Skip to main content
SpringerLink
Log in

Regularizing Flows for Constrained Matrix-Valued Images

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

Abstract

Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differential-geometric framework to define PDEs acting on some manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space (viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structure-preserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging.

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

Similar content being viewed by others

References

  1. R. Abraham, J.E. Marsden, and T.S. Ratiu, Manifolds, Tensors, Analysis, and Applications, Springer, 1988, Draft. 3rd edition 2001 from http://www.cds.caltech.edu/ marsden/ bib src/mta/Book/.

  2. S.I. Amari, "Natural gradient works efficiently in learning," Neural Computation,Vol. 10, pp. 251–276, 1998.

    Google Scholar 

  3. G. Aubert and P. Kornprobst, "Mathematical problems in image Processing," Applied Mathematical Sciences,Vol. 147, Springer, 2001.

  4. F. Barbaresco, "Spatial denoising of statistical parameters es-timation by Beltrami diffusion on embedding Siegel space," PSIP'2003-Physics in Signal & Image Processing Grenoble, France, 2003.

  5. M. Bertalmio, G. Sapiro, L.-T. Cheng, and S. Osher, "Variational problems and PDE's on implicit surfaces," in Paragios [28].

  6. W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry,2nd edition. Academic Press, 1986.

  7. C.A. Botsaris, "Constrained optimization along geodesics," Journal of Mathematical Analysis and Applications,Vol. 79, pp. 295–306, 1981.

    Google Scholar 

  8. T. Chan and J. Shen, "Variational restoration of non-flat image features: models and algorithms," SIAM J. Appl. Math,Vol. 61, No. 4, pp. 1338–1361, 2000.

    Google Scholar 

  9. C. Chefd'hotel, D. Tschumperlé, R. Deriche, and O. Faugeras, Constrained Flows of Matrix Valued Functions: Application to Diffusion Tensor Regularization, Springer, May 2002, pp. 251–265.

  10. A. Chorin, T. Hugues, M. McCraken, and J. Marsden, "Product formulas and numerical algorithms," Communications on Pure and Applied Mathematics,Vol. 31, pp. 205–256, 1978.

    Google Scholar 

  11. O. Coulon, D.C. Alexander, and S.R. Arridge, "A regularization scheme for diffusion tensor magnetic resonance Images," in In-ternational Conference on Information Processing in Medical Imaging,Davis, USA, 2001, pp. 92–105.

  12. P.E. Crouch and R. Grossman, "Numerical integration of or-dinary differential equations on Manifolds," J. Nonlinear Sci., Vol. 3, pp. 1–33, 1993.

    Google Scholar 

  13. Y.-J. Dai, M. Shoji, and H. Urakawa, "Harmonic maps into Lie groups and homogeneous spaces," Differential Geometry and its Applications,Vol. 7, pp. 143–160, 1997.

    Google Scholar 

  14. A. Edelman, T.A. Arias, and S.T. Smith, "The geometry of algorithms with orthogonality constraints," SIAM J. Matrix Anal. Appl.,Vol. 20, No. 2, pp. 303–353, 1998.

    Google Scholar 

  15. J. Gallier and D. Xu, "Computing exponentials of skew sym-metric matrices and logarithms of orthogonal matrices," International Journal of Robotics and Automation,Vol. 18, No. 1, pp. 10–20, 2003.

    Google Scholar 

  16. G. Golub and C. Van Loan, Matrix Computations. 2nd edition, North Oxford Academic, 1986.

  17. E. Hairer, C. Lubich, and G. Wanner, "Geometric numerical integration: Structure preserving algorithms for ordinary differential equations," Springer Series in Computational Mathematics, Vol. 31, Springer, 2002.

  18. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978.

  19. U. Helmke and J.B. Moore, "Optimization and dynamical systems," Communications and Control Engineering Series, Springer, 1994.

  20. R. Horn and C. Johnson, "Matrix Analysis," Cambridge University Press, 1985.

  21. A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, and A. Zanna, "Lie-group methods," Acta Numerica, pp. 215–365, 2000.

  22. V. Kac (Ed.), Infinite Dimensional Lie Groups with Applications. Mathematical Sciences Research Institute Publications, Springer, 1985, Vol. 4.

    Google Scholar 

  23. R. Kimmel and N. Sochen, "Orientation diffusion or how to comb a porcupine?, Special issue on PDEs in Image Processing, Computer Vision, and Computer Graphics, Journal of Visual Communication and Image Representation,Vol. 13, pp. 238–248, 2002.

    Google Scholar 

  24. D. Lewis and P.J. Olver, Geometric Integration Algorithms on Homogeneous Manifolds, Preprint, 2001.

  25. F. Mémoli, G. Sapiro, and S. Osher, "Solving variational problems and partial differential equations mapping into general target Manifolds," Computational and Applied Mathematics Report, UCLA, 2002.

  26. R. Mneimné and F. Testard, Introduction a la theorie des groupes de lie classiques. Hermann, 1986.

  27. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces Springer, 2002.

  28. N. Paragios (Ed.), in Proceedings of the IEEE Workshop on Variational and Level Set Methods in Computer Vision,Vancouver, Canada, IEEE Computer Society, July 2001.

    Google Scholar 

  29. P. Perona, "Orientation diffusion," IEEE Transactions on Image Processing,Vol. 7, No. 3, 457–467, 1998.

    Google Scholar 

  30. C. Poupon, "Detection des faisceaux de fibres de la substance blanche pour l etude de la connectivite anatomique cerebrale," Ph.D. thesis, ENST, 1999.

  31. G. Sapiro, "Geometric partial differential equations and image Analysis," Cambridge University Press, 2001.

  32. S.T. Smith, "Geometric optimization for adaptative filtering," Ph.D. thesis, Harvard University, 1993.

  33. S.T. Smith, "Optimization techniques on Riemannian manifolds," Fields Institute Communications, American Mathematical Society, Providence, RI, 1994, Vol. 3, pp. 113 146.

    Google Scholar 

  34. N. Sochen, R. Kimmel, and R. Malladi, "A general framework for low level Vision," IEEE Transactions on Image Processing, Vol. 17, No. 3, pp. 310–318, 1998.

    Google Scholar 

  35. B. Tang, G. Sapiro, and V. Caselles, "Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case," International Journal of Computer Vision,Vol. 36, No. 2, pp. 149–161, 2000.

    Google Scholar 

  36. B. ter Haar Romeny (Ed.), "Geometry driven diffusion in computer Vision," Computational Imaging and Vision, Kluwer Aca-demic Publishers, 1994.

  37. A. Trouvé, "Diffeomorphisms groups and pattern matching in image Analysis," International Journal of Computer Vision, Vol. 28, No. 2, pp. 213–221, 1998.

    Google Scholar 

  38. D. Tschumperlé and R. Deriche, "Diffusion tensor regularization with constraints preservation," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR'2001), IEEE Computer Society, 2001.

  39. D. Tschumperlé and R. Deriche, "Orthonormal vector sets regularization with PDE's and Applications," International Journal of Computer Vision,Vol. 50, No. 3, pp. 237–252, 2002.

    Google Scholar 

  40. K. Uhlenbeck, "Harmonic maps into Lie groups (classical solutions of the chiral model)," J. Differential Geometry,Vol. 30, pp. 1–50, 1989.

    Google Scholar 

  41. H. Urakawa, "Calculus of variations and harmonic maps," Translations of Mathematical Monographs, No. 132, American Mathematical Society, 1993.

  42. B. Vemuri, Y. Chen, M. Rao, T. McGraw, T. Mareci, and Z. Wang, "Fiber tract mapping from diffusion tensor MRI," in Paragios [28].

  43. L.A. Vese and S.J. Osher, "Numerical methods for p-harmonic flows and applications to image Processing," Computational and Applied Mathematics Report, UCLA, 2001.

  44. J. Weickert, Anisotropic Diffusion in Image ProcessingTeubner Verlag, 1998.

  45. J. Weickert and T. Brox, Diffusion and Regularization of Vector and Matrix-Valued Images, Preprint 58, Univeritat des Saarlandes, 2002.

  46. C. F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.A. Jolesz, and R. Kikinis, "Processing and visualization for diffusion tensor MRI, Medical Image Analysis," Vol. 6, No. 2, pp. 93–108, 2002.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Odyssée Lab., INRIA Sophia-Antipolis, 2004 route des Lucioles, 06902, Sophia-Antipolis, France

    C. Chefd'hotel

  2. Odyssée Lab., INRIA Sophia-Antipolis, 2004 route des Lucioles, 06902, Sophia-Antipolis, France

    D. Tschumperlé

  3. Odyssée Lab., INRIA Sophia-Antipolis, 2004 route des Lucioles, 06902, Sophia-Antipolis, France

    R. Deriche

  4. Odyssée Lab., INRIA Sophia-Antipolis, 2004 route des Lucioles, 06902, Sophia-Antipolis, France

    O. Faugeras

Authors
  1. C. Chefd'hotel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Tschumperlé
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Deriche
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. O. Faugeras
    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

Chefd'hotel, C., Tschumperlé, D., Deriche, R. et al. Regularizing Flows for Constrained Matrix-Valued Images. Journal of Mathematical Imaging and Vision 20, 147–162 (2004). https://doi.org/10.1023/B:JMIV.0000011920.58935.9c

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JMIV.0000011920.58935.9c

Search

Navigation