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Efficient , Robust , Nonlinear , and Guaranteed Positive Definite Diffusion Tensor EstimationPowerPoint Presentation

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

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### 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 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?

- 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

y

Best feasible point

Best feasible point on gradient descent path

x

Forbidden minimizer

Efficient?

- Padé approx e2x(1x)/(1+x) for eFD:

- 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?

- 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 !

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 !

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

- C software included in AFNI package:
- http://afni.nimh.nih.gov
- 25625654333 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

- 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

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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