Journal of Zhejiang University SCIENCE C

, Volume 13, Issue 2, pp 90–98 | Cite as

Diffusion tensor interpolation profile control using non-uniform motion on a Riemannian geodesic



Tensor interpolation is a key step in the processing algorithms of diffusion tensor imaging (DTI), such as registration and tractography. The diffusion tensor (DT) in biological tissues is assumed to be positive definite. However, the tensor interpolations in most clinical applications have used a Euclidian scheme that does not take this assumption into account. Several Riemannian schemes were developed to overcome this limitation. Although each of the Riemannian schemes uses different metrics, they all result in a ‘fixed’ interpolation profile that cannot adapt to a variety of diffusion patterns in biological tissues. In this paper, we propose a DT interpolation scheme to control the interpolation profile, and explore its feasibility in clinical applications. The profile controllability comes from the non-uniform motion of interpolation on the Riemannian geodesic. The interpolation experiment with medical DTI data shows that the profile control improves the interpolation quality by assessing the reconstruction errors with the determinant error, Euclidean norm, and Riemannian norm.

Key words

Diffusion tensor (DT) DT imaging (DTI) DT interpolation Interpolation profile control Riemannian geodesic 

CLC number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alexander, D.C., Barker, G.J., 2005. Optimal imaging parameters for fiber-orientation estimation in diffusion MRI. NeuroImage, 27(2):357–367. [doi:10.1016/j.neuroimage.2005.04.008]CrossRefGoogle Scholar
  2. Alexander, D.C., Pierpaoli, C., Basser, P.J., Gee, J.C., 2001. Spatial transformations of diffusion tensor MR images. IEEE Trans. Med. Imag., 20(11):1131–1139. [doi:10.1109/42.963816]CrossRefGoogle Scholar
  3. Arsigny, V., Fillard, P., Pennec, X., Ayache, N., 2006. Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med., 56(2):411–421. [doi:10.1002/mrm.20965]CrossRefGoogle Scholar
  4. Bansal, R., Staib, L.H., Xu, D.R., Laine, A.F., Royal, J., Peterson, B.S., 2008. Using perturbation theory to compute the morphological similarity of diffusion tensors. IEEE Trans. Med. Imag., 27(5):589–607. [doi:10.1109/TMI.2007.912391]CrossRefGoogle Scholar
  5. Basser, P.J., Mattiello, J., Bihan, D.L., 1994. MR diffusion tensor spectroscopy and imaging. Biophys. J., 66(1):259–267. [doi:10.1016/S0006-3495(94)80775-1]CrossRefGoogle Scholar
  6. Batchelor, P.G., Moakher, M., Atkinson, D., Calamante, F., Connelly, A., 2005. A rigorous framework for diffusion tensor calculus. Magn. Reson. Med., 53(1):221–225. [doi:10.1002/mrm.20334]CrossRefGoogle Scholar
  7. Chefd’Hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O., 2004. Regularizing flows for constrained matrix-valued images. J. Math. Imag. Vis., 20(1–2):147–162. [doi:10.1023/B:JMIV.0000011920.58935.9c]CrossRefGoogle Scholar
  8. Filley, C.M., 2001. The Behavioral Neurology of White Matter. Oxford University Press, New York, p.299.Google Scholar
  9. Fletcher, P.T., Sarang, J., 2007. Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process., 87(2):250–262. [doi:10.1016/j.sigpro.2005.12.018]CrossRefMATHGoogle Scholar
  10. Hoptman, M.J., Nierenberg, J., Bertisch, H.C., Catalano, D., Ardekani, B.A., Branch, C.A., DeLisi, L.E., 2008. A DTI study of white matter microstructure in individuals at high genetic risk for schizophrenia. Schizophr. Res., 106(2–3): 115–124. [doi:10.1016/j.schres.2008.07.023]CrossRefGoogle Scholar
  11. Jones, D.K., Simmons, A., Williams, S.C.R., Horsfield, M.A., 1999. Non-invasive assessment of axonal fibre connectivity in the human brain via diffusion tensor MRI. Magn. Reson. Med., 42(1):37–41. [doi:10.1002/(SICI)1522-2594 (199907)42:1<37::AID-MRM7>3.0.CO;2-O]CrossRefGoogle Scholar
  12. Kindlmann, G., Estépar, R.S.J., Niethammer, M., Haker, S., Westin, C.F., 2007. Geodesic-loxodromes for diffusion tensor interpolation and difference measurement. LNCS, 4791:1–9. [doi:10.1007/978-3-540-75757-3_1]Google Scholar
  13. Pajevic, S., Basser, P.J., 2003. Parametric and non-parametric statistical analysis of DT-MRI. J. Magn. Reson., 161(1): 1–14. [doi:10.1016/S1090-7807(02)00178-7]CrossRefGoogle Scholar
  14. Peng, H., Orlichenkob, A., Dawe, R.J., Agam, G., Zhang, S., Arfanakis, K., 2009. Development of a human brain diffusion tensor template. NeuroImage, 46(4):967–980. [doi:10.1016/j.neuroimage.2009.03.046]CrossRefGoogle Scholar
  15. Pennec, X., Fillard, P., Ayache, N., 2006. A Riemannian framework for tensor computing. Int. J. Comput. Vis., 66(1):41–66. [doi:10.1007/s11263-005-3222-z]MathSciNetCrossRefGoogle Scholar
  16. Roosendaal, S.D., Geurts, J.J.G., Vrenken, H., Hulst, H.E., Cover, K.S., 2009. Regional DTI differences in multiple sclerosis patients. NeuroImage, 44(4):1397–1403. [doi:10.1016/j.neuroimage.2008.10.026]CrossRefGoogle Scholar
  17. Schonberg, T., Pianka, P., Hendler, T., Pasternak, O., Assaf, Y., 2006. Characterization of displaced white matter by brain tumors using combined DTI and fMRI. NeuroImage, 30(4):1100–1111. [doi:10.1016/j.neuroimage.2005.11.015]CrossRefGoogle Scholar
  18. Snook, L., Plewes, C., Beaulieu, C., 2007. Voxel based versus region of interest analysis in diffusion tensor imaging of neurodevelopment. NeuroImage, 34(1):243–252. [doi:10.1016/j.neuroimage.2006.07.021]CrossRefGoogle Scholar
  19. Stejskal, E.O., Tanner, J.E., 1965. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys., 42(1):288–292. [doi:10.1063/1.1695690]CrossRefGoogle Scholar
  20. Zhang, H., Yushkevich, P.A., Alexander, D.C., Gee, J.C., 2006. Deformable registration of diffusion tensor MR images with explicit orientation optimization. Med. Image Anal., 10(5):764–785. [doi:10.1016/]CrossRefGoogle Scholar
  21. Zhou, Y.X., Dougherty, J.H., Hubner, K.F., Bai, B., Cannon, R.L., Hutson, R.K., 2008. Abnormal connectivity in the posterior cingulate and hippocampus in early Alzheimer’s disease and mild cognitive impairment. Alzheim. Dement., 4(4):265–270. [doi:10.1016/j.jalz.2008.04.006]CrossRefGoogle Scholar

Copyright information

© Journal of Zhejiang University Science Editorial Office and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.MOE Key Laboratory of Biomedical EngineeringZhejiang UniversityHangzhouChina
  2. 2.Department of ElectronicsKim Chaek University of TechnologyPyongyangDPR of Korea

Personalised recommendations