The Annals of Applied Statistics

Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space

Sofia C. Olhede and Brandon Whitcher

Full-text: Open access


High angular resolution diffusion imaging data is the observed characteristic function for the local diffusion of water molecules in tissue. This data is used to infer structural information in brain imaging. Nonparametric scalar measures are proposed to summarize such data, and to locally characterize spatial features of the diffusion probability density function (PDF), relying on the geometry of the characteristic function. Summary statistics are defined so that their distributions are, to first-order, both independent of nuisance parameters and also analytically tractable. The dominant direction of the diffusion at a spatial location (voxel) is determined, and a new set of axes are introduced in Fourier space. Variation quantified in these axes determines the local spatial properties of the diffusion density. Nonparametric hypothesis tests for determining whether the diffusion is unimodal, isotropic or multi-modal are proposed. More subtle characteristics of white-matter microstructure, such as the degree of anisotropy of the PDF and symmetry compared with a variety of asymmetric PDF alternatives, may be ascertained directly in the Fourier domain without parametric assumptions on the form of the diffusion PDF. We simulate a set of diffusion processes and characterize their local properties using the newly introduced summaries. We show how complex white-matter structures across multiple voxels exhibit clear ellipsoidal and asymmetric structure in simulation, and assess the performance of the statistics in clinically-acquired magnetic resonance imaging data.

Article information

Ann. Appl. Stat. Volume 5, Number 2B (2011), 1293-1327.

First available: 13 July 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Olhede, Sofia C.; Whitcher, Brandon. Nonparametric tests of structure for high angular resolution diffusion imaging in Q -space. The Annals of Applied Statistics 5 (2011), no. 2B, 1293--1327. doi:10.1214/10-AOAS441.

Export citation


  • Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, 10th ed. Dover, New York.
  • Alexander, D. C. (2005). Multiple-fibre reconstruction algorithms for diffusion MRI. Ann. New York Acad. Sci. 1046 113–133.
  • Alexander, D. C., Barker, G. J. and Arridge, S. R. (2002). Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data. Magnetic Resonance in Medicine 48 331–340.
  • Basser, P. J. (2002). Relationships between diffusion tensor and q-space MRI. Magnetic Resonance in Medicine 47 392–397.
  • Basser, P. J., Mattiello, J. and Bihan, D. L. (1994). Estimation of the effective self-diffusion tensor from the NMR spin-echo. Journal of Magnetic Resonance B 103 247–254.
  • Basser, P. J. and Pierpaoli, C. (1996). Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance B 111 209–219.
  • Behrens, T. E. J., Johansen-Berg, H., Jbabdi, S., Rushworth, M. F. S. and Woolrich, M. W. (2007). Probabilistic tractography with multiple fibre orientations: What can we gain? NeuroImage 34 1077–1088.
  • Callaghan, P. T. (1993). Principles of Nuclear Magnetic Resonance Microscopy. Clarendon Press, Oxford.
  • Chen, Y., Guo, W., Zheng, Q., Rao, M. and Liu, Y. (2005). Apparent diffusion coefficient approximation and diffusion anisotropy characterization in DWI. In IPMI 2005 246–257. Springer, Berlin.
  • Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R. (2006). Apparent diffusion coefficients from high angular resolution diffusion imaging: Estimation and applications. Magnetic Resonance in Medicine 56 395–410.
  • Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R. (2007). Regularized, fast and robust analytical Q-ball imaging. Magnetic Resonance in Medicine 58 497–510.
  • Frank, L. R. (2001). Characterization of anisotropy in high angular resolution diffusion-weighted MRI. Magnetic Resonance in Medicine 45 935–939.
  • Gradshteyn, I. S. and Ryzhik, I. M. (2000). Tables of Integrals, Series and Products, 6th ed. Academic Press, New York.
  • Gudbjartsson, H. and Patz, S. (1995). The Rician distribution of noisy MRI data. Magnetic Resonance in Medicine 34 910–914.
  • Hess, C. P., Mukherjee, P., Han, E. T., Xu, D. and Vigneron, D. B. (2006). Q-ball reconstruction of multimodal fibre orientations using the spherical harmonic basis. Magnetic Resonance in Medicine 56 104–117.
  • Hosey, T., Williams, G. and Ansorge, R. (2005). Inference of multiple fiber orientations in high angular resolution diffusing imaging. Magnetic Resonance in Medicine 54 1480–1489.
  • Jansons, K. M. and Alexander, D. C. (2003). Persistent angular structure: New insights from diffusion magnetic resonance imaging data. Inverse Problems 19 1031–1046.
  • Jensen, J. H., Helpern, J. A., Ramani, A., Lu, H. and Kaczynski, K. (2005). Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magnetic Resonance in Medicine 53 1432–1440.
  • Jian, B. and Vemuri, B. C. (2007). A unified computational framework for deconvolution to reconstruct multiple fibres from diffusion weighted MRI. IEEE Transactions on Medical Imaging 26 1464–1471.
  • Kaden, E., Knösche, T. R. and Anwander, A. (2007). Parametric spherical deconvolution: Inferring anatomical connectivity using diffusion MR imaging. NeuroImage 37 474–488.
  • Khachaturian, M. H., Wisco, J. J. and Tuch, D. S. (2007). Boosting the sampling efficiency of q-ball imaging using multiple wavevector fusion. Magnetic Resonance in Medicine 57 289–296.
  • Matérn, B. (1960). Spatial variation. In Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations 39. Statens Skogsforskningsinstitut, Stockholm, Sweden.
  • Mori, S. and van Zijl, C. M. (2002). Fiber tracking: Principles and strategies—a technical review. NMR in Biomedicine 15 468–480.
  • Olhede, S. C. and Whitcher, B. (2008a). HARDI wavelet and non-parametric estimation. Technical report, University College London.
  • Olhede, S. C. and Whitcher, B. (2008b). A statistical framework to characterise microstructure in high angular resolution diffusion imaging. In 5th IEEE International Symposium on Biomedical Imaging 899–902. IEEE, Piscataway, New Jersey.
  • Özarslan, E., Vemuri, B. C. and Mareci, T. M. (2005). Generalized scalar measures for diffusion MRI using trace, variance and entropy. Magnetic Resonance in Medicine 53 866–876.
  • Özarslan, E., Shephard, T. M., Vemuri, B. C., Blackband, S. J. and Mareci, T. M. (2006). Resolution of complex tissue microarchitecture using the diffusion orientation transform. NeuroImage 31 1086–1103.
  • Parker, G. J. M. and Alexander, D. C. (2005). Probabilistic anatomical connectivity derived from the microscopic persistent angular structure of cerebral tissue. Philos. Trans. Royal Soc. London Ser. B 360 893–902.
  • Rao, M., Chen, Y., Vemuri, B. C. and Wang, F. (2004). Cumulative residual entropy: A new measure of information. IEEE Transactions on Information Theory 50 1220–1228.
  • Savadjiev, P., Campbell, J. S. W., Pike, G. B. and Siddiqi, K. (2006). 3D curve inference for diffusion MRI regularization and fibre tractography. Medical Image Analysis 10 799–813.
  • Seunarine, K., Cook, P. A., Hall, M. G., Embleton, K. V., Parker, G. J. M. and Alexander, D. C. (2007). Exploiting peak anisotropy for tracking through complex structure. In Mathematical Methods in Biomedical Image Analysis 1. IEEE Computer Society, Rio de Janeiro.
  • Tournier, J. D., Calamante, F., Gadian, D. G. and Connelly, A. (2004). Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23 1176–1185.
  • Tuch, D. S. (2004). Q-ball imaging. Magnetic Resonance in Medicine 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. Magnetic Resonance in Medicine 48 577–582.
  • Wedeen, V. J., Hagmann, P., Tseng, W. Y., Reese, T. G. and Weisskoff, R. M. (2005). Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magnetic Resonance in Medicine 54 1377–1386.
  • Wu, Y. C. and Alexander, A. L. (2007). Hybrid diffusion imaging. NeuroImage 36 617–629.

Supplemental materials