Diffeomorphism Invariant Riemannian Framework for Ensemble Average Propagator Computing

  • Jian Cheng
  • Aurobrata Ghosh
  • Tianzi Jiang
  • Rachid Deriche
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6892)


Background: In Diffusion Tensor Imaging (DTI), Riemannian framework based on Information Geometry theory has been proposed for processing tensors on estimation, interpolation, smoothing, regularization, segmentation, statistical test and so on. Recently Riemannian framework has been generalized to Orientation Distribution Function (ODF) and it is applicable to any Probability Density Function (PDF) under orthonormal basis representation. Spherical Polar Fourier Imaging (SPFI) was proposed for ODF and Ensemble Average Propagator (EAP) estimation from arbitrary sampled signals without any assumption.

Purpose: Tensors only can represent Gaussian EAP and ODF is the radial integration of EAP, while EAP has full information for diffusion process. To our knowledge, so far there is no work on how to process EAP data. In this paper, we present a Riemannian framework as a mathematical tool for such task.

Method: We propose a state-of-the-art Riemannian framework for EAPs by representing the square root of EAP, called wavefunction based on quantum mechanics, with the Fourier dual Spherical Polar Fourier (dSPF) basis. In this framework, the exponential map, logarithmic map and geodesic have closed forms, and weighted Riemannian mean and median uniquely exist. We analyze theoretically the similarities and differences between Riemannian frameworks for EAPs and for ODFs and tensors. The Riemannian metric for EAPs is diffeomorphism invariant, which is the natural extension of the affine-invariant metric for tensors. We propose Log-Euclidean framework to fast process EAPs, and Geodesic Anisotropy (GA) to measure the anisotropy of EAPs. With this framework, many important data processing operations, such as interpolation, smoothing, atlas estimation, Principal Geodesic Analysis (PGA), can be performed on EAP data.

Results and Conclusions: The proposed Riemannian framework was validated in synthetic data for interpolation, smoothing, PGA and in real data for GA and atlas estimation. Riemannian median is much robust for atlas estimation.


Parameter Space Probability Density Function Probability Density Function Geodesic Distance Orientation Distribution Function 
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.


  1. 1.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, USA (2000)zbMATHGoogle Scholar
  2. 2.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine 56, 411–421 (2006)CrossRefGoogle Scholar
  3. 3.
    Assemlal, H.E., Tschumperlé, D., Brun, L.: Efficient and robust computation of PDF features from diffusion MR signal. Medical Image Analysis 13, 715–729 (2009)CrossRefGoogle Scholar
  4. 4.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscropy and imaging. Biophysical Journal 66, 259–267 (1994)CrossRefGoogle Scholar
  5. 5.
    Buss, S.R., Fillmore, J.P.: Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics 20, 95–126 (2001)CrossRefGoogle Scholar
  6. 6.
    Cheng, J., Ghosh, A., Jiang, T., Deriche, R.: Riemannian median and its applications for orientation distribution function computing. In: ISMRM (2010)Google Scholar
  7. 7.
    Cheng, J., Ghosh, A., Jiang, T., Deriche, R.: A riemannian framework for orientation distribution function computing. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009. LNCS, vol. 5761, pp. 911–918. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Cheng, J., Ghosh, A., Jiang, T., Deriche, R.: Model-free and analytical EAP reconstruction via spherical polar fourier diffusion MRI. In: Jiang, T., Navab, N., Pluim, J.P.W., Viergever, M.A. (eds.) MICCAI 2010. LNCS, vol. 6361, pp. 590–597. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Fletcher, P.T.: Statistical Variability in Nonlinear Spaces Application to Shape Analysis and DT-MRI. Ph.D. thesis, University of North Carolina (2004)Google Scholar
  10. 10.
    Fletcher, P.T., Venkatasubramanian, S., Joshi, S.: The geometric median on riemannian manifolds with application to robust atlas estimation. NeuroImage 45, S143–S152 (2009)Google Scholar
  11. 11.
    Fletcher, P., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87, 250–262 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    Goh, A., Lenglet, C., Thompson, P., Vidal, R.: A nonparametric Riemannian framework for processing high angular resolution diffusion images and its applications to ODF-based morphometry. NeuroImage (2011)Google Scholar
  13. 13.
    Lenglet, C., Rousson, M., Deriche, R.: Statistics on the manifold of multivariate normal distributions theory and application to diffusion tensor MRI processingy. Journal of Mathematical Imaging and Vision 25(3), 423–444 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pennec, X., Fillard, P., Ayache, N.: A riemannian framework for tensor computing. International Journal of Computer Vision 66, 41–66 (2006)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jian Cheng
    • 1
    • 2
  • Aurobrata Ghosh
    • 2
  • Tianzi Jiang
    • 1
  • Rachid Deriche
    • 2
  1. 1.Center for Computational Medicine, LIAMA, Institute of AutomationChinese Academy of SciencesChina
  2. 2.Athena Project Team, INRIA Sophia Antipolis-MéditerranéeFrance

Personalised recommendations