Detecting%20sparse%20signals%20and%20sparse%20connectivity%20in%20scale-space,%20with%20applications%20to%20the%20'bubbles'%20task%20in%20an%20fMRI%20experiment

1
Detecting sparse signals and sparse connectivity
in scale-space, with applications to the
'bubbles' task in an fMRI experiment

• Keith Worsley, Nicholas Chamandy, McGill
• Jonathan Taylor, Stanford and Université de
  Montréal
• Robert Adler, Technion
• Philippe Schyns, Fraser Smith, Glasgow
• Frédéric Gosselin, Université de Montréal
• Arnaud Charil, Montreal Neurological Institute

2
Astrophysics
3
Sloan Digital Sky Survey, data release 6, Aug. 07
4
What is bubbles?
5
Nature (2005)
6
Subject is shown one of 40 faces chosen at
random
Happy
Sad
Fearful
Neutral
7
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
8
Your turn

• Trial 2

Subject response Fearful CORRECT
9
Your turn

• Trial 3

Subject response Happy INCORRECT (Fearful)
10
Your turn

• Trial 4

Subject response Happy CORRECT
11
Your turn

• Trial 5

Subject response Fearful CORRECT
12
Your turn

• Trial 6

Subject response Sad CORRECT
13
Your turn

• Trial 7

Subject response Happy CORRECT
14
Your turn

• Trial 8

Subject response Neutral CORRECT
15
Your turn

• Trial 9

Subject response Happy CORRECT
16
Your turn

• Trial 3000

Subject response Happy INCORRECT (Fearful)
17
Bubbles analysis

• E.g. Fearful (3000/4750 trials)

Trial 1 2 3 4
5 6 7 750
Sum
Correct trials
Thresholded at proportion of correct
trials0.68, scaled to 0,1
Use this as a bubble mask
Proportion of correct bubbles (sum correct
bubbles) /(sum all bubbles)
18
Results

• Mask average face
• But are these features real or just noise?
• Need statistics

Happy Sad
Fearful Neutral
19
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

ZN(0,1) statistic
Trial 1 2 3 4
5 6 7 750
Response 0 1 1 0
1 1 1 1
20
Results

• Thresholded at Z1.64 (P0.05)
• Multiple comparisons correction?
• Need random field theory

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
21
Euler Characteristic Heuristic
Euler characteristic (EC) blobs - holes (in
2D) Excursion set Xt s Z(s) t, e.g. for
neutral face
EC 0 0 -7 -11
13 14 9 1 0
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)
0

• Exact expression for E(EC) for all thresholds,
• E(EC) P(max Z t) is extremely accurate.

-10
-20

-4
-3
-2
-1
0
1
2
3
4
Threshold, t
22
The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)


FWHM
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Results, corrected for search

• Random field theory threshold Z3.92 (P0.05)
• 3.82 3.80 3.81
  3.80
• Saddle-point approx (2007) Z? (P0.05)
• Bonferroni Z4.87 (P0.05) nothing

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
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Theorem (1981,1995)
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Steiner-Weyl Tube Formula (1930)
Morse Theory method (1981, 1995)

• Put a tube of radius r about the search
• region ?S

• EC has a point-set representation

r
Tube(?S,r)
?S

• Find volume, expand as a power series
• in r, pull off coefficients

• For a Gaussian random field

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Tube(?S,r)
r
?S
Steiner-Weyl Volume of Tubes Formula (1930)
Lipschitz-Killing curvatures are just intrinisic
volumes or Minkowski functionals in the
(Riemannian) metric of the variance of the
derivative of the process
27
S
S
S
Edge length ?
Lipschitz-Killing curvature of triangles
Lipschitz-Killing curvature of union of triangles
28
Non-isotropic data?
ZN(0,1)
s2
s1

• Can we warp the data to isotropy? i.e. multiply
  edge lengths by ??
• Globally no, but locally yes, but we may need
  extra dimensions.
• Nash Embedding Theorem dimensions D
  D(D1)/2 D2 dimensions 5.
• Better idea replace Euclidean distance by the
  variogram
• d(s1, s2)2 Var(Z(s1)
  - Z(s2)).

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(No Transcript)
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Non-isotropic data
ZN(0,1)
ZN(0,1)
s2
s1
Edge length ?(s)
Lipschitz-Killing curvature of triangles
Lipschitz-Killing curvature of union of triangles
31
We need independent identically distributed
random fields e.g. residuals from a linear model

Lipschitz-Killing curvature of triangles
Lipschitz-Killing curvature of union of triangles
32
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
4
15.2
2
10.2
0
-2
6.8
-60
-40
-20
0
20
40
60
w FWHM (mm, on log scale)
One 15mm signal
34
8
22.7
6
4
15.2
2
10.2
0
-2
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
15mm signal is best detected with a 15mm
smoothing filter
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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
0
-2
6.8
-60
-40
-20
0
20
40
60
w FWHM (mm, on log scale)
Two 10mm signals 20mm apart
34
8
22.7
6
4
15.2
2
10.2
0
-2
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
But if the signals are too close together they
are detected as a single signal half way between
them
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Scale space can even separate two signals at the
same location!
8mm and 150mm signals at the same location
10
5
0
-60
-40
-20
0
20
40
60
170
20
76
15
34
w FWHM (mm, on log scale)
10
15.2
5
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
35
Scale space Lipschitz-Killing curvatures
36
Rotation spaceTry all rotated elliptical
filters
Unsmoothed data
Threshold Z5.25 (P0.05)
Maximum filter
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Beautiful symmetry
Steiner-Weyl Tube Formula (1930)
Taylor Gaussian Tube Formula (2003)

• Put a tube of radius r about the search region
  ?S and rejection region Rt

Z2N(0,1)
Rt
r
Tube(Rt,r)
Tube(?S,r)
r
?S
Z1N(0,1)
t
t-r

• Find volume or probability, expand as a power
  series in r, pull off coefficients

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Z2N(0,1)
Rejection region Rt
Tube(Rt,r)
r
Z1N(0,1)
t
t-r
Taylors Gaussian Tube Formula (2003)
39
EC densities for some standard test statistics

• Using Morse theory method (1981, 1995)
• T, ?2, F (1994)
• Scale space (1995, 2001)
• Hotellings T2 (1999)
• Correlation (1999)
• Roys maximum root, maximum canonical correlation
  (2007)
• Wilks Lambda (2007) (approximation only)
• Using Gaussian Kinematic Formula
• T, ?2, F are now one line
• Likelihood ratio tests for cone alternatives (e.g
  chi-bar, beta-bar) and nonnegative least-squares
  (2007)

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Accuracy of the P-value approximation
The expected EC gives all the polynomial terms in
the expansion for the P-value.
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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 1 2 3 4
5 6 7 3000
fMRI
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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 P0.05, n3000, c0.113, T6.22

Cao Worsley, Annals of Applied Probability
(1999)
43
Discussion modeling

• The random response is Y1 (correct) or 0
  (incorrect), or YfMRI
• The regressors are Xjbubble mask at pixel j, j1
  240x38091200 (!)
• Logistic regression or ordinary regression
• logit(E(Y)) or E(Y) b0X1b1X91200b91200
• But there are only n3000 observations (trials)
  
• Instead, since regressors are independent, fit
  them one at a time
• logit(E(Y)) or E(Y) b0Xjbj
• However the regressors (bubbles) are random with
  a simple known distribution, so turn the problem
  around and condition on Y
• E(Xj) c0Ycj
• 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)b0X1b1X91200
  b91200X1X2b1,2