Random shapes in brain mapping and astrophysics using an idea from geostatistics

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Random shapes in brain mapping and
astrophysicsusing an idea from geostatistics

• Keith Worsley,
• McGill
• Jonathan Taylor,
• Stanford and Université de Montréal
• Arnaud Charil,
• Montreal Neurological Institute

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CfA red shift survey, FWHM13.3
100
80
60
"Meat ball"
40
topology
"Bubble"
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topology
0
Euler Characteristic (EC)
-20
-40
"Sponge"
-60
topology
CfA
-80
Random
Expected
-100
-5
-4
-3
-2
-1
0
1
2
3
4
5
Gaussian threshold
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Brain imaging Detect sparse regions of
activation Construct a test statistic image
for detecting activation Activated regions test
statistic gt threshold Choose threshold to
control false positive rate to say 0.05 i.e.
P(max test statistic gt threshold)
0.05 Bonferroni???
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Example test statistic
Z1N(0,1)
Z2N(0,1)
s2
s1
Rejection regions,
Excursion sets,
Threshold t
Z2
Search Region, S
Z1
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Euler characteristic heuristic
Search Region, S
Excursion sets, Xt
EC 1 7 6 5 2
1 1 0
Observed
Expected
Euler characteristic, EC
Threshold, t
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Tube(?S,r)
Radius, r
Tube(Rt,r)
Radius, r
Z2
r
Rt
r
?S
Z1
Probability
Area
Radius of Tube, r
Radius of Tube, r
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Z2N(0,1)
Rejection region Rt
r
Tube(Rt,r)
Z1N(0,1)
t
t-r
Taylors Gaussian Kinematic Formula
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r
Tube(?S,r)
?S
Steiner-Weyl Volume of Tubes Formula
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Edge length ?
Lipschitz-Killing curvature of simplices
FWHM/v(4log2)
Lipschitz-Killing curvature of union of simplices
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Non-isotropic data?
ZN(0,1)
s2
s1
Can we warp the data to isotropy? i.e. multiply
edge lengths by ?? Locally yes, but we may need
extra dimensions. Nash Embedding
Theorem dimensions D D(D1)/2 D2
dimensions 5
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Warping to isotropy not needed only warp the
triangles
ZN(0,1)
ZN(0,1)
s2
s1
Edge length ?
Lipschitz-Killing curvature of simplices
FWHM/v(4log2)
Lipschitz-Killing curvature of union of simplices
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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
Unbiased!
Lipschitz-Killing curvature of union of simplices
Unbiased!
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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

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n425 subjects, correlation -0.568
Average cortical thickness
Average lesion volume
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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 R (D dimensions), one in S (E dimensions)
• sample size n
• MS lesion data P0.05, c0.325, T7.07

Cao Worsley, Annals of Applied Probability
(1999)
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Normalization

• LDlesion density, CTcortical 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) )

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Histogram
threshold
threshold
Conditional histogram scaled to same max at
each distance
threshold
threshold