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

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Efficient, Robust, Nonlinear, and Guaranteed
Positive Definite Diffusion TensorEstimation

• Robert W Cox Daniel R Glen
• SSCC / NIMH / NIH / DHHS / USA / EARTH

ISMRM 2006 Seattle 09 May 2006
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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

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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

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2D Cartoon Example
y
Best feasible point
Best feasible point on gradient descent path
x
Forbidden minimizer
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Guaranteed PD ?

• Descent direction that keeps PD-ness

• Find M that gives fastest descent rate

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Efficient ?

• Padé approx e?2x ?(1?x)/(1x) 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? ) lt E(s) , also try step 2?
• If E(s2? ) lt E(s? ), keep for next step

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Robust ?

• Iterate D(s) to convergence using weights wq1
  (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

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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

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More Results !
Fractional Anisotropy
Angular Deviation
FA0.0 ??1o
FA0.6 ??6o

• Angular deviation between principal eigenvector
  of D computed with linearized and current method
• Angles only displayed where FA gt 0.2 (i.e., in
  WM)

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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 lt 4 lt 6
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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

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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/