Efficient robust nonlinear and guaranteed positive definite diffusion tensor estimation
Download
1 / 12

Efficient , Robust , Nonlinear , and Guaranteed Positive Definite Diffusion Tensor Estimation - PowerPoint PPT Presentation


  • 57 Views
  • Uploaded on

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 ?.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Efficient , Robust , Nonlinear , and Guaranteed Positive Definite Diffusion Tensor Estimation' - adelie


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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/


ad