Visualizing Diffusion Tensor Dissimilarity using an ICA Based Perceptual Colour Metric

1. Visualizing Diffusion Tensor Dissimilarity using an ICA Based Perceptual Colour Metric Mark S. Drew and Ghassan Hamarneh Vision and Media Lab Medical Image Analysis Lab Simon Fraser University {mark,hamarneh}@cs.sfu.ca

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

3. 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.

4. 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.

5. Because tensor is diffusion, turns out that it’s positive semi-definite: diagonal Λ  0 (The diffusion at each sample location is represented by a 3x3 covariance.) D is symmetric => U is orthogonal, real.

6. 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)

7. 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.

8. (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. So far, we looked at each voxel separately. => An approach to whole-brain analysis has been to compute ICA components

9. 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.

10. 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

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

12. So matrix F is the set of filters that corresponds to basis B: 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 pseudoinverse to find coefficients.

13. 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

14. 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: 1 5 3 2 4 6

15. #1 all-positive since D is diagonally-dominant #1 most important (ordered by variance) sizes of these coefficients c :

16. So, algorithm proposed here: • map 6D  perceptual color: L*, a*, b*: • ---------------------------------------------------- • map c1↦ L* ; • How to map remaining 5D into a*,b*? • Formv’ = (v – c1 b1), perform ICA again; • Repeat in remaining 4D space.

18. Now map L*, a*, b* to nonlinear sRGB display space: • L*, a*, b* ↦ XYZ tristimulus values (nonlinear transform) • XYZ ↦ sRGBlinear (linear, plus clipping) • sRGBlinear ↦sRGB (if-statement + gamma-correction)

19. Test on a synthetic phantom: ↦ (plus 5% Gaussian noise)

20. Some results: slice 5 slice 50 slice 24 slice 26 slice 25

21. 55 slices: scalar T1 tensor DT tensor enhances perception of organization & connectivity

22. CC standard subdivision, ↦ tone curves for histogram equalization Test: can this really help in visualization? -- Consider Corpus Callosum segmentation: vertical slice (sagittal)

23. Compare to FA (histeq’d): FA previously used to distinguish the seven segments: Apply Tone curves to whole brain:

24. sRGB colour: F-statistic=32.5 At least one confidence interval does not overlap, from segment to segment – facilitating differentiating them.  Which measure can best discriminate regions of distinct diffusion properties? 95% confidence intervals overlap: can’t differentiate segments.  FA — F-statistic=29.9 log(D) — F-statistic=22.6

25. 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): substantial change in CIELAB between segments.

26. 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

27. Thanks! To Natural Sciences and Engineering Research Council of Canada