Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions

  • Evan Schwab
  • Bijan Afsari
  • René Vidal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7511)


Current methods in high angular resolution diffusion imaging (HARDI) estimate the probability density function of water diffusion as a continuous-valued orientation distribution function (ODF) on the sphere. However, such methods could produce an ODF with negative values, because they enforce non-negativity only at finitely many directions. In this paper, we propose to enforce non-negativity on the continuous domain by enforcing the positive semi-definiteness of Toeplitz-like matrices constructed from the spherical harmonic representation of the ODF. We study the distribution of the eigenvalues of these matrices and use it to derive an iterative semi-definite program that enforces non-negativity on the continuous domain. We illustrate the performance of our method and compare it to the state-of-the-art with experiments on synthetic and real data.


diffusion imaging orientation distribution functions spherical harmonics Toeplitz matrices eigenvalue distribution theorem 


  1. 1.
    Frank, L.R.: Characterization of anisotropy in high angular resolution diffusion-weighted MRI. Magnetic Resonance in Medicine 47(6), 1083–1099 (2002)CrossRefGoogle Scholar
  2. 2.
    Özarslan, E., Mareci, T.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magnetic Resonance in Medicine 50, 955–965 (2003)CrossRefGoogle Scholar
  3. 3.
    Hess, C.P., Mukherjee, P., Han, E.T., Xu, D., Vigneron, D.B.: Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis. Magnetic Resonance in Medicine 56(1), 104–117 (2006)CrossRefGoogle Scholar
  4. 4.
    Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast and robust analytical Q-ball imaging. Mag. Res. in Med. 58(3), 497–510 (2007)CrossRefGoogle Scholar
  5. 5.
    Jian, B., Vemuri, B.: A unified computational framework for deconvolution to reconstruct multiple fibers from diffusion weighted MRI. IEEE Transactions on Medical Imaging 26(11), 1464–1471 (2007)CrossRefGoogle Scholar
  6. 6.
    Tristan-Vega, A., Westin, C.F., Aja-Fernandez, S.: Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging. NeuroImage 47(2), 638–650 (2009)CrossRefGoogle Scholar
  7. 7.
    Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N.: Reconstruction of the orientation distribution function in single- and multiple-shell q-ball imaging within constant solid angle. Magnetic Resonance in Medicine 64(2), 554–566 (2010)Google Scholar
  8. 8.
    Qi, L., Yu, G., Wu, E.: Higher order positive semidefinite diffusion tensor imaging. SIAM J. Imaging Sci. 3 (2010)Google Scholar
  9. 9.
    Barmpoutis, A., Vemuri, B.: A unified framework for estimating diffusion tensors of any order with symmetric positive-definite constraints. In: IEEE International Symposium on Biomedical Imaging, pp. 1385–1388 (2010)Google Scholar
  10. 10.
    Goh, A., Lenglet, C., Thompson, P., Vidal, R.: Estimating Orientation Distribution Functions with Probability Density Constraints and Spatial Regularity. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009, Part I. LNCS, vol. 5761, pp. 877–885. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35(4), 1459–1472 (2007)CrossRefGoogle Scholar
  12. 12.
    Grenander, U., Szego, G.: Toeplitz Forms and their Applications. University of California Press (1958)Google Scholar
  13. 13.
    Shirdhonkar, S., Jacobs, D.: Non-negative lighting and specular object recognition. In: IEEE Conference on Computer Vision and Pattern Recognition (2005)Google Scholar
  14. 14.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (1985)Google Scholar
  15. 15.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 1.21 (April, 2011),
  16. 16.
    Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control. LNCIS, vol. 371, pp. 95–110. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    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 56(1), 1181–1201 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Evan Schwab
    • 1
  • Bijan Afsari
    • 1
  • René Vidal
    • 1
  1. 1.Center for Imaging ScienceJohns Hopkins UniversityUSA

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