The Annals of Applied Statistics

Fiber direction estimation, smoothing and tracking in diffusion MRI

Raymond K. W. Wong, Thomas C. M. Lee, Debashis Paul, Jie Peng, and Alzheimer’s Disease Neuroimaging Initiative

Full-text: Open access


Diffusion magnetic resonance imaging is an imaging technology designed to probe anatomical architectures of biological samples in an in vivo and noninvasive manner through measuring water diffusion. The contribution of this paper is threefold. First, it proposes a new method to identify and estimate multiple diffusion directions within a voxel through a new and identifiable parametrization of the widely used multi-tensor model. Unlike many existing methods, this method focuses on the estimation of diffusion directions rather than the diffusion tensors. Second, this paper proposes a novel direction smoothing method which greatly improves direction estimation in regions with crossing fibers. This smoothing method is shown to have excellent theoretical and empirical properties. Last, this paper develops a fiber tracking algorithm that can handle multiple directions within a voxel. The overall methodology is illustrated with simulated data and a data set collected for the study of Alzheimer’s disease by the Alzheimer’s Disease Neuroimaging Initiative (ADNI).

Article information

Ann. Appl. Stat., Volume 10, Number 3 (2016), 1137-1156.

Received: January 2015
Revised: September 2015
First available in Project Euclid: 28 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Diffusion tensor imaging direction smoothing multi-tensor model fiber tracking tractography


Wong, Raymond K. W.; Lee, Thomas C. M.; Paul, Debashis; Peng, Jie; Disease Neuroimaging Initiative, Alzheimer’s. Fiber direction estimation, smoothing and tracking in diffusion MRI. Ann. Appl. Stat. 10 (2016), no. 3, 1137--1156. doi:10.1214/15-AOAS880.

Export citation


  • Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2006). Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56 411–421.
  • Bammer, R., Holdsworth, S. J., Veldhuis, W. B. and Skare, S. T. (2009). New methods in diffusion-weighted and diffusion tensor imaging. Magn. Reson. Imaging Clin. N. Am. 17 175–204.
  • Basser, P. J., Pajevic, S., Pierpaoli, C., Duda, J. and Aldroubi, A. (2000). In vivo fiber tractography using DT-MRI data. Magn. Reson. Med. 44 625–632.
  • Beaulieu, C. (2002). The basis of anisotropic water diffusion in the nervous system—A technical review. NMR Biomed. 15 435–455.
  • Behrens, T. E. J., Woolrich, M. W., Jenkinson, M., Johansen-Berg, H., Nunes, R. G., Clare, S., Matthews, P. M., Brady, J. M. and Smith, S. M. (2003). Characterization and propagation of uncertainty in diffusion-weighted MR imaging. Magn. Reson. Med. 50 1077–1088.
  • Behrens, T. E. J., Berg, H. J., Jbabdi, S., Rushworth, M. F. S. and Woolrich, M. W. (2007). Probabilistic diffusion tractography with multiple fibre orientations: What can we gain? Neuroimage 34 144–155.
  • Carmichael, O., Chen, J., Paul, D. and Peng, J. (2013). Diffusion tensor smoothing through weighted Karcher means. Electron. J. Stat. 7 1913–1956.
  • Chanraud, S., Zahr, N., Sullivan, E. V. and Pfefferbaum, A. (2010). MR diffusion tensor imaging: A window into white matter integrity of the working brain. Neuropsychol. Rev. 20 209–225.
  • Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R. (2007). Regularized, fast, and robust analytical Q-ball imaging. Magn. Reson. Med. 58 497–510.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. Chapman & Hall, London.
  • Fillard, P., Pennec, X., Arsigny, V. and Ayache, N. (2007). Clinical DT-MRI estimation, smoothing, and fiber tracking with log-Euclidean metrics. Medical Imaging, IEEE Transactions on 26 1472–1482.
  • Fletcher, P. T. and Joshi, S. (2007). Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87 250–262.
  • Friman, O., Farneback, G. and Westin, C.-F. (2006). A Bayesian approach for stochastic white matter tractography. Medical Imaging, IEEE Transactions on 25 965–978.
  • Gudbjartsson, H. and Patz, S. (1995). The Rician distribution of noisy MRI data. Magn. Reson. Med. 34 910–914.
  • Hosey, T., Williams, G. and Ansorge, R. (2005). Inference of multiple fiber orientations in high angular resolution diffusion imaging. Magn. Reson. Med. 54 1480–1489.
  • Kaufman, L. and Rousseeuw, P. J. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.
  • Koch, M. A., Norris, D. G. and Hund-Georgiadis, M. (2002). An investigation of functional and anatomical connectivity using magnetic resonance imaging. Neuroimage 16 241–250.
  • Mori, S. (2007). Introduction to Diffusion Tensor Imaging. Elsevier, Amsterdam.
  • Mori, S. and van Zijl, P. C. M. (2002). Fiber tracking: Principles and strategies—A technical review. NMR Biomed. 15 468–480.
  • Mori, S., Crain, B. J., Chacko, V. P. and Van Zijl, P. (1999). Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Annals of Neurology 45 265–269.
  • Mukherjee, P., Berman, J. I., Chung, S. W., Hess, C. P. and Henry, R. G. (2008). Diffusion tensor MR imaging and fiber tractography: Theoretic underpinnings. AJNR Am. J. Neuroradiol. 29 632–641.
  • Nimsky, C., Ganslandt, O. and Fahlbusch, R. (2006). Implementation of fiber tract navigation. Neurosurgery 58 ONS–292–303; discussion ONS–303–4.
  • Parker, G. J. M. and Alexander, D. C. (2003). Probabilistic Monte Carlo based mapping of cerebral connections utilising whole-brain crossing fibre information. In Information Processing in Medical Imaging 684–695. Springer, Berlin.
  • Pennec, X., Fillard, P. and Ayache, N. (2006). A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66 41–66.
  • Rousseeuw, P. J. (1987). Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 20 53–65.
  • Scherrer, B. and Warfield, S. K. (2010). Why multiple b-values are required for multi-tensor models. Evaluation with a constrained log-Euclidean model. In 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro 1389–1392. IEEE, Rotterdam.
  • Schwartzman, A., Dougherty, R. F. and Taylor, J. E. (2008). False discovery rate analysis of brain diffusion direction maps. Ann. Appl. Stat. 2 153–175.
  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
  • Sporns, O. (2011). Networks of the Brain. MIT Press, Cambridge, MA.
  • Tabelow, K., Voss, H. U. and Polzehl, J. (2012). Modeling the orientation distribution function by mixtures of angular central Gaussian distributions. J. Neurosci. Methods 203 200–211.
  • Tournier, J., Calamante, F., Gadian, D. G., Connelly, A. et al. (2004). Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23 1176–1185.
  • Tournier, J., Calamante, F., Connelly, A. et al. (2007). Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35 1459–1472.
  • Tuch, D. S. (2002). Diffusion MRI of complex tissue structure. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
  • Tuch, D. S. (2004). Q-ball imaging. Magn. Reson. Med. 52 1358–1372.
  • Tuch, D. S., Reese, T. G., Wiegell, M. R., Makris, N., Belliveau, J. W. and Wedeen, V. J. (2002). High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magn. Reson. Med. 48 577–582.
  • Weinstein, D., Kindlmann, G. and Lundberg, E. (1999). Tensorlines: Advection-diffusion based propagation through diffusion tensor fields. In Proceedings of the Conference on Visualization 249–253.
  • Wiegell, M. R., Larsson, H. B. and Wedeen, V. J. (2000). Fiber crossing in human brain depicted with diffusion tensor MR imaging1. Radiology 217 897–903.
  • Wong, R. K. W., Lee, T. C. M., Paul, D. and Peng, J. (2016). Supplement to “Fiber direction estimation, smoothing and tracking in diffusion MRI.” DOI:10.1214/15-AOAS880SUPP.
  • Yuan, Y., Zhu, H., Lin, W. and Marron, J. S. (2012). Local polynomial regression for symmetric positive definite matrices. J. R. Stat. Soc. Ser. B. Stat. Methodol. 74 697–719.
  • Zhu, H., Zhang, H., Ibrahim, J. G. and Peterson, B. S. (2007). Statistical analysis of diffusion tensors in diffusion-weighted magnetic resonance imaging data. J. Amer. Statist. Assoc. 102 1085–1102.

See also

Supplemental materials