1. L’imagerie numérique et son analyse à des fins cliniques: quelques applications
Keith Worsley, Math + Stats,
Arnaud Charil, Montreal Neurological Institute, McGill
Philippe Schyns, Fraser Smith, Psychology, Glasgow
Jonathan Taylor,
Stanford and Université de Montréal

2. What is ‘bubbles’?

3. Nature (2005)

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

5. … 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

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

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

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

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

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

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

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

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

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

15. 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)

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

17. 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 …

18. Comparison Both depend on average correct bubbles, rest is ~ constant
Z=(Average correct bubbles
average incorrect bubbles)
/ pooled sd
Proportion correct bubbles
= Average correct bubbles
/ (average all bubbles * 4)

19. 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

21. 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

22. Euler Characteristic = #blobs - #holes
Excursion set {Z > threshold} for neutral face
EC =
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z > t).
Exact expression for E(EC) for all thresholds,
E(EC) ~ P(max Z > t) is extremely accurate.

23. The details …

24. 2
Tube(S,r) r S

25. 3

26. B A

28. 6
Λ is big
TubeΛ(S,r) S r
Λ is small

29. 2 ν
U(s1) s1 S S
Tube
Tube s2 s3
U(s3)
U(s2)

30. Z2 R r
Tube(R,r) Z1
N2(0,I)

31. Tube(R,r) R z
t-r t z1
Tube(R,r) r R R z2 z3

34. Summary

36. Random field theory results
For searching in D (=2) dimensions, P-value of max Z is (Adler, 1981; W, 1995):
P(max Z > z)
~ E( Euler characteristic of thresholded set )
= Resels × Euler characteristic density (+ boundary)
Resels (=Lipschitz-Killing curvature/c) is
Image area / (bubble FWHM)2 = 146.2
Euler characteristic density(×c) is
(4 log(2))D/2 zD-1 exp(-z2/2) / (2π)(D+1)/2
See forthcoming book Adler, Taylor (2007)

37. Results, corrected for search
Thresholded at Z=3.92 (P=0.05)
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

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. Discussion: thresholding
Thresholding in advance is vital, since we cannot store all the ~1 billion 5D Z values
Resels=(image resels = 146.2) × (fMRI resels = )
for P=0.05, threshold is Z = 6.22 (approx)
The threshold based on Gaussian RFT can be improved using new non-Gaussian RFT based on saddle-point approximations (Chamandy et al., 2006)
Model the bubbles as a smoothed Poisson point process
The improved thresholds are slightly lower, so more activation is detected
Only keep 5D local maxima
Z(pixel, voxel) > Z(pixel, 6 neighbours of voxel)
> Z(4 neighbours of pixel, voxel)

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
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 …

42. 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

43. 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
Cao & Worsley, Annals of Applied Probability (1999)

44. 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) )

45. ‘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

46. Science (2004)

47. fMRI activation detected by correlation between subjects at the same voxel
The average nonselective time course across all activated regions obtained during the first 10 min of the movie for all five subjects. Red line represents the across subject average time course. There is a striking degree of synchronization among different individuals watching the same movie.
Voxel-by-voxel intersubject correlation between the source subject (ZO) and the target subject (SN). Correlation maps are shown on unfolded left and right hemispheres (LH and RH, respectively). Color indicates the significance level of the intersubject correlation in each voxel. Black dotted lines denote borders of retinotopic visual areas V1, V2, V3, VP, V3A, V4/V8, and estimated border of auditory cortex (A1).The face-, object-, and building-related borders (red, blue, and green rings, respectively) are also superimposed on the map. Note the substantial extent of intersubject correlations and the extension of the correlations beyond visual and auditory cortices.

48. What are the subjects watching during high activation? Faces …

49. … buildings …

50. … hands

51. Thresholding? Homologous correlation random field
Correlation between 2 equally smooth fields at the same location, searched over all locations in R (in D dimensions)
P-values are larger than for the usual correlation field (correlation between a field and a scalar)
E.g. resels=1000, df=100, threshold=5, usual P=0.051, homologous P=0.139
Cao & Worsley, Annals of
Applied Probability (1999)

52. Detecting Connectivity between Images: the 'Bubbles' Task in fMRI
Keith Worsley, McGill
Phillipe Schyns, Fraser Smith, Glasgow

53. Subject is shown one of 40 faces chosen at random …
Happy
Sad
Fearful
Neutral
… but face is only revealed through random ‘bubbles’
E.g. first trial: “Sad” expression:
Subject is asked the expression: “Neutral”
Response: Incorrect=0
75 random bubble centres
Smoothed by a
Gaussian ‘bubble’
What the
subject sees
Sad