Advertisement

Journal of Mathematical Imaging and Vision

, Volume 49, Issue 2, pp 317–334 | Cite as

DTI Segmentation and Fiber Tracking Using Metrics on Multivariate Normal Distributions

  • Minyeon Han
  • F. C. Park
Article

Abstract

Existing clustering-based methods for segmentation and fiber tracking of diffusion tensor magnetic resonance images (DT-MRI) are based on a formulation of a similarity measure between diffusion tensors, or measures that combine translational and diffusion tensor distances in some ad hoc way. In this paper we propose to use the Fisher information-based geodesic distance on the space of multivariate normal distributions as an intrinsic distance metric. An efficient and numerically robust shooting method is developed for computing the minimum geodesic distance between two normal distributions, together with an efficient graph-clustering algorithm for segmentation. Extensive experimental results involving both synthetic data and real DT-MRI images demonstrate that in many cases our method leads to more accurate and intuitively plausible segmentation results vis-à-vis existing methods.

Keywords

Magnetic resonance imaging Diffusion tensor Image segmentation Fiber Tracking Multivariate normal distribution Riemannian geometry 

Notes

Acknowledgements

This research was supported in part by the Center for Advanced Intelligent Manipulation, the Biomimetic Robotics Research Center, the BK21+ program at SNU-MAE, and SNU-IAMD.

References

  1. 1.
    Abou-Moustafa, K.T., Ferrie, F.P.: A note on metric properties for some divergence measures: the Gaussian case. J. Mach. Learn. Res. 25, 1–15 (2012) Google Scholar
  2. 2.
    Alexander, D., Gee, J., Bajcsy, R.: Similarity measures for matching diffusion tensor images. In: British Machine Vision Conference, vol. 99, pp. 93–102 (1999) Google Scholar
  3. 3.
    Arsigny, V., et al.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56, 411–421 (2006) CrossRefGoogle Scholar
  4. 4.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. J. Biophys. 66, 259–267 (1994) CrossRefGoogle Scholar
  5. 5.
    Calvo, M., Oller, J.: A distance between multivariate normal distributions based in an embedding into a Siegel group. J. Multivar. Anal. 35, 223–242 (1990) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Coulon, O., Alexander, D.C., Arridge, S.: Diffusion tensor magnetic resonance image regularization. Med. Image Anal. 8(1), 47–68 (2004) CrossRefGoogle Scholar
  7. 7.
    Descoteaux, M., Lenglet, C., Deriche, R.: Diffusion tensor sharpening improves white matter tractography. Proc. SPIE 6512, 65121J (2007) CrossRefGoogle Scholar
  8. 8.
    Eriksen, P.S.: Geodesics connected with the Fisher metric on the multivariate normal manifold. In: Proc. GST Workshop, Lancaster, pp. 225–229 (1987) Google Scholar
  9. 9.
    Fletcher, P.T., et al.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Imai, T., Takaesu, A., Wakayama, M.: Remarks on geodesics for multivariate normal models. Surv. Math. Ind. B(6), 125–130 (2011) MathSciNetGoogle Scholar
  11. 11.
    Jbabdi, S., Bellec, P., Toro, R., Daunizeau, J., Pélégrini-Issac, M., Benali, H.: Accurate anisotropic fast marching for diffusion-based geodesic tractography. Int. J. Biomed. Imaging 2008, 320195 (2008) CrossRefGoogle Scholar
  12. 12.
    Kaufman, L., Rousseeuw, P.J.: Clustering by means of medoids. In: Dodge, Y. (ed.) Statistical Data Analysis Based on the L 1 Norm and Related Methods, pp. 405–416. North-Holland, Amsterdam (1987) Google Scholar
  13. 13.
    Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing. J. Math. Imaging Vis. 25, 423–444 (2006) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Lovric, M., Min-Oo, M., Ruh, E.A.: Multivariate normal distributions parametrized as a Riemannian symmetric space. J. Multivar. Anal. 74(1), 36–48 (2000) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Maitra, R., Peterson, A.D., Ghosh, A.P.: A systematic evaluation of different methods for initializing the k-means clustering algorithm. In: IEEE Trans. Knowledge and Data Engineering (2010) Google Scholar
  16. 16.
    Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26, 735–747 (2005) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Moakher, M., Batchelor, P.G.: Symmetric positive-definite matrices: from geometry to applications and visualization. In: Visualization and Processing of Tensor Fields, pp. 285–298 (2006) CrossRefGoogle Scholar
  18. 18.
    O’Donnell, L., Haker, S., Westin, C.-F.: New approaches to estimation of white matter connectivity in diffusion tensor MRI: elliptic PDEs and geodesics in a tensor-warped space. In: Proc. Med. Image Comput. Comp. Assisted Intervention, vol. 2488, pp. 459–466 (2002) Google Scholar
  19. 19.
    Otsu, N.: A threshold selection method from gray-level histogram. IEEE Trans. Syst. Man Cybern. SMC-9(1), 62–66 (1979) MathSciNetGoogle Scholar
  20. 20.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–46 (2006) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sepasian, N., ten Thije Boonkkamp, J.H.M., Ter Haar Romeny, B.M., Vilanova, A.: Multivalued geodesic ray-tracing for computing brain connections using diffusion tensor imaging. SIAM J. Imaging Sci. 5, 483–504 (2012) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Stat. 11, 211–233 (1984) MATHMathSciNetGoogle Scholar
  23. 23.
    Smith, S.T.: Covariance, subspace, and intrinsic Cramér–Rao bounds. IEEE Trans. Signal Process. 53(5), 1610–1630 (2005) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Tsuda, K., Akaho, S., Asai, K.: The em algorithm for kernel matrix completion with auxiliary data. J. Mach. Learn. Res. 4, 67–81 (2003) MathSciNetGoogle Scholar
  25. 25.
    Wang, Z., Vemuri, B.: DTI segmentation using an information theoretic tensor dissimilarity measure. IEEE Trans. Med. Imaging 24(10), 1267–1277 (2005) CrossRefGoogle Scholar
  26. 26.
    Wiegell, M., Tuch, D., Larson, H., Wedeen, V.: Automatic segmentation of thalamic nuclei from diffusion tensor magnetic resonance imaging. NeuroImage 19, 391–402 (2003) CrossRefGoogle Scholar
  27. 27.
    Ziyan, U., Tuch, D., Westin, C.F.: Segmentation of thalamic nuclei from DTI using spectral clustering. In: Proc. Med. Image Comput. Comp. Assisted Intervention, pp. 807–814 (2006) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Robotics LaboratorySeoul National UniversitySeoulKorea

Personalised recommendations