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Metric Selection and Diffusion Tensor Swelling

  • Ofer Pasternak
  • Nir Sochen
  • Peter J. Basser
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

The measurement of the distance between diffusion tensors is the foundation on which any subsequent analysis or processing of these quantities, such as registration, regularization, interpolation, or statistical inference is based. Euclidean metrics were first used in the context of diffusion tensors; then geometric metrics, having the practical advantage of reducing the “swelling effect,” were proposed instead. In this chapter we explore the physical roots of the swelling effect and relate it to acquisition noise. We find that Johnson noise causes shrinking of tensors, and suggest that in order to account for this shrinking, a metric should support swelling of tensors while averaging or interpolating. This interpretation of the swelling effect leads us to favor the Euclidean metric for diffusion tensor analysis. This is a surprising result considering the recent increase of interest in the geometric metrics.

Keywords

Fractional Anisotropy Diffusion Tensor Imaging Diffusion Tensor Riemannian Metrics Anisotropic Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank Liz Salak for reviewing the manuscript. OP wishes to acknowledge with thanks that part of the research on which this publication is based was supported by a Fulbright Post-doctoral Scholar Fellowship, awarded by the Fulbright commission for Israel, the United States-Israel Educational Foundation. PJB was supported by the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development.

References

  1. 1.
    Andersson, J.L.: Maximum a posteriori estimation of diffusion tensor parameters using a Rician noise model: why, how and but. NeuroImage 42(4), 1340–1356 (2008)Google Scholar
  2. 2.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006). doi:10.1002/mrm.20965Google Scholar
  3. 3.
    Assaf, Y., Pasternak, O.: Diffusion tensor imaging (DTI)-based white matter mapping in brain research: a review. J. Mol. Neurosci. 34(1), 51–61 (2008)Google Scholar
  4. 4.
    Basser, P.J., Pierpaoli, C.: A simplified method to measure the diffusion tensor from seven MR images. Magn. Reson. Med. 39, 928–934 (1998)Google Scholar
  5. 5.
    Basser, P.J., Jones, D.K.: Diffusion-tensor MRI: theory, experimental design and data analysis—a technical review. NMR Biom. 15, 456–467 (2002)Google Scholar
  6. 6.
    Basser, P.J., Pajevic, S.: A normal distribution for tensor-valued random variables to analyze diffusion tensor MRI data. IEEE TMI 22, 785–794 (2003)Google Scholar
  7. 7.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994)Google Scholar
  8. 8.
    Batchelor, P.G., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Magn. Reson. Med. 53, 221–225 (2005)Google Scholar
  9. 9.
    Crank, J.: The Mathematics of Diffusion. Oxford University Press, New York, USA (1975)Google Scholar
  10. 10.
    Einstein, A.: Investigations on the Theory of the Brownian Movement. Dover, New York (1926)Google Scholar
  11. 11.
    Eisenhart, L.: Differential Geometry. Princeton University Press, Princeton (1940)Google Scholar
  12. 12.
    Fillard, P., Arsigny, V., Pennec, X., Ayache, N.: Clinical DT-MRI estimation, smoothing and fiber tracking with log-Euclidean metrics. IEEE Trans. Med. Imaging 26(11), 1472–1482 (2007). doi:10.1109/TMI.2007.899173. PMID: 18041263Google Scholar
  13. 13.
    Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)Google Scholar
  14. 14.
    Gur, Y., Pasternak, O., Sochen, N.: Fast gl(n)-invariant framework for tensors regularization. Int. J. Comput. Vis. 85(3) (2009)Google Scholar
  15. 15.
    Jian, B., Vemuri, B.C., Ozarslan, E., Carney, P.R., Mareci, T.H.: A novel tensor distribution model for the diffusion-weighted MR signal. NeuroImage 37(1), 164–176 (2007)Google Scholar
  16. 16.
    Kindlmann, G., Ennis, D.B., Whitaker, R.T., Westin, C.F.: Diffusion tensor analysis with invariant gradients and rotation tangents. IEEE Trans. Med. Imaging 26(11), 1483–1499 (2007). doi:10.1109/TMI.2007.907277Google Scholar
  17. 17.
    Koay, C.G., Basser, P.J.: Analytically exact correction scheme for signal extraction from noisy magnitude MR signals. J. Magn. Reson. 179(2), 317–322 (2006)Google Scholar
  18. 18.
    Kubicki, M., McCarley, R., Westin, C.F., Park, H.J., Maier, S., Kikinis, R., Jolesz, F.A., Shenton, M.E.: A review of diffusion tensor imaging studies in schizophrenia. J. Psychiatr. Res. 41, 15–30 (2007)Google Scholar
  19. 19.
    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(3), 423–444 (2006)Google Scholar
  20. 20.
    de Luis-Garcia, R., Alberola-Lopez, C., Kindlmann, G., Westin, C.F.: Automatic segmentation of white matter structures from DTI using tensor invariants and tensor orientation. In: Proc 17th Annual Meeting ISMRM. International Society for Magnetic Resonance in Medicine, Honolulu, USA (2009)Google Scholar
  21. 21.
    Maaß, H.: Siegel’s Modular Forms and Dirichlet Series. Springer, Berlin (1971)Google Scholar
  22. 22.
    Moakher, M.: On the averaging of symmetric positive-definite tensors. J. Elast. 82, 273–296 (2006)Google Scholar
  23. 23.
    Pajevic, S., Basser, P.J.: Parametric and non-parametric statistical analysis of DT-MRI data. J. Magn. Reson. 161(1), 1–14 (2003)Google Scholar
  24. 24.
    Pasternak, O., Sochen, N., Basser, P.: The effect of metric selection on the analysis of diffusion tensor MRI data. Neuroimage 49, 2190–2204 (2010)Google Scholar
  25. 25.
    Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006). doi:10.1007/s10851-006-6228-4Google Scholar
  26. 26.
    Pierpaoli, C., Marenco, S., Rohde, G., Jones, D., Barnett, A.: Analyzing the contribution of cardiac pulsation to the variability of quantities derived from the diffusion tensor. In: Proc 11th Annual Meeting ISMRM, p. 70. International Society for Magnetic Resonance in Medicine, Toronto, Canada (2003)Google Scholar
  27. 27.
    Rohde, G.K., Barnett, A.S., Basser, P.J., Marenco, S., Pierpaoli, C.: Comprehensive approach for correction of motion and distortion in diffusion-weighted MRI. Magn. Reson. Med. 51(1), 103–114 (2004)Google Scholar
  28. 28.
    Skare, S., Anderson, J.: On the effects of gating in diffusion imaging of the brain using single shot EPI. Magn. Reson. Imaging 19, 1125–1128 (2001)Google Scholar
  29. 29.
    Tarantola, A.: Elements for Physics: Quantities, Qualities, and Intrinsic Theories. Springer, Berlin (2006)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Brigham and Women’s HospitalHarvard Medical SchoolBostonUSA
  2. 2.Department of Applied MathematicsTel Aviv UniversityTel AvivIsrael
  3. 3.The Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD)National Institutes of HealthBethesdaUSA

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