EfficientEfficient,, RobustRobust,, NonlinearNonlinear,, andand Guaranteed Positive Guaranteed Positive DefiniteDefinite Diffusion TensorDiffusion Tensor

EstimationEstimation

Robert W Cox Robert W Cox & & DDaniel R Glenaniel R GlenSSCC / NIMH / NIH / DHHS / USA / EARTH

ISMRM 2006 – Seattle – 09 May 2006

NonlinearNonlinear ??

I (q) =J exp(−b(q) • D)+noise(q)

−log I (q ) J( ) ≈ b(q ) • D − noise(q) ⋅eb(q ) • D / J

• 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

eb(q ) •D

b(q )

matrix dot product

Positive DefinitePositive Definite ??• Weighted LSq error functional E

• Given D, linear solve for base image J

• Gradient descent on D to minimize E

E(D, J ) = 1

2wqq∑ J exp(−b(q) • D)−I (q)⎡⎣ ⎤⎦

2

J (D) = wqI(q) exp(−b(q) • D)

q∑⎡⎣ ⎤⎦ wq exp(−2b(q) • D)q∑⎡⎣ ⎤⎦

∇DE ≡ F(D, J ) = − wqq∑ J exp(−b ( q ) • D) − I ( q )[ ]b ( q )

• Oops!Oops! Minimizer D still may not be PD

∂D(s) ∂s = −F D, J (D)( ) [initialize at s = 0 with log-linear LSq]

2D Cartoon Example2D Cartoon ExampleE(x, y) = 1

2 (x−2)2 + 1

2 (y+1)2 with x> 0,y> 0

gradient descent: ∂

∂s

x

y

⎛⎝⎜

⎞⎠⎟

=2 −x

−1 −y

⎛⎝⎜

⎞⎠⎟=straight line

modified descent: ∂

∂s

x

y

⎛⎝⎜

⎞⎠⎟

=x 0

0 y

⎛⎝⎜

⎞⎠⎟

(2 −x)

−(1 + y)

⎛⎝⎜

⎞⎠⎟=curve

x

y

Best feasible point

Best feasible point on gradient descent path

Forbidden minimizer

Guaranteed PDGuaranteed PD ??• Descent direction that keeps PD-ness

E(D +δG) =E(D) + F • δG +O(δ 2 ) [for arb sym G, δ = 1]

let D→ e−δMDe−δMT

for some matrix M (D remains PD)

≈D−δ(MD+DMT )

so E→ E(D)−δF • (MD+DMT ) +O(δ 2 )

• Find M that gives fastest descent rate⇒ maximize F • (MD + DMT ) / (M • M)1/2

⇒ M = FD ⇒ ∂D(s) ∂s = − FD2 + D2F( )

⇒ ∂E / ∂s = −2 FD2

= −2FD • FD ≤ 0

EfficientEfficient ??• Padé approx e2x (1x)/(1+x) for eδ FD : D(s +δ ) =[H−H+

−1]D(s)[H−H+−1]T and δ =stepsize in s

where H± =I ±1

2δFD and H−H+

−1 ≈e−δ FD

• Guarantees D remains PD for any δ• And is O(δ

2) accurate method for ODE

• Choose δ to ensure E decreases quickly

• If E(s+δ ) < E(s) , also try step 2δ

• If E(s+2δ ) < E(s+δ ), keep for next step

RobustRobust ??• 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 !!

• Colorized Fractional Anistropy of D• Voxels with negative eigenvalues are colored black• Problem is worst where D is most anisotropic

Linearized Method Current Method

More Results 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 Anisotropy Angular Deviation

FA=0.0=1o

FA=0.6=6o

MiscellanyMiscellany• C software included in AFNIAFNI 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

ConclusionsConclusions• 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 NOTNOT implemented a nonlinear NONNON-

guaranteed PD solver for comparison• Have NOT looked at local minima issue

Finally … ThanksFinally … ThanksMM Klosek. MM Klosek.

JS Hyde. A Jesmanowicz. BD Ward.JS Hyde. A Jesmanowicz. BD Ward.EC Wong. KM Donahue. EC Wong. KM Donahue.

PA Bandettini. T Ross. RM Birn. J Ratke.PA Bandettini. T Ross. RM Birn. J Ratke.ZS Saad. G Chen. ZS Saad. G Chen.

RC Reynolds. PP Christidis.RC Reynolds. PP Christidis. K Bove-Bettis. LR Frank. K Bove-Bettis. LR Frank. DS Cohen. DA Jacobson.DS Cohen. DA Jacobson.

Former students from MCW.Former students from MCW.Et alii …Et alii …

http://afni.nimh.nih.gov/pub/tmp/ISMRM2006/