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Visualizing Diffusion Tensor Visualizing Diffusion Tensor Dissimilarity using an Dissimilarity using an ICA Based Perceptual ICA Based Perceptual

Colour MetricColour Metric

Mark S. Drew and Ghassan HamarnehVision and Media Lab Medical Image Analysis Lab

Simon Fraser University{mark,hamarneh}@cs.sfu.ca

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Diffusion tensor imaging

Best reference: Wandell, NIPS 2001 tutorial: Diffusion tensor imaging and fiber tractography of human brain pathways

-Diffusion “Tensor” — a 3 x 3 matrix at each voxel: xx, xy, xz, yx=xy, yy, yz, zx=xz, zy=yz, zz

-Data is MR signal — enscapulates information on water diffusion, at the voxel.

- Brownian motion, but tends to follows “tracts” can use to discover structure∴

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Higher diffusion

along tracts ⇛

= Complementary but different information from more traditional T1- and T2-weighted spin-echo MR data.

⇛ Useful for tasks such as inverse problem of locating epilepsy trigger locations from EEG values – use tensor values as electrical conductivity tensor in generalized Poisson equation.

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xx, xy, xz, yx=xy, yy, yz, zx=xz, zy=yz, zz

measurements in multiple directions

So, 9 variables at each voxel,

But symmetric, => 6 independent variables.

Traditional visualization: use eigenvectors of each tensor.

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zzyzxz

yzyyxy

xzxyxxTUU

D is symmetric => U is orthogonal, real.

D

Because tensor is diffusion, turns out that it’spositive semi-definite: diagonal Λ 0

(The diffusion at each sample location is represented by a 3x3 covariance.)

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The 3 columns of U are normalized 3-vectors,

So typically visualize D by showing an ellipsoid, with axes along the eigenvectors, and lengths e.values.

(or other glyphs,e.g. superquadrics)

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What about colour?

Simple approaches have been used

and, little attention has been paid to forming colours that actually correspond to a difference metric within the structure being imaged!

-Simply map the principal eigenvector at each pixel into {R,G,B}

- colour the three components of the first eigenvector according to a pre-determined colour map

- multiply the 3x3 matrix times a “probe vector”, and map to {R,G,B}; etc.

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So far, we looked at each voxel separately.

=> An approach to whole-brain analysis has been to compute ICA components

(i.e., Independent Component Analysis (see Drew and Bergner, CIC12, 2004 for a tutorial)

seems to extract main, noise-free diffusion signal as largest component; then other effects (eddy currents); then noise.

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Can we map main signal to brightness, and modulate by assigning colour to other components?

=> That way, we code the main information into the visual channel with the most acuity, and reserve colour as a modulating factor encoding the remaining ICA information.

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Can we map a meaningful metric into a perceptually meaningful metric in colour space?

The Log-Euclidean metric for DT data: [Arsigny et al. MICCAI’05]

-taking logs shown to provide reasonable metric for differences between DTs at different voxels-but wish to maintain positive semi-definite property- so define “Log” function, via

Log(D) = U diag ( log ( diag ( ) ) ) UT

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Want to go to 6-vector representation,for simplicity. But maintain same difference-measure as for 9 components:

3,2

3,1

2,1

3,1

2,2

1,1

2

2

2)(

E

E

E

E

E

E

Evecv )(, DLogE

Ev

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ICA on 6-vectors, for practical data, generates a rank-6 basis, B : i.e., B is 6x6.

But B is not orthogonal, so must form pseudoinverseto find coefficients.

So matrix F is the set of filters that corresponds to basis B:

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Diffusion Tensor Data and Perceptual Colour

Consider brain scans (we’ll use 256x256x55 voxels, from http://lbam.med.jhmi.edu/)

slice 25

T1-weighted DT: 1,1 component

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DT is zero where there is no diffusion;so form 2D convex hull to use foreground DT signal:

=> v = vec(Log(D))So coeff’s:

c = F v => let’s look at coeff’s:

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2

3

4

5

6

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sizes of these coefficients c :

#1 most important (ordered by variance)

#1 all-positive sin

ce D is d

iagonally-d

omin

ant

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So, algorithm proposed here:

map 6D perceptual color: L*, a*, b*:----------------------------------------------------

map c1 ↦ L* ;

How to map remaining 5D into a*,b*? Form v’ = (v – c1 b1), perform ICA again; Repeat in remaining 4D space.

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Now map L*, a*, b* to nonlinear sRGB display space:

(1)L*, a*, b* ↦ XYZ tristimulus values (nonlinear transform)

(2) XYZ sRGB↦ linear (linear,

plus clipping)

(3) sRGBlinear ↦ sRGB (if-statement + gamma-correction)

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Test on a synthetic phantom:

↦

(plus 5% Gaussian noise)

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Some results:

slice 5 slice 50

slice 24slice 26

slice 25

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55 slices:

scalar T1 tensor DTtensor enhances perception of organization & connectivity

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CC

Test: can this really help in visualization? --

Consider Corpus Callosum segmentation:

vertical slice (sagittal)

standard subdivision,

↦ tone curves for

histogram equalization

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Apply Tone curves to whole brain:

Compare to FA (histeq’d):

FA previously used to distinguish the seven segments:

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1

23

123

)(

i i

i iFA

Which measure can best discriminate regions of distinct diffusion properties?

FA — F-statistic=29.9log(D) — F-statistic=22.6

At least one confidence interval

does not overlap, from segment

to segment –

facilitating differentiating them.

sRGB colour: F-statistic=32.5

95% confidence intervals

overlap: can’t differentiate segments.

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How do the 7 segments look in L*,a*,b*?

CIELAB coordinates for the means in the seven CC segments (coloured using the mean sRGB colour from the histeq CC): substantialchange in CIELAB between segments.

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Future: …! Cleaner visualization : We'd like to segment, e.g. using extension of the Mean-Shift segmentation algorithm;

should allow for easier evaluation for diagnosis, by medical experts

! Other methods of assigning CIELAB using distance-preserving dimensionality reduction:

We've used ICA = a linear method (as is PCA)

Non-linear mappings:

-- MDS Multidimensional Scaling for assigning location in a low-D space

-- LLE Locally Linear Embedding (based on proximity matrices via a graph);

-- Isomap - another graph-based method for nonlinear dimensionality reduction

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Thanks!To Natural Sciences and

Engineering Research Council of Canada