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Efficient , Robust , Nonlinear , and Guaranteed Positive Definite Diffusion Tensor Estimation

This paper presents a robust and efficient method for estimating diffusion tensors using a nonlinear approach that ensures positive definiteness. The method is based on minimizing the error between the observed image data and the estimated diffusion tensor. The results show the effectiveness of the proposed method in accurately estimating diffusion tensors in white matter regions.

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Efficient , Robust , Nonlinear , and Guaranteed Positive Definite Diffusion Tensor Estimation

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  1. Efficient,Robust,Nonlinear,andGuaranteed Positive DefiniteDiffusion TensorEstimation Robert W Cox & Daniel R Glen SSCC/NIMH/NIH/DHHS/USA/EARTH ISMRM 2006 – Seattle – 09 May 2006

  2. Nonlinear? • Nonlinear relationship between image data I(q) and D = what we want to know matrix dot product • Ignore noise, transform to linear system for D and solve via OLS? • Oops! Noise level depends nonlinearly on unknowns. In WM, varies strongly with directionality of

  3. Positive Definite? • Weighted LSq error functional E • Given D, linear solve for base image J • Gradient descent on D to minimize E • Oops! Minimizer D still may not be PD

  4. 2D Cartoon Example y Best feasible point Best feasible point on gradient descent path x Forbidden minimizer

  5. Guaranteed PD? • Descent direction that keeps PD-ness • Find M that gives fastest descent rate

  6. Efficient? • Padé approx e2x(1x)/(1+x) for eFD: • Guarantees D remains PD for any  • And is O(2) accurate method for ODE • Choose  to ensure E decreases quickly • If E(s+)<E(s) , also try step 2 • If E(s+2) < E(s+), keep for next step

  7. Robust? • Iterate D(s) to convergence using weights wq=1(most voxels go pretty fast) • Compute residuals (mismatch from data) • And standard deviation of residuals • Reduce weight wq if data point q has “too large” residual (relative to std.deviation) • If had to re-weight, start over • Using final D(s) from first round as starting point for this second round

  8. Some Results ! Linearized Method Current Method • Colorized Fractional Anistropy of D • Voxels with negative eigenvalues are colored black • Problem is worst where D is most anisotropic

  9. More Results ! Fractional Anisotropy Angular Deviation FA=0.0 =1o FA=0.6 =6o • Angular deviation between principal eigenvector of D computed with linearized and current method • Angles only displayed where FA > 0.2 (i.e., in WM)

  10. Miscellany • C software included in AFNI package: • http://afni.nimh.nih.gov • 25625654333 min vs 20 s(iMac Intel) • NIfTI-1 format for file interchange (someday?) • Potential improvements: • {Isotropic D} {Spheroidal D}{General D} • Replace weighted LSq with a sub-quadratic robust error metric (residual) • Simultaneously estimate image registration parameters along with D # Params: 1 < 4 < 6

  11. Conclusions • You may as well use a nonlinear & guaranteed PD solver, since the CPU time penalty is small • And the software is free free free • Significant impact in 1-2% of WM voxels • Importance for applications yet to be evaluated by us • Have NOT implemented a nonlinear NON-guaranteed PD solver for comparison • Have NOT looked at local minima issue

  12. Finally … Thanks MM Klosek. JS Hyde. A Jesmanowicz. BD Ward. EC Wong. KM Donahue. PA Bandettini. T Ross. RM Birn. J Ratke. ZS Saad. G Chen. RC Reynolds. PP Christidis. K Bove-Bettis. LR Frank. DS Cohen. DA Jacobson. Former students from MCW. Et alii … http://afni.nimh.nih.gov/pub/tmp/ISMRM2006/

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