Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization

  • C. Chefd’hotel
  • D. Tschumperlé
  • R. Deriche
  • O. Faugeras
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2350)


Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows acting on constrained datasets. We focus our interest on flows of matrix-valued functions undergoing orthogonal and spectral constraints. The corresponding evolution PDE’s are found by minimization of cost functionals, and depend on the natural metrics of the underlying constrained manifolds (viewed as Lie groups or homogeneous spaces). Suitable numerical schemes that fit the constraints are also presented. We illustrate this theoretical framework through a recent and challenging problem in medical imaging: the regularization of diffusion tensor volumes (DTMRI).


Tangent Space Homogeneous Space Tensor Orientation Constraint Preserve Image Inpainting 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    L. Alvarez, R. Deriche, J. Weickert, and J. Sánchez. Dense disparity map estimation respecting image discontinuities: A PDE and scale-space based approach. International Journal of Visual Communication and Image Representation, 2000.Google Scholar
  2. 2.
    Y. Amit. A nonlinear variational problem for image matching. SIAM Journal on Scientific Computing, 15(1), January 1994.Google Scholar
  3. 3.
    M. Bertalmio, L.T. Cheng, S. Osher, and G. Sapiro. Variational problems and partial differential equations on implicit surfaces: The framework and examples in image processing and pattern formation. UCLA Research Report, June 2000.Google Scholar
  4. 4.
    M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester. Image inpainting. In Kurt Akeley, editor, Proceedings of the SIGGRAPH, pages 417–424. ACM Press, ACM SIGGRAPH, Addison Wesley Longman, 2000.Google Scholar
  5. 5.
    T. Chan and J. Shen. Variational restoration of non-flat image features: Models and algorithms. Research Report, Computational and applied mathematics, department of mathematics Los Angeles, June 1999.Google Scholar
  6. 6.
    T. Chan and J. Shen. Mathematical models for local deterministic inpaintings. Technical Report 00-11, Department of Mathematics, UCLA, Los Angeles, March 2000.Google Scholar
  7. 7.
    P. Charbonnier, G. Aubert, M. Blanc-Féraud, and M. Barlaud. Two deterministic half-quadratic regularization algorithms for computed imaging. In Proceedings of the International Conference on Image Processing, volume II, pages 168–172, 1994.CrossRefGoogle Scholar
  8. 8.
    C. Chefd’hotel, D. Tschumperlé, O. Faugeras and R. Deriche. Geometric Integration of Constraint Preserving Flows and Applications to Image Processing. INRIA Research Report, to appear, 2002.Google Scholar
  9. 9.
    O. Coulon, D.C. Alexander, and S.R. Arridge. A geometrical approach to 3d diffusion tensor magnetic resonance image regularisation. Technical Report, Department of Computer Science, University College London., 2001.Google Scholar
  10. 10.
    O. Coulon, D.C. Alexander, and S.R. Arridge. A regularization scheme for diffusion tensor magnetic resonance images. In XVIIth International Conferenceon Information Processing in Medical Imaging, 2001.Google Scholar
  11. 11.
    R. Deriche, P. Kornprobst, and G. Aubert. Optical flow estimation while preserving its discontinuities: A variational approach. In Proceedings of the 2nd Asian Conference on Computer Vision, volume 2, pages 71–80, Singapore, December 1995.Google Scholar
  12. 12.
    O. Faugeras and R. Keriven. Variational principles, surface evolution, PDE’s, level set methods and the stereo problem. IEEE Transactions on Image Processing, 7(3):336–344, March 1998.Google Scholar
  13. 13.
    G. Golub and C. Van Loan. Matrix computations. The John Hopkins University Press, Baltimore, Maryland, second edition, 1989.zbMATHGoogle Scholar
  14. 14.
    G. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer Academic Publishers, 1995.Google Scholar
  15. 15.
    U. Helmke and J. Moore. Optimization and Dynamical Systems. Springer, 1994.Google Scholar
  16. 16.
    V. Kac, editor. Infinite Dimensional Lie Groups with Applications. Mathematical Sciences Research Institute Publications 4. Springer, 1985.Google Scholar
  17. 17.
    R. Kimmel and N. Sochen. Orientation diffusion or how to comb a porcupine. Technical Report 2000-02, CIS, 2000. Accepted to special issue on PDEs in Image Processing, Computer Vision, and Computer Graphics, Journal of Visual Communication and Image Representation, 2000.Google Scholar
  18. 18.
    W. Klingenberg. Riemannian Geometry. de Gruyter Studies in Mathematics 1. Walter de Gruyter, 1982.Google Scholar
  19. 19.
    P. Kornprobst, R. Deriche, and G. Aubert. Nonlinear operators in image restoration. In Proceedings of the International Conference on Computer Vision and Pattern Recognition, pages 325–331, Puerto Rico, June 1997. IEEE Computer Society, IEEE.Google Scholar
  20. 20.
    S. Lang. Differential Manifolds. Springer, 1985.Google Scholar
  21. 21.
    D. Le Bihan. Methods and applications of diffusion MRI. In I.R. Young, editor, Magnetic Resonance Imaging and Spectroscopy in Medicine and Biology. John Wiley and Sons, 2000.Google Scholar
  22. 22.
    D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42:577–684, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    P. Perona. Orientation diffusions. IEEE Transactions on Image Processing, 7(3):457–467, March 1998.Google Scholar
  24. 24.
    P. Perona and J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7):629–639, July 1990.Google Scholar
  25. 25.
    L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.zbMATHCrossRefGoogle Scholar
  26. 26.
    G. Sapiro. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, 2001.Google Scholar
  27. 27.
    S. Smith. Optimization Techniques on Riemannian Manifolds. Hamiltonian and Gradient Flows, Algorithms and Control. American Mathematical Society, 1994, Fields Institute for Research in Mathematical Sciences, A. Bloch editor, 1994.Google Scholar
  28. 28.
    N. Sochen, R. Kimmel, and R. Malladi. A geometrical framework for low level vision. IEEE Transaction on Image Processing, Special Issue on PDE based Image Processing, 7(3):310–318, 1998.zbMATHMathSciNetGoogle Scholar
  29. 29.
    B. Tang, G. Sapiro, and V. Caselles. Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case. The International Journal of Computer Vision, 36(2):149–161, February 2000.Google Scholar
  30. 30.
    A. Trouvé. Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision, 28(3):213–21, 1998.CrossRefGoogle Scholar
  31. 31.
    D. Tschumperlé and R. Deriche. Constrained and unconstrained PDE’s for vector image restoration. In Ivar Austvoll, editor, Proceedings of the 10th Scandinavian Conference on Image Analysis, pages 153–160, Bergen, Norway, June 2001.Google Scholar
  32. 32.
    D. Tschumperlé and R. Deriche. Diffusion tensor regularization with constraints preservation. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Kauai Marriott, Hawaii, December 2001.Google Scholar
  33. 33.
    D. Tschumperlé and R. Deriche. Regularization of orthonormal vector sets using coupled PDE’s. In Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods (VLSM’01), July 2001.Google Scholar
  34. 34.
    B. Vemuri, Y. Chen, M. Rao, T. McGraw, T. Mareci, and Z. Wang. Fiber tract mapping from diffusion tensor MRI. In Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision (VLSM’01), July 2001.Google Scholar
  35. 35.
    J. Weickert. Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart, 1998.zbMATHGoogle Scholar
  36. 36.
    C.-F. Westin, S. Maier, B. Khidhir, P. Everett, F. Jolesz, and R. Kikinis. Image processing for diffusion tensor magnetic resonance imaging. In Proceedings of the Second International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI’99), Springer-Verlag, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • C. Chefd’hotel
    • 1
  • D. Tschumperlé
    • 1
  • R. Deriche
    • 1
  • O. Faugeras
    • 1
  1. 1.INRIASophia-AntipolisFrance

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