1. Random shapes in brain mapping and astrophysics using an idea from geostatistics
Keith Worsley,
McGill
Jonathan Taylor,
Stanford and Université de Montréal
Arnaud Charil,
Montreal Neurological Institute

3. CfA red shift survey, FWHM=13.3
100 80 60
"Meat ball" 40
topology
"Bubble" 20
topology
Euler Characteristic (EC)
-20
-40
"Sponge"
-60
topology
CfA
-80
Random
Expected
-100 -5 -4 -3 -2 -1 1 2 3 4 5
Gaussian threshold

4. Brain imaging Detect sparse regions of “activation”
Construct a test statistic image for detecting activation
Activated regions: test statistic > threshold
Choose threshold to control false positive rate to say 0.05
i.e. P(max test statistic > threshold) = 0.05
Bonferroni???

5. ¹ Â = m a x Z c o s + i n X = f s : ¹ Â ¸ g R = f Z : ¹ Â ¸ g
Example test statistic: Â = m a x 2 Z 1 c o s + i n
Z1~N(0,1)
Z2~N(0,1) s2 s1
Excursion sets,
Rejection regions, X t = f s : Â g R t = f Z : Â g
Threshold t Z2
Search
Region, S Z1

6. Euler characteristic heuristic
0.5 1
1.5 2
2.5 3
3.5 4 -2 6 8 10
Euler characteristic heuristic
Search Region, S
Excursion sets, Xt
EC=
Observed P ( m a x s 2 S Â ) t E C = : 5 3 7
Expected
Euler characteristic, EC
Threshold, t E ( C S \ X t ) = D d L

7. L i p s c h t z - K l n g u r v a e ( S ) E C d e n s i t y ½ ( ) E (
Tube(λS,r)
Radius, r
Tube(Rt,r)
Radius, r E ( C S \ X t ) = D d L Z2 r Rt r λS = S d @ Z s p 4 l o g 2 F W H M Z1 L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ( ) j T u b e ( S ; r ) P ( T u b e R t ; r )
Area L 2 ( S )
Probability p 2 1 ( t ) r 2 L 1 ( S ) r ( t ) L ( S ) r 2 2 ( t ) r
Radius of Tube, r
Radius of Tube, r j T u b e ( S ; r ) = D X d 2 + 1 L P ( T u b e R t ; r ) = 1 X d 2 !

8. E C d e n s i t y ½ ( ) o f h ¹ Â a c P ( Z ; T u b e R r ) = X ¼ ½ !
Z2~N(0,1) E C d e n s i t y ( ) o f h  a c
Rejection region Rt r
Tube(Rt,r)
Z1~N(0,1)
t-r t
Taylor’s Gaussian Kinematic Formula: P ( Z 1 ; 2 T u b e R t r ) = X d ! + z 4 3 8 .

9. L i p s c h t z - K l n g u r v a e ( S ) d A r e a ( T u b ¸ S ; ) =
Tube(λS,r) λS
Steiner-Weyl Volume of Tubes Formula: A r e a ( T u b S ; ) = D X d 2 + 1 L P i m t E C R s l p 4 o g

10. H o w t ¯ n d L i p s c h z - K l g u r v a e ( S ) L ( ² ) = 1 , ¡ N
Edge length × λ
Lipschitz-Killing curvature of simplices L ( ) = 1 , N e d g l n t h 2 p r i m a
FWHM/√(4log2)
Lipschitz-Killing curvature of union of simplices L ( S ) = P + N 1 2

11. Non-isotropic data? Z~N(0,1) s2 s1 Can we warp the data to isotropy?= S d @ Z s p 4 l o g 2 F W H M s1
Can we warp the data to isotropy?
i.e. multiply edge lengths by λ?
Locally yes, but we may need extra dimensions.
Nash Embedding Theorem:
dimensions ≤ D + D(D+1)/2
D=2: dimensions ≤ 5

13. Warping to isotropy not needed – only warp the triangles
Z~N(0,1)
Z~N(0,1) s2 = S d @ Z s p 4 l o g 2 F W H M s1
Edge length × λ
Lipschitz-Killing curvature of simplices L ( ) = 1 , N e d g l n t h 2 p r i m a
FWHM/√(4log2)
Lipschitz-Killing curvature of union of simplices L ( S ) = P + N 1 2

14. E s t i m a n g L p c h z - K l u r v e ( S ) d L ( ² ) = 1 , ¡ N e d2 3 4 5 6 7 8 9 n E s t i m a n g L p c h z - K l u r v e d ( S )
We need independent & identically distributed random fields
e.g. residuals from a linear model …
Replace coordinates of the simplices in S⊂RealD by
(Z1,…,Zn) / ||(Z1,…,Zn)|| in Realn
Lipschitz-Killing curvature of simplices L ( ) = 1 , N e d g l n t h 2 p r i m a
Unbiased!
Lipschitz-Killing curvature of union of simplices L ( S ) = P + N 1 2
Unbiased!

15. MS lesions and cortical thickness
Idea: MS lesions interrupt neuronal signals, causing thinning in down-stream cortex
Data: n = 425 mild MS patients
Lesion density, smoothed 10mm
Cortical thickness, smoothed 20mm
Find connectivity i.e. find voxels in 3D, nodes in 2D with high
correlation(lesion density, cortical thickness)
Look for high negative correlations …

16. n=425 subjects, correlation = -0.56810 20 30 40 50 60 70 80
1.5 2
2.5 3
3.5 4
4.5 5
5.5
Average cortical thickness
Average lesion volume

17. Thresholding? Cross correlation random field
Correlation between 2 fields at 2 different locations, searched over all pairs of locations
one in R (D dimensions), one in S (E dimensions)
sample size n
MS lesion data: P=0.05, c=0.325, T=7.07
Cao & Worsley, Annals of Applied Probability (1999)

18. Normalization LD=lesion density, CT=cortical thickness
Simple correlation:
Cor( LD, CT )
Subtracting global mean thickness:
Cor( LD, CT – avsurf(CT) )
And removing overall lesion effect:
Cor( LD – avWM(LD), CT – avsurf(CT) )

19. ‘Conditional’ histogram: scaled to same max at each distance
0.5 1
1.5 2
2.5
x 10 5
correlation
Same hemisphere 50
100
150
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.4
0.6
0.8
distance (mm)
Different hemisphere
Histogram
threshold
threshold
‘Conditional’ histogram: scaled to same max at each distance
threshold
threshold