Bayesian Heteroscedastic Regression for Diffusion Tensor Imaging

  • Bertil WegmannEmail author
  • Anders Eklund
  • Mattias Villani
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


We propose a single-diffusion tensor model with heteroscedastic noise and a Bayesian approach via a highly efficient Markov Chain Monte Carlo ( MCMC) algorithm for inference. The model is very flexible since both the noise-free signal and the noise variance are functions of diffusion covariates, and the relevant covariates in the noise are automatically selected by Bayesian variable selection. We compare the estimated diffusion tensors from our model to a homoscedastic counterpart with no covariates in the noise, and to commonly used linear and nonlinear least squares methods. The estimated single-diffusion tensors within each voxel are compared with respect to fractional anisotropy (FA) and mean diffusivity (MD). Using data from the Human Connectome Project, our results show that the noise is clearly heteroscedastic, especially the posterior variance for MD is substantially underestimated by the homoscedastic model, and inferences from the homoscedastic model are on average spuriously precise. Inferences from commonly used ordinary and weighted least squares methods (OLS and WLS) show that it is not adequate to estimate the single-diffusion tensor from logarithmic measurements.



The authors would like to thank professor Hans Knutsson for his code for visualizing diffusion directions.

Anders Eklund was supported by the Information Technology for European Advancement (ITEA) 3 Project BENEFIT (better effectiveness and efficiency by measuring and modelling of interventional therapy) and by the Swedish research council (grant 2015-05356, “Learning of sets of diffusion MRI sequences for optimal imaging of micro structures“). Anders Eklund and Bertil Wegmann were supported by the Swedish research council (grant 2013-5229, “Statistical analysis of fMRI data”).

Data collection and sharing for this project was provided by the Human Connectome Project (HCP; Principal Investigators: Bruce Rosen, M.D., Ph.D., Arthur W. Toga, Ph.D., Van J. Weeden, MD). HCP funding was provided by the National Institute of Dental and Craniofacial Research (NIDCR), the National Institute of Mental Health (NIMH), and the National Institute of Neurological Disorders and Stroke (NINDS). HCP data are disseminated by the Laboratory of Neuro Imaging at the University of Southern California.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bertil Wegmann
    • 1
    Email author
  • Anders Eklund
    • 1
    • 2
    • 3
  • Mattias Villani
    • 1
  1. 1.Division of Statistics and Machine Learning, Department of Computer and Information ScienceLinköping UniversityLinkopingSweden
  2. 2.Division of Medical Informatics, Department of Biomedical EngineeringLinköping UniversityLinkopingSweden
  3. 3.Center for Medical Image Science and Visualization (CMIV)Linköping UniversityLinkopingSweden

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