Theoretical characterization of angular resolution for linear ODF estimation

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Tuesday, June 27, 2017: 12:45 PM  - 2:45 PM 

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Submission Type:

Abstract Submission 

On Display:

Monday, June 26 & Tuesday, June 27 


Divya Varadarajan1, Justin Haldar1


1University of Southern California, Los Angeles, USA

First Author:

Divya Varadarajan    -  Lecture Information | Contact Me
University of Southern California
Los Angeles, USA


White matter tractography is a powerful tool for medicine and neuroscience, but fiber tracking accuracy is often confounded by crossing fibers. Despite much progress in the development of advanced q-space sampling [1-3] and orientation distribution function (ODF) estimation strategies [4,5], the design of an optimal ODF measurement approach still remains a longstanding unsolved problem. We address this issue by introducing a new model-free theoretical framework for characterizing resolution in ODF estimation. This framework can be used to compare and evaluate different q-space sampling schemes and ODF estimation methods, without requiring modeling assumptions or extensive empirical evaluations. Building off of previous work [4,6,7], our approach relies on a novel theoretical relationship between the estimated ODF and the true ensemble average propagator (EAP), a probability distribution describing local molecular diffusion.


The ODF with constant solid angle correction [2], is defined as
where u is an orientation unit vector, and fΔ(x) is the EAP.

Extending previous work [7] and neglecting noise for simplicity (noise is easily included), we can derive that any linearly estimated ODF Ô(u) is related to the EAP by:
Ô(u)= ∫fΔ(x)g(u,x)dx,
where g(u,x) is the "EAP response", and is easily evaluated using similar techniques to [7]. The EAP response can be interpreted as a point spread function (PSF) for ODF estimation that depends only on the q-space sampling scheme and the ODF estimation method. We hypothesize that measures like the main-lobe width of the EAP response will indicate resolution (like with conventional PSFs) and strongly correlate with empirical measures of angular resolution. If true, then it becomes possible to optimize data sampling and ODF estimation based on the EAP response.

To test our hypothesis, we calculated EAP responses for two ODF estimation schemes (FRACT [6] and SHORE [8]) and several multi-shell sampling schemes (b=[1000,2000,3000], [2000,3000,4000], [3000,4000,5000] and [4000,5000,6000] s/mm^2). We used a single-shell (b=3000 s/mm^2) acquisition for FRACT and same diffusion orientations as the human connectome project (HCP) 3-shell protocol [9]. Main-lobe width was calculated as the full-width at half maximum at the peak of the EAP response along the axis perpendicular to u.

Empirical simulations and real HCP data were used to assess the correlation between the EAP response characteristics and the empirically-observed ODF resolution. ODF estimation and visualization were implemented using BrainSuite [10] ( We quantified empirical ODF resolution by simulating multiple voxels, each containing two diffusion tensors with equal volume fraction and diffusion coefficients in the range 1-3 mm^2/s, and measuring the minimum angle of separation (MAS) between the tensors at which the two distinct orientations are unresolvable.


Fig. 1 shows that both MAS and EAP response main-lobe width were correlated, suggesting that the EAP response predicts angular resolution. Fig. 2 shows the EAP response of FRACT and SHORE for the HCP data. FRACT has a lower main-lobe width, predicting higher angular resolution than SHORE for this data. This prediction is consistent with qualitative examination of the ODFs. Note that our goal in this comparison was to demonstrate the usefulness of the EAP response -- FRACT is not always better than SHORE, and the performance difference is specific to the q-space sampling scheme and the set of estimation parameters that were used.
Supporting Image: Fig1.JPG
Supporting Image: Fig2.JPG


We have proposed using EAP responses to assess angular resolution in linear ODF measurement. We demonstrated that the proposed metric correlated with empirical angular resolution, and believe that such a theoretical approach will prove useful for improving ODF measurement methods.

Imaging Methods:

Diffusion MRI 2

Modeling and Analysis Methods:

Diffusion MRI Modeling and Analysis 1
Exploratory Modeling and Artifact Removal
Methods Development

Poster Session:

Poster Session - Tuesday


Design and Analysis
White Matter

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[1] Tuch, D. S.; Reese, T. G.; Wiegell, M. R.; Makris, N.; Belliveau, J. W. & Wedeen, V. J. (2002), ‘High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity’, Magn. Reson. Med., 48, 577-582
[2] Descoteaux, M.; Deriche, R.; LeBihan, D.; Mangin, J.-F. & Poupon, C.(2011), ‘Multiple q-shell diffusion propagator imaging’, Med. Image Anal., Elsevier, 15, 603-621
[3] Wedeen, V. J.; Hagmann, P.; Tseng, W.-Y. I.; Reese, T. G. & Weisskoff, R. M.( 2005), ‘Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging’, Magn. Reson. Med., 54, 1377-1386
[4] Tuch, D. S.(2004), ‘Q-Ball imaging’, Magn. Reson. Med., 52, 1358-1372
[5] Tournier, J.-D.; Calamante, F.; Gadian, D. G. & Connelly, A.(2005), ‘Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution’, NeuroImage, 2004, 23, 1176-1185
[6] Haldar, J. P. & Leahy, R. M.(2013), ‘Linear transforms for Fourier data on the sphere: Application to high angular resolution diffusion MRI of the brain’, NeuroImage, 71, 233-247
[7] Varadarajan, D. & Haldar, J. P. (2016), ‘A theoretical framework for sampling and reconstructing ensemble average propagators in diffusion MRI’, Proc. Int. Soc. Magn. Reson. Med., 204
[8] Özarslan, E.; Koay, C.; Shepherd, T. M.; Blackband, S. J. & Basser, P. J. (2009), ‘Simple harmonic oscillator based reconstruction and estimation for three-dimensional q-space’, MRI Proc. Int. Soc. Magn. Reson. Med., 1396
[9] Van Essen, D. C.; Smith, S. M.; Barch, D. M.; Behrens, T. E.; Yacoub, E.; Ugurbil, K.; & WU-Minn HCP Consortium. (2013), ‘The WU-Minn human connectome project: an overview’, Neuroimage, 80, 62-79.
[10] Shattuck, D. W. & Leahy, R. M. BrainSuite: An automated cortical surface identification tool Med. Image Anal., 2002, 6, 129-142