Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE's

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

Abstract

We are interested in PDE's (Partial Differential Equations) in order to smooth multi-valued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE's, we point out the pros and cons of the different equations proposed in the literature. Then, we introduce a new tensor-driven PDE, regularizing images while taking the curvatures of specific integral curves into account. We show that this constraint is particularly well suited for the preservation of thin structures in an image restoration process. A direct link is made between our proposed equation and a continuous formulation of the LIC's (Line Integral Convolutions by Cabral and Leedom (1993). It leads to the design of a very fast and stable algorithm that implements our regularization method, by successive integrations of pixel values along curved integral lines. Besides, the scheme numerically performs with a sub-pixel accuracy and preserves then thin image structures better than classical finite-differences discretizations. Finally, we illustrate the efficiency of our generic curvature-preserving approach – in terms of speed and visual quality – with different comparisons and various applications requiring image smoothing : color images denoising, inpainting and image resizing by nonlinear interpolation.

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

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Alvarez, L., Guichard, F., Lions, P.L., and Morel, J.M. 1993. Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis, 123(3):199–257.

    Google Scholar 

  • Alvarez, L. and Mazorra, L. 1994. Signal and image restoration using shock filters and anisotropic diffusion. SIAM Journal of Numerical Analysis, 31(2):590–605.

    Google Scholar 

  • Ashikhmin, M. 2001. Synthesizing Natural Textures. ACM Symposium on Interactive 3D Graphics, Research Triangle Park, NorthCarolina, pp. 217–226.

  • Aubert, G. and Kornprobst, P. 2002. Mathematical problems in image processing: Partial differential equations and the calculus of variations, Applied Mathematical Sciences, vol. 147, Springer-Verlag.

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

  • Becker, J., Preusser, T., and Rumpf, M. 2000. PDE methods in flow simulation post processing. Computing and Visualization in Science, 3(3):159–167.

    Google Scholar 

  • Bertalmio, M., Cheng, L.T., Osher, S., and Sapiro, G. 2001. Variational problems and partial differential equations on implicit surfaces. Computing and Visualization in Science, 174(2):759–780.

    Google Scholar 

  • Bertalmio, M., Sapiro, G., Caselles, V., and Ballester, C. 2000. Image inpainting. ACM SIGGRAPH, International Conference on Computer Graphics and Interactive Techniques, pp. 417–424.

  • Bertalmio, M., Vese, L., Sapiro, G., and Osher, S. 2003. Simultaneous structure and texture image inpainting. IEEE Transactions on Image Processing, 12(8):882–889.

    Google Scholar 

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

    Google Scholar 

  • Cabral, B. and Leedom, L.C. 1993. Imaging vector fields using line integral convolution. SIGGRAPH'93, in Computer Graphics, 27, 263–272.

    Google Scholar 

  • Carmona, R. and Zhong, S. 1998. Adaptive smoothing respecting feature directions. IEEE Transactions on Image Processing, 7(3):353–358.

    Google Scholar 

  • Chambolle, A. and Lions, P.L. 1997. Image recovery via total variation minimization and related problems. Nümerische Mathematik, 76(2):167–188.

    Google Scholar 

  • Chan, T. and Shen, J. 2000. Variational restoration of non-flat image features: Models and algorithms. SIAM Journal of Applied Mathematics, 61(4):1338–1361.

    Google Scholar 

  • Chan, T. and Shen, J. 2001. Non-texture inpaintings by curvature-driven diffusions. Journal of Visual Communication and Image Representation, 12(4):436–449.

    Google Scholar 

  • Charbonnier, P., Blanc-Féraud, L., Aubert, G., and Barlaud, M. 1997. Deterministic edge-preserving regularization in computed imaging. IEEE Transactions on Image Processing, 6(2):298–311.

    Google Scholar 

  • Chef’dhotel, C., Tschumperlé, D., Deriche, R., and Faugeras, O. 2004. Regularizing flows for constrained matrix-valued images. Journal of Mathematical Imaging and Vision, 20(2):147–162.

    Google Scholar 

  • Coulon, O., Alexander, D.C., and Arridge, S.R. 2001. A regularization scheme for diffusion tensor magnetic resonance images. 17th International Conference on Information Processing in Medical Imaging, LNCS, 2082:92–105.

  • Criminisi, A., Perez, P., and Toyama, K. 2003. Object Removal by Exemplar-based Inpainting. IEEE Conference on Computer Vision and Pattern Recognition, 2:721–728.

  • Deriche, R. and Faugeras, O. 1997. Les EDP en traitement des images et vision par ordinateur. Traitement du Signal, 13(6).

  • Di Zenzo, S. 1986. A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing, 33:116–125.

  • Jia, J. and Tang, C.K. 2003. Image Repairing: Robust Image Synthesis by Adaptive ND Tensor Voting. IEEE Conference on Computer Vision and Pattern Recognition, 1:643–650.

  • Kimmel, R., Malladi, R., and Sochen, N. 2000. Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. International Journal of Computer Vision, 39(2):111–129.

    Google Scholar 

  • Kimmel, R. and Sochen, N. 2002. Orientation diffusion or how to comb a porcupine. Journal of Visual Communication and Image Representation, 13:238–248.

    Google Scholar 

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

    Google Scholar 

  • Kornprobst, P., Deriche, R., and Aubert, G. 1997. Nonlinear operators in image restoration. IEEE Conference on Computer Vision and Pattern Recognition, 325–331.

  • Krissian, K. 2000. Multiscale Analysis: Application to Medical Imaging and 3D Vessel Detection. Ph.D. Thesis, INRIA-Sophia Antipolis/France.

  • Lindeberg, T. 1994. Scale-Space Theory in Computer Vision. Kluwer Academic Publishers.

  • Masnou, S. and Morel, J.-M. 1998. Level lines based disocclusion. IEEE International Conference on Image Processing, 3:259–263.

  • Nielsen, M., Florack, L., and Deriche, R. 1997. Regularization, scale-space and edge detection filters. Journal of Mathematical Imaging and Vision, 7(4):291–308.

    Google Scholar 

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

    Google Scholar 

  • Perona, P. 1998. Orientation diffusions. IEEE Transactions on Image Processing, 7(3):457–467.

    Google Scholar 

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

    Google Scholar 

  • Preusser, T. and Rumpf, M. 1999. Anisotropic nonlinear diffusion in flow visualization. IEEE Visualization Conference.

  • Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. 1992. “Runge-Kutta Method.” In Numerical Recipes in FORTRAN: The Art of Scientific Computing, Cambridge University Press, pp. 704–716.

  • Rudin, L., Osher, S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268.

    Google Scholar 

  • Sapiro, G. 2001. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press.

  • Sapiro, G. and Ringach, D.L. 1996. Anisotropic diffusion of multi-valued images with applications to color filtering. IEEE Transactions on Image Processing, 5(11):1582–1585.

    Google Scholar 

  • Sochen, N., Kimmel, R., and Bruckstein, A.M. 2001. Diffusions and confusions in signal and image processing. Journal of Mathematical Imaging and Vision, 14(3):195–209.

    Google Scholar 

  • Stalling, D. and Hege, H.C. 1995. Fast and Resolution Independent Line Integral Convolution. ACM SIGGRAPH, 22nd Annual Conference on Computer Graphics and Interactive Technique, pp. 249–256.

  • Tang, B., Sapiro, G., and Caselles, V. 1999. Direction diffusion. IEEE International Conference on Computer Vision, pp. 1245.

  • Tang, B., Sapiro, G., and Caselles, V. 2000. Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case. International Journal of Computer Vision, 36(2):149–161.

    Google Scholar 

  • Tomasi, C. and Manduchi, R. 1998. Bilateral filtering for gray and color images. IEEE International Conference on Computer Vision, 839–846.

  • Tschumperlé, D. 2002. PDE's based Regularization of Multi-valued images and applications. PhD Thesis, Université de Nice-Sophia Antipolis/France.

  • Tschumperlé, D. and Deriche, R. 2002. Orthonormal vector sets regularization with PDE's and applications. International Journal of Computer Vision, 50:237–252.

    Google Scholar 

  • Tschumperlé, D. and Deriche, R. 2001. Diffusion tensor regularization with constraints preservation. IEEE Conference on Computer Vision and Pattern Recognition, 1:948–953.

  • Tschumperlé, D. and Deriche, R. 2002. Diffusion PDE's on vector-valued images: Local approach and geometric viewpoint. IEEE Signal Processing Magazine, 19(5):16–25.

    Google Scholar 

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

  • Tschumperlé, D. The CImg Library: http://cimg.sourceforge.net. The C++ Template Image Processing Library.

  • Vemuri, B., Chen, Y., Rao, M., McGraw, T., Mareci, T., and Wang, Z. 2001. Fiber tract mapping from diffusion tensor MRI. IEEE Workshop on Variational and Level Set Methods in Computer Vision.

  • Weickert, J. 1994. Anisotropic diffusion filters for image processing based quality control. 7th European Conference on Mathematics in Industry, pp. 355–362.

  • Weickert, J. 1998. Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart.

  • Weickert, J. 1999. Coherence-enhancing diffusion of colour images. Image and Vision Computing, 17:199–210.

    Google Scholar 

  • Weickert, J. and Brox, T. 2002. Diffusion and regularization of vector and matrix-valued images. Inverse Problems, Image Analysis, and Medical Imaging, vol. 313 of Contemporary Mathematics, pp. 251–268.

  • Weickert, J., 2003. Coherence-enhancing shock filters. Pattern Recognition, 25th DAGM Symposium, LNCS, 2781:1–8.

  • Wei, L.Y. and Levoy, M., 2000. Fast texture synthesis using tree-structured vector quantization. ACM SIGGRAPH, International Conference on Computer Graphics and Interactive Techniques, pp. 479–488.

Download references

Author information

Authors and Affiliations

  1. Equipe Image/GREYC (UMR CNRS 6072), 6 Bd du Maréchal Juin, 14050, Caen Cedex, France

    David TschumperlÉ

Authors
  1. David TschumperlÉ
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

TschumperlÉ, D. Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE's. Int J Comput Vision 68, 65–82 (2006). https://doi.org/10.1007/s11263-006-5631-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-006-5631-z

Keywords

Search

Navigation