliet, L. J. (2005). Separable bilateral filtering for fast video preprocessing. In IEEE International Conference on Multimedia and Expo (ICME), 2005(1) (pp. 1–4).

  • Rudin, L. I., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1–4), 259–268.

    Article  MathSciNet  MATH  Google Scholar 

  • Saad, Y. (2003). Iterative methods for sparse linear systems (2nd ed.). Philadelphia, PA: SIAM.

    Book  MATH  Google Scholar 

  • Sapiro, G. (2001). Geometric partial differential equations and image analysis. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Sapiro, G., & Ringach, D. L. (1996). Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Transactions on Image Processing, 5(11), 1582–1586.

    Article  MATH  Google Scholar 

  • Schmaltz, C., Peter, P., Mainberger, M., Ebel, F., Weickert, J., & Bruhn, A. (2014). Understanding, optimising, and extending data compression with anisotropic diffusion. International Journal of Computer Vision, 108(3), 222–240.

    Article  MathSciNet  Google Scholar 

  • Schwarz, H. R. (1988). Numerische Mathematik (pp. 43–45). Stuttgart: Teubner.

    MATH  Google Scholar 

  • Stalling, D., & Hege, H. C. (1995). Fast and resolution independent line integral convolution. In Proceedings of SIGGRAPH’95 (pp. 249–256).

  • Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations (2nd ed.). Philadelphia: SIAM.

    Book  MATH  Google Scholar 

  • Sun, Q., Hossack, J., Tang, J., & Acton, S. (2004). Speckle reducing anisotropic diffusion for 3D ultrasound images. Computerized Medical Imaging and Graphics, 28(8), 461–470.

    Article  Google Scholar 

  • Tomasi, C., & Manduchi, R. (1998). Bilateral filtering for gray and color images. In Proceedings of the IEEE International Conference on Computer Vision (pp. 839–846).

  • Tschumperlé, D. (2006). Fast anisotropic smoothing of multi-valued images using curvature-preserving PDEs. International Journal of Computer Vision, 68(1), 65–82.

    Article  Google Scholar 

  • Tschumperlé, D., & Deriche, R. (2005). Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(4), 506–517.

    Article  Google Scholar 

  • Vogel, C. R., & Oman, M. E. (1998). Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Transactions on Image Processing, 7(6), 813–824.

    Article  MathSciNet  MATH  Google Scholar 

  • Weickert, J. (1998). Anisotropic diffusion in image processing. Teubner, Stuttgart. Freely avaliable from http://www.mia.uni-saarland.de/weickert/Papers/book.pdf.

  • Weickert, J. (1999). Coherence-enhancing diffusion of colour images. Image and Vision Computing, 17, 201–212.

    Article  Google Scholar 

  • Weickert, J., Romeny, B., & Viergever, M. A. (1998). Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing, 7(3), 398–410.

    Article  Google Scholar 

  • Weickert, J., Grewenig, S., Schroers, C., & Bruhn, A. (2016). Cyclic schemes for PDE-based image analysis. International Journal of Computer Vision, 118(3), 275–299. https://doi.org/10.1007/s11263-015-0874-1.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Yang, G. Z., Burger, P., Firmin, D. N., & Underwood, S. R. (1996). Structure adaptive anisotropic image filtering. Image and Vision Computing, 14(2), 135–145.

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Prof. C. Germain (IMS, Bordeaux), Prof. G.L. Vignoles, and O. Coindreau (LCTS), Snecma Propulsion Solide, and the ESRF (European Synchrotron Radiation Facility) ID 19 team for providing the C/C composite 3D image.

Author information

Authors and Affiliations

  1. Robotics and Control Unit, AIMEN Technology Center, 36410, O Porriño, Spain

    Roi Méndez-Rial

  2. Department of Signal Theory and Communications and AtlanTTic, ETSE Telecomunicación, University of Vigo, 36310, Vigo, Spain

    Julio Martín-Herrero

Authors
  1. Roi Méndez-Rial
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julio Martín-Herrero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roi Méndez-Rial.

Additional information

Communicated by Julien Mairal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Méndez-Rial, R., Martín-Herrero, J. Separable Anisotropic Diffusion. Int J Comput Vis 126, 651–670 (2018). https://doi.org/10.1007/s11263-017-1060-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-017-1060-4

Keywords

Search

Navigation