1. Winter Conference Borgafjäll
Detecting changes in brain scale, shape and connectivity via the geometry of random fields
Jin Cao, Lucent
Nick Chamandy, Google
Khalil Shafie, Northern Colorado
Jonathan Taylor, Stanford
Keith Worsley, McGill

2. EC heuristic A t h i g r e s o l d , x c u n X = f : T ( ) ¸ m p E C 1y P a 2 S \ 9 8 5 I D

3. I n t r i s c v o l u m e ¹ ( S ) F w h b a y @ x C , = ¡ D 2 ¼ £ Z f\ X t ) = D d I n t r i s c v o l u m e d ( S ) F w h b a y @ x C , = D 2 Z 1 f g ; j . 3 E V : E C d e n s i t y ( ) M o r h : S \ X = 1 f T g @ 2 + b u a D P F G m l , Z p

4. Beautiful symmetry: L i p s c h t z - K l n g u r v a e ( S ) E C d e\ X t ) = D d L R
Beautiful symmetry: 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 ( R )
Steiner-Weyl Tube Formula (1930)
Taylor Kinematic Formula (2003)
Put a tube of radius r about the search region λS and rejection region Rt: = S d @ Z s
Z2~N(0,1) Rt r
Tube(λS,r)
Tube(Rt,r) r λS
Z1~N(0,1)
t-r t
Find volume or probability, expand as a power series in r, pull off coefficients: 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 !

5. EC density of the T-statistic field, v df - After lots of messy algebra (Morse) - Two lines (Taylor’s Gaussian Tube Formula) ( t ) = Z 1 + 2 u d 3

6. Multivariate linear models for random field dataY ( s ) n q i t h e o b r v a l d m x p 2 S
< D . A y w u : = X B + E ; N I W c 6 g H T
# regressors
p= p>1
q= T F
# variables
q>1 Hotelling’s T2 Wilks’ Λ
Pillai’s trace
Roy’s max root
etc. N e d n u l i s t r b o : P ( m a x 2 S T )

7. Deformation Based Morphometry (DBM) D’Arcy Thompson (1917) On Growth and Form2 Y ( s ) s 1
n1 = 17 non-missile brain trauma patients, 3-14 days in coma,
n2 = 19 age and gender matched controls
Y(s) = vector (q=3) deformations needed to warp each MRI to an atlas standard
Locate damage: find regions where deformations are different, i.e. shape change

8. Hotelling’s T2 random field, v df - After lots of messy algebra (Morse)( t ) = Z 1 + 2 q u d ; 3 :

9. ? Last case W e s h a l n o w ¯ d P - v u p r x i m t ( b q E C y ) f
# regressors
p= p>1
q= T✔ F✔
# variables
q>1 Hotelling’s T2✔ Wilks’ Λ
Pillai’s trace
Roy’s max root
etc. ? W e s h a l n o w d P - v u p r x i m t ( b q E C y ) f R , T = g Y X 1 I : j c .

10. Roy’s union-intersection principleM a k e i t n o u v r l m d b y p g c q 1 : ( Y s ) = X B + E ; H L F h - f .
< R x T 2 N w S C , P D
Almost the EC density of the Roy’s maximum root field ✔
Why ½? F(s,u)=F(s,-u) ✔

11. Cross correlation random fieldX ( s ) , 2 S
< M a n d Y T N b 1 v c o r f G u i m l . D h C ; = : P x E g j ! + k

12. Maximum canonical cross correlation random field( s ) n p , 2 S
< M a d Y q T N b m r i c o f G u l . D h x C ; = v 1 w g P j k + ! z :

13. What is ‘bubbles’?

14. Nature (2005)

15. Subject is shown one of 40 faces chosen at random …
Happy
Sad
Fearful
Neutral

16. … but face is only revealed through random ‘bubbles’
First trial: “Sad” expression
Subject is asked the expression: “Neutral”
Response: Incorrect
75 random bubble centres
Smoothed by a
Gaussian ‘bubble’
What the
subject sees
Sad

17. Your turn …
Trial 2
Subject response:
“Fearful”
CORRECT

18. Your turn …
Trial 3
Subject response:
“Happy”
INCORRECT
(Fearful)

19. Your turn …
Trial 4
Subject response:
“Happy”
CORRECT

20. Your turn …
Trial 5
Subject response:
“Fearful”
CORRECT

21. Your turn …
Trial 6
Subject response:
“Sad”
CORRECT

22. Your turn …
Trial 7
Subject response:
“Happy”
CORRECT

23. Your turn …
Trial 8
Subject response:
“Neutral”
CORRECT

24. Your turn …
Trial 9
Subject response:
“Happy”
CORRECT

25. Your turn …
Trial 3000
Subject response:
“Happy”
INCORRECT
(Fearful)

26. E.g. Fearful (3000/4=750 trials):
Bubbles analysis
E.g. Fearful (3000/4=750 trials):
Trial
… + 750
= Sum
Correct
trials
Thresholded at
proportion of
correct trials=0.68,
scaled to [0,1]
Use this
as a
bubble
mask
Proportion of correct bubbles
=(sum correct bubbles)
/(sum all bubbles)

27. Happy Sad Fearful Neutral
Results
Mask average face
But are these features real or just noise?
Need statistics …
Happy Sad Fearful Neutral

28. Very similar to the proportion of correct bubbles:
Statistical analysis
Correlate bubbles with response (correct = 1, incorrect = 0), separately for each expression
Equivalent to 2-sample Z-statistic for correct vs. incorrect bubbles, e.g. Fearful:
Very similar to the proportion of correct bubbles:
Z~N(0,1)
statistic
Trial …
Response …

29. Happy Sad Fearful Neutral
Results
Thresholded at Z=1.64 (P=0.05)
Multiple comparisons correction?
Need random field theory …
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

30. Euler Characteristic Heuristic
Euler characteristic (EC) = #blobs - #holes (in 2D)
Excursion set Xt = {s: Z(s) ≥ t}, e.g. for neutral face:
EC = 30
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z ≥ t).
Observed
Expected 20 10
EC(Xt)
-10
Exact expression for E(EC) for all thresholds,
E(EC) ~ P(max Z ≥ t) is extremely accurate.
-20 -4 -3 -2 -1 1 2 3 4
Threshold, t

31. EC densities of Z above t
The result I f Z ( s ) N ; 1 i a n o t r p c G u d m e l , 2
< w h = V @ P x S E C \ : g z + A 3 v F W H M 4 ( Z t ) 1 2 L ( S ) 1 2
Lipschitz-Killing
curvatures of S
(=Resels(S)×c)
EC densities of Z above t
Z(s)
white noise
filter = *
FWHM

32. Results, corrected for search
Random field theory threshold: Z=3.92 (P=0.05)
Saddle-point approx (2007): Z=↑ (P=0.05)
Bonferroni: Z=4.87 (P=0.05) – nothing
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

33. Scale space: smooth Z(s) with range of filter widths w
= continuous wavelet transform
adds an extra dimension to the random field: Z(s,w)
Scale space, no signal 34 8
22.7 6
15.2 4 2
10.2 -2
6.8
-60
-40
-20 20 40 60
w = FWHM (mm, on log scale)
One 15mm signal 34 8
22.7 6
15.2 4 2
10.2 -2
6.8
-60
-40
-20 20 40 60
Z(s,w)
s (mm)
15mm signal is best detected with a 15mm smoothing filter

34. Matched Filter Theorem (= Gauss-Markov Theorem):
“to best detect signal + white noise,
filter should match signal”
10mm and 23mm signals 34 8
22.7 6 4
15.2 2
10.2 -2
6.8
-60
-40
-20 20 40 60
w = FWHM (mm, on log scale)
Two 10mm signals 20mm apart 34 8
22.7 6
15.2 4 2
10.2 -2
6.8
-60
-40
-20 20 40 60
Z(s,w)
s (mm)
But if the signals are too close together they are
detected as a single signal half way between them

35. Scale space can even separate two signals at the same location!
8mm and 150mm signals at the same location 10 5
-60
-40
-20 20 40 60
170 20 76 15
w = FWHM (mm, on log scale) 34 10
15.2 5
6.8
-60
-40
-20 20 40 60
Z(s,w)
s (mm)

36. Scale space Lipschitz-Killing curvatureso s e f i a k r n l w t h R 2 = 1 d B . T c m Z ( ; ) D N : L z - K g v [ ] + b X j ! 4 @ F G , P x

37. Rotation space: Try all rotated elliptical filters
Unsmoothed data
Threshold
Z=5.25 (P=0.05)
Maximum filter

38. Bubbles task in fMRI scanner
Correlate bubbles with BOLD at every voxel:
Calculate Z for each pair (bubble pixel, fMRI voxel)
a 5D “image” of Z statistics …
Trial …
fMRI

39. Thresholding? Cross correlation random field
Correlation between 2 fields at 2 different locations,
searched over all pairs of locations, one in S, one in T:
Bubbles data: P=0.05, n=3000, c=0.113, T=6.22 P m a x s 2 S ; t T C ( ) c E f : g = d i X j L n h 1 ! + b k l

40. Discussion: modeling
The random response is Y=1 (correct) or 0 (incorrect), or Y=fMRI
The regressors are Xj=bubble mask at pixel j, j=1 … 240x380=91200 (!)
Logistic regression or ordinary regression:
logit(E(Y)) or E(Y) = b0+X1b1+…+X91200b91200
But there are only n=3000 observations (trials) …
Instead, since regressors are independent, fit them one at a time:
logit(E(Y)) or E(Y) = b0+Xjbj
However the regressors (bubbles) are random with a simple known distribution, so turn the problem around and condition on Y:
E(Xj) = c0+Ycj
Equivalent to conditional logistic regression (Cox, 1962) which gives exact inference for b1 conditional on sufficient statistics for b0
Cox also suggested using saddle-point approximations to improve accuracy of inference …
Interactions? logit(E(Y)) or E(Y)=b0+X1b1+…+X91200b91200+X1X2b1,2+ …

41. MS lesions and cortical thickness
Idea: MS lesions interrupt neuronal signals, causing thinning in down-stream cortex
Data: n = 425 mild MS patients
5.5 5
4.5 4
Average cortical thickness (mm)
3.5 3
2.5
Correlation = ,
T = (423 df) 2
1.5 10 20 30 40 50 60 70 80
Total lesion volume (cc)

42. MS lesions and cortical thickness at all pairs of points
Dominated by total lesions and average cortical thickness, so remove these effects as follows:
CT = cortical thickness, smoothed 20mm
ACT = average cortical thickness
LD = lesion density, smoothed 10mm
TLV = total lesion volume
Find partial correlation(LD, CT-ACT) removing TLV via linear model:
CT-ACT ~ 1 + TLV + LD
test for LD
Repeat for all voxels in 3D, nodes in 2D
~1 billion correlations, so thresholding essential!
Look for high negative correlations …
Threshold: P=0.05, c=0.300, T=6.48

43. Cluster extent rather than peak height (Friston, 1994)
Choose a lower level, e.g. t=3.11 (P=0.001)
Find clusters i.e. connected components of excursion set
Measure cluster extent
by resels
Distribution:
fit a quadratic to the
peak:
Distribution of maximum cluster extent:
Bonferroni on N = #clusters ~ E(EC). Z
D=1 L D ( c l u s t e r )
extent t
Peak
height L D ( c l u s t e r ) Y Â k s

44. References
Adler, R.J. and Taylor, J.E. (2007). Random fields and geometry. Springer.
Adler, R.J., Taylor, J.E. and Worsley, K.J. (2008). Random fields, geometry, and applications. In preparation.