Skip to main content
Springer Nature Link
Log in

A Riemannian Approach to Diffusion Tensor Images Segmentation

  • Conference paper
Information Processing in Medical Imaging (IPMI 2005)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 3565))

  • 2549 Accesses

Abstract

We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images. Our approach is grounded on the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions. We introduce a variational formulation, in the level set framework, to estimate the optimal segmentation according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. Moreover, we must respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the diffusion tensor image. We validate our algorithm on synthetic data and report interesting results on real datasets. We focus on two structures of the white matter with different properties and respectively known as the corpus callosum and the corticospinal tract.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Le Bihan, D., Breton, E., Lallemand, D., Grenier, P., Cabanis, E., Laval-Jeantet, M.: MR imaging of intravoxel incoherent motions: Application to diffusion and perfusion in neurologic disorders. Radiology, 401–407 (1986)

    Google Scholar 

  2. Basser, P., Mattiello, J., Le Bihan, D.: MR diffusion tensor spectroscopy and imaging. Biophysica, 259–267 (1994)

    Google Scholar 

  3. Tschumperlé, D., Deriche, R.: Variational frameworks for DT-MRI estimation, regularization and visualization. In: Proc. ICCV, pp. 116–122 (2003)

    Google Scholar 

  4. Mori, S., Crain, B., Chacko, V., Zijl, P.V.: Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Annals of Neurology 45, 265–269 (1999)

    Article  Google Scholar 

  5. Behrens, T., Johansen-Berg, H., Woolrich, M., Smith, S., Wheeler-Kingshott, C., Boulby, P., Barker, G., Sillery, E., Sheehan, K., Ciccarelli, O., Thompson, A., Brady, J., Matthews, P.: Non-invasive mapping of connections between human thalamus and cortex using diffusion imaging. Nat. Neuroscience 6, 750–757 (2003)

    Article  Google Scholar 

  6. Lenglet, C., Deriche, R., Faugeras, O.: Inferring white matter geometry from diffusion tensor MRI: Application to connectivity mapping. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 127–140. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  7. Corouge, I., Gouttard, S., Gerig, G.: A statistical shape model of individual fiber tracts extracted from diffusion tensor MRI. In: Barillot, C., Haynor, D.R., Hellier, P. (eds.) MICCAI 2004. LNCS, vol. 3217, pp. 671–679. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  8. Zhukov, L., Museth, K., Breen, D., Whitaker, R., Barr, A.: Level set segmentation and modeling of DT-MRI human brain data. Journal of Electronic Imaging 12(1), 125–133 (2003)

    Article  Google Scholar 

  9. Wiegell, M., Tuch, D., Larson, H., Wedeen, V.: Automatic segmentation of thalamic nuclei from diffusion tensor magnetic resonance imaging. NeuroImage 19, 391–402 (2003)

    Article  Google Scholar 

  10. Feddern, C., Weickert, J., Burgeth, B., Welk, M.: Curvature-driven PDE methods for matrix-valued images. Technical Report 104, Department of Mathematics, Saarland University, Saarbrücken, Germany (2004)

    Google Scholar 

  11. Rousson, M., Lenglet, C., Deriche, R.: Level set and region based surface propagation for diffusion tensor MRI segmentation. In: Proc. Computer Vision Approaches to Medical Image Analysis, ECCV Workshop, pp. 123–134 (2004)

    Google Scholar 

  12. Wang, Z., Vemuri, B.C.: Tensor field segmentation using region based active contour model. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 304–315. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  13. Wang, Z., Vemuri, B.: An affine invariant tensor dissimilarity measure and its application to tensor-valued image segmentation. In: Proc. CVPR, pp. 228–233 (2004)

    Google Scholar 

  14. Lenglet, C., Rousson, M., Deriche, R.: Segmentation of 3D probability density fields by surface evolution: Application to diffusion MRI. In: Barillot, C., Haynor, D.R., Hellier, P. (eds.) MICCAI 2004. LNCS, vol. 3216, pp. 18–25. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  15. Jonasson, L., Bresson, X., Hagmann, P., Cuisenaire, O., Meuli, R., Thiran, J.: White matter fiber tract segmentation in DT-MRI using geometric flows. Medical Image Analysis (2004) (in press)

    Google Scholar 

  16. Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on multivariate normal distributions: A geometric approach and its application to diffusion tensor MRI. Research Report 5242, INRIA (2004)

    Google Scholar 

  17. Rao, C.: Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)

    MATH  MathSciNet  Google Scholar 

  18. Burbea, J., Rao, C.: Entropy differential metric, distance and divergence measures in probability spaces: A unified approach. Journal of Multivariate Analysis 12, 575–596 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  19. Skovgaard, L.: A Riemannian geometry of the multivariate normal model. Technical Report 81/3, Statistical Research Unit, Danish Medical Research Council, Danish Social Science Research Council (1981)

    Google Scholar 

  20. Burbea, J.: Informative geometry of probability spaces. Expositiones Mathematica 4, 347–378 (1986)

    MATH  MathSciNet  Google Scholar 

  21. Eriksen, P.: Geodesics connected with the fisher metric on the multivariate manifold. Technical Report 86-13, Inst. of Elec. Systems, Aalborg University (1986)

    Google Scholar 

  22. Calvo, M., Oller, J.: An explicit solution of information geodesic equations for the multivariate normal model. Statistics and Decisions 9, 119–138 (1991)

    MATH  MathSciNet  Google Scholar 

  23. Förstner, W., Moonen, B.: A metric for covariance matrices. Technical report, Stuttgart University, Dept. of Geodesy and Geoinformatics (1999)

    Google Scholar 

  24. Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26(3), 735–747 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Fletcher, P., Joshi, S.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Proc. Computer Vision Approaches to Medical Image Analysis, ECCV Workshop, pp. 87–98 (2004)

    Google Scholar 

  26. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Research Report 5255, INRIA (2004)

    Google Scholar 

  27. Atkinson, C., Mitchell, A.: Rao’s distance measure. Sankhya: The Indian Journal of Stats 43, 345–365 (1981)

    MATH  MathSciNet  Google Scholar 

  28. Karcher, H.: Riemannian centre of mass and mollifier smoothing. Comm. Pure Appl. Math 30, 509–541 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pennec, X.: Probabilities and statistics on Riemannian manifolds: A geometric approach. Research Report 5093, INRIA (2004)

    Google Scholar 

  30. Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and shape statistics. Research Report 4820, INRIA (2003)

    Google Scholar 

  31. Rousson, M.: Cues integrations and front evolutions in image segmentation. PhD thesis, Université de Nice-Sophia Antipolis (2004)

    Google Scholar 

  32. Leclerc, Y.: Constructing simple stable description for image partitioning. International Journal of Computer Vision 3, 73–102 (1989)

    Article  Google Scholar 

  33. Zhu, S., Yuille, A.: Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 18, 884–900 (1996)

    Article  Google Scholar 

  34. Paragios, N., Deriche, R.: Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. Journal of Visual Communication and Image Representation 13, 249–268 (2002)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Chan, T., Vese, L.: Active contours without edges. IEEE Transactions on Image Processing 10, 266–277 (2001)

    Article  MATH  Google Scholar 

  37. Zhao, H., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. Journal of Computational Physics 127, 179–195 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  38. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. The International Journal of Computer Vision 22, 61–79 (1997)

    Article  MATH  Google Scholar 

  39. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. In: Proc. ICCV, pp. 259–268 (1987)

    Google Scholar 

  40. Rousson, M., Deriche, R.: A variational framework for active and adaptative segmentation of vector valued images. In: Proc. IEEE Workshop on Motion and Video Computing, pp. 56–62 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I.N.R.I.A, 2004 route des lucioles, 06902, Sophia-Antipolis, France

    Christophe Lenglet, Rachid Deriche & Olivier Faugeras

  2. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540, USA

    Mikaël Rousson

  3. CMRR, University of Minnesota, 2021 6th Street SE, Minneapolis, MN, 55455, USA

    Stéphane Lehericy & Kamil Ugurbil

Authors
  1. Christophe Lenglet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikaël Rousson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rachid Deriche
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Olivier Faugeras
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Stéphane Lehericy
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Kamil Ugurbil
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, The University of Iowa, IA 52242, Iowa City

    Gary E. Christensen

  2. Department of Electrical and Computer Engineering, University of Iowa, IA 52242, Iowa City, USA

    Milan Sonka

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lenglet, C., Rousson, M., Deriche, R., Faugeras, O., Lehericy, S., Ugurbil, K. (2005). A Riemannian Approach to Diffusion Tensor Images Segmentation. In: Christensen, G.E., Sonka, M. (eds) Information Processing in Medical Imaging. IPMI 2005. Lecture Notes in Computer Science, vol 3565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11505730_49

Download citation

  • DOI: https://doi.org/10.1007/11505730_49

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26545-0

  • Online ISBN: 978-3-540-31676-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation