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

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/