The geometry of random images in astrophysics and brain mapping

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The geometry of random images in astrophysics and
brain mapping

• Keith Worsley, McGill University, Montreal
• Jin Cao, Lucent Technologies
• David Siegmund, Stanford
• Khalil Shafie, Shahid Beheshti University, Tehran
• Jonathan Taylor, McGill and Stanford
• Louis Collins, Jason Lerch, David MacDonald,
  Tomas Paus, Alex Zijdenbos, Alan Evans,
• Montreal Neurological Institute
• www.math.mcgill.ca/keith

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fMRI data 120 scans, 3 scans each of hot, rest,
warm, rest, hot, rest,
X (hot warm effect) / S.d. N(0,1) if no
effect
4
One cycle, averaged over 10 cycles
892
fMRI data
890
Hot
Simple idea fit b1 sin b2 cos test b10,
b20, using X Z12 Z22 where Zi bi/sd(bi)
888
886
delay and spreading 5(?) secs
884
882
Warm
880
878
Rest
Rest
876
874
872
0
5
10
15
20
25
30
35
40
45
Time, seconds
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PCA_IMAGE PCA of time ? space
1 exclude first frames
2 drift
3 long-range correlation or anatomical effect
remove by converting to of brain
4 signal?
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FMRILM fits a linear model for fMRI time series
with AR(p) errors

• Linear model
• ?
  ?
• Yt (stimulust HRF) b driftt c errort
• AR(p) errors
• ? ?
  ?
• errort a1 errort-1 ap errort-p s WNt

unknown parameters
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FMRIDESIGN example pain perception
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FMRILM first step estimate the autocorrelation
?

• AR(1) model errort a1 errort-1 s WNt
• Fit the linear model using least squares
• errort Yt fitted Yt
• â1 Correlation ( errort , errort-1)
• Estimating errorts changes their correlation
  structure slightly, so â1 is slightly biased
• Raw autocorrelation Smoothed 15mm Bias
  corrected â1
• 
  -0.05 0

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FMRILM second step refit the linear model
Pre-whiten Yt Yt â1 Yt-1, then fit using
least squares
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Higher order AR model? Try AR(3)
a
a
a
1
2
3
0.3
0.2
AR(1) seems to be adequate
0.1
0
has little effect on the T statistics
-0.1
AR(1)
AR(2)
AR(3)
5
0
-5
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Milky way
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Gaussianized T field?
EC
Gaussian threshold
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EC for WMAP, unsmoothed, FWHM 1.6o
2500
Observed
2000
Expected
10
1500
0
1000
500
-10
-8
-7
-6
-5
-4
0
EC
20
-500
-1000
10
-1500
0
4
5
6
7
8
-2000
-2500
-5
-4
-3
-2
-1
0
1
2
3
4
5
Gaussian threshold
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Observed Expected EC, with Poisson sd in tails
EC
Gaussian threshold
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EC for WMAP smoothed to FWHMo (expected)
no smoothing
scale space
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Simulation (expected)
no smoothing
scale space
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Scale space WMAP data
2000
Observed
Expected
o
o

1.6

20.5
1500
1000
10
0
500
EC
-10
-4.5
-4
-3.5
0
20
10
-500
0
3.5
4
4.5
5
-1000
-5
-4
-3
-2
-1
0
1
2
3
4
5
Gaussian threshold
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Scale space smooth X(t) with a range of filter
widths, s continuous wavelet transform adds an
extra dimension to the random field X(t, s)
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
S FWHM (mm, on log scale)
One 15mm signal
34
8
22.7
6
4
15.2
2
10.2
0
-2
6.8
-60
-40
-20
0
20
40
60
t (mm)
15mm signal best detected with a 15mm smoothing
filter
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Matched Filter Theorem ( Gauss-Markov Theorem)
to best detect a 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
S 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
-60
-40
-20
0
20
40
60
t (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
113.7
20
76
50.8
15
S FWHM (mm, on log scale)
34
10
22.7
15.2
5
10.2
6.8
-60
-40
-20
0
20
40
60
t (mm)
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FWHM 6.8mm
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FWHM 9mm
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FWHM 11mm
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FWHM 15mm
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FWHM 20mm
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FWHM 26mm
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FWHM 34mm
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FWHM
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FWHM
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FWHM
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FWHM
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FWHM
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FWHM
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FWHM
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FWHM
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fMRI data 120 scans, 3 scans each of hot, rest,
warm, rest, hot, rest,
X (hot warm effect) / S.d. N(0,1) if no
effect
42
One cycle, averaged over 10 cycles
892
fMRI data
890
Hot
Simple idea fit b1 sin b2 cos test b10,
b20, using X Z12 Z22 where Zi bi/sd(bi)
888
886
delay and spreading 5(?) secs
884
882
Warm
880
878
Rest
Rest
876
874
872
0
5
10
15
20
25
30
35
40
45
Time, seconds