efficient robust nonlinear and guaranteed positive definite diffusion tensor estimation
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Efficient , Robust , Nonlinear , and Guaranteed Positive Definite Diffusion Tensor Estimation. Robert W Cox & D aniel R Glen SSCC / NIMH / NIH / DHHS / USA / EARTH. ISMRM 2006 – Seattle – 09 May 2006. Nonlinear ?.

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efficient robust nonlinear and guaranteed positive definite diffusion tensor estimation

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

nonlinear
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
positive definite
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
2d cartoon example
2D Cartoon Example

y

Best feasible point

Best feasible point on gradient descent path

x

Forbidden minimizer

guaranteed pd
Guaranteed PD?
  • Descent direction that keeps PD-ness
  • Find M that gives fastest descent rate
efficient
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
robust
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
some results
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
more results
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)
miscellany
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

conclusions
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
finally thanks
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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