Skip to main content
SpringerLink
Log in

DTI Segmentation and Fiber Tracking Using Metrics on Multivariate Normal Distributions

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

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.

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

Fig. 1
Fig. 2
Fig. 3
Algorithm 1
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Algorithm 2
Algorithm 3
Algorithm 4
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Notes

  1. In our experiments, if \(\operatorname{dist}(B', C)\) is less than 0.05, log C (B′) is well approximated by B′−C. We use this constant value in Algorithm 1.

References

  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. 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. Arsigny, V., et al.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56, 411–421 (2006)

    Article  Google Scholar 

  4. Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. J. Biophys. 66, 259–267 (1994)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Coulon, O., Alexander, D.C., Arridge, S.: Diffusion tensor magnetic resonance image regularization. Med. Image Anal. 8(1), 47–68 (2004)

    Article  Google Scholar 

  7. Descoteaux, M., Lenglet, C., Deriche, R.: Diffusion tensor sharpening improves white matter tractography. Proc. SPIE 6512, 65121J (2007)

    Article  Google Scholar 

  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. Fletcher, P.T., et al.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Imai, T., Takaesu, A., Wakayama, M.: Remarks on geodesics for multivariate normal models. Surv. Math. Ind. B(6), 125–130 (2011)

    MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Otsu, N.: A threshold selection method from gray-level histogram. IEEE Trans. Syst. Man Cybern. SMC-9(1), 62–66 (1979)

    MathSciNet  Google Scholar 

  20. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–46 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Stat. 11, 211–233 (1984)

    MATH  MathSciNet  Google Scholar 

  23. Smith, S.T.: Covariance, subspace, and intrinsic Cramér–Rao bounds. IEEE Trans. Signal Process. 53(5), 1610–1630 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  25. Wang, Z., Vemuri, B.: DTI segmentation using an information theoretic tensor dissimilarity measure. IEEE Trans. Med. Imaging 24(10), 1267–1277 (2005)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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 

Download references

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.

Author information

Authors and Affiliations

  1. Robotics Laboratory, Seoul National University, Seoul, 151-744, Korea

    Minyeon Han & F. C. Park

Authors
  1. Minyeon Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. C. Park
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minyeon Han.

Appendix: Triangle Inequality Counterexample

Appendix: Triangle Inequality Counterexample

In [1] it is claimed that the distance metrics (2) and (3) satisfy all the metric axioms including the triangle inequality. The following simple counterexample shows that this is not the case. Consider the following three normal distributions a, b and c on \(\mathcal{N}(1)\):

$$\begin{aligned} \begin{aligned} &\mu_a = 0,\qquad \sigma_a = 1, \\ &\mu_b = 1,\qquad \sigma_b = 1, \\ &\mu_c = 2,\qquad \sigma_c = 0.01. \end{aligned} \end{aligned}$$
(34)

The distances calculated using the proposed metrics are given in Table 2.

Table 2 Distances between a, b and c
Full size table

Both metrics violate the triangle inequality, i.e., \(\operatorname{dist}(a,b) + \operatorname{dist}(b,c) \ngeq \operatorname{dist}(c,a)\).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Han, M., Park, F.C. DTI Segmentation and Fiber Tracking Using Metrics on Multivariate Normal Distributions. J Math Imaging Vis 49, 317–334 (2014). https://doi.org/10.1007/s10851-013-0466-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-013-0466-z

Keywords

Search

Navigation