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 ? Nonlinear relationship between image data I (q) and D = what we want to know Ignore noise, transform to linear system for D and solve via OLS? Oops! Oops! Noise level depends nonlinearly on unknowns. In WM, varies strongly with directionality of matrix dot product

Positive Definite ? Weighted LSq error functional E Given D, linear solve for base image J Gradient descent on D to minimize E Oops! Oops! Minimizer D still may not be PD

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

Guaranteed PD ? Descent direction that keeps PD-ness Find M that gives fastest descent rate

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 ? Iterate D(s) to convergence using weights w q =1 (most voxels go pretty fast) Compute residuals (mismatch from data) And standard deviation of residuals Reduce weight w q 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 ! Colorized Fractional Anistropy of D Voxels with negative eigenvalues are colored black Problem is worst where D is most anisotropic Linearized MethodCurrent Method

More Results ! Angular deviation between principal eigenvector of D computed with linearized and current method Angles only displayed where FA > 0.2 (i.e., in WM) Fractional AnisotropyAngular Deviation FA=0.0  =1 o FA=0.6  =6 o

Miscellany AFNI C software included in AFNI package:  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 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 NOTNON 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 …