random shapes in brain mapping and astrophysics using an idea from geostatistics
DESCRIPTION
Random shapes in brain mapping and astrophysics using an idea from geostatistics. Keith Worsley, McGill Jonathan Taylor , Stanford and Universit é de Montr é al Arnaud Charil, Montreal Neurological Institute. CfA red shift survey, FWHM=13.3. 100. 80. 60. "Meat ball". 40. - PowerPoint PPT PresentationTRANSCRIPT
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Random shapes in brain mapping and astrophysics
using an idea from geostatistics
Keith Worsley,McGill
Jonathan Taylor, Stanford and Université de Montréal
Arnaud Charil, Montreal Neurological Institute
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100CfA red shift survey, FWHM=13.3
Gaussian threshold
Eul
er C
hara
cter
istic
(E
C)
"Bubble"topology
"Sponge"topology
"Meat ball" topology
CfARandomExpected
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Brain imaging
Detect sparse regions of “activation”
Construct a test statistic image for detecting activation
Activated regions: test statistic > threshold
Choose threshold to control false positive rate to say 0.05
i.e. P(max test statistic > threshold) = 0.05
Bonferroni???
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Z1~N(0,1) Z2~N(0,1)
Z1
Z2
Rejection regions,Excursion sets,
SearchRegion,
S
Threshold t
s2
s1
Example test statistic: ¹Â = max0· µ· ¼=2
Z1 cosµ+ Z2 sinµ
X t = fs : ¹Â ¸ tg Rt = fZ : ¹Â ¸ tg
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Eule
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, EC
Threshold, t
Excursion sets, Xt
Observed
Expected
EC= 1 7 6 5 2 1 1 0
Search Region, S
Euler characteristic heuristic
P(maxs2S
¹Â(s) ¸ t)
¼E(E C) = 0:05
) t = 3:75
E(EC(S \ X t)) =DX
d=0
Ld(S)½d(t)
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Tube(λS,r) Radius, r Tube(Rt,r) Radius, r
Z2
Z1
Are
a
Radius of Tube, r
Pro
babili
ty
Radius of Tube, r
r
λS
r
Rt
E(E C(S \ X t)) =DX
d=0
Ld(S)½d(t)
L2(S)
2L1(S)r¼L0(S)r2
P(Tube(Rt;r))
P(Tube(Rt;r)) =1X
d=0
(2¼)d=2½d(t)rd=d!
p2¼½1(t)r
½0(t)¼½2(t)r2
jTube(¸S;r)j
jTube(¸S;r)j =DX
d=0
¼d
¡ (d=2+1)L D ¡ d(S)rd
Lipschitz-K illing curvature Ld(S) EC density ½d(t)
¸ = Sdµ
@Z@s
¶
=
p4log2
FWHM
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P (Z1;Z2 2 Tube(Rt;r)) =1X
d=0
(2¼)d=2½d(t)rd=d!
= ½0(t) + (2¼)1=2½1(t)r + (2¼)½2(t)r2=2+¢¢¢
=
Z 1
t¡ r(2¼)¡ 1=2e¡ z2=2dz +e¡ (t¡ r )2=2=4
½0(t) =
Z 1
t(2¼)¡ 1=2e¡ z2=2dz + e¡ t2=2=4
½1(t) = (2¼)¡ 1e¡ t2=2 +(2¼)¡ 1=2e¡ t2=2t=4
½2(t) = (2¼)¡ 3=2e¡ t2=2t + (2¼)¡ 1e¡ t2=2(t2 ¡ 1)=8
...
Tube(Rt,r)
Z1~N(0,1)
r
Rejection region Rt
tt-r
Z2~N(0,1)
Taylor’s Gaussian Kinematic Formula:
EC density ½d(t)of the ¹Â statistic
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Tube(λS,r)
λS
r
Steiner-Weyl Volume of Tubes Formula:
Lipschitz-K illingcurvature L d(S)
Area(Tube(¸S;r)) =DX
d=0
¼d=2
¡ (d=2+1)LD ¡ d(S)rd
= L2(S) + 2L1(S)r +¼L0(S)r2
= Area(¸S) + Perimeter(¸S)r +EC(¸S)¼r2
L0(S) = EC(¸S) = Resels0(S)
L1(S) = 12Perimeter(¸S) =
p4log2 Resels1(S)
L2(S) = Area(¸S) = 4log2 Resels2(S)
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. . . . . . . ... . . . . . .. . . . . .. . . . Lipschitz-Killing curvature of simplices
Lipschitz-Killing curvature of union of simplices
FW
HM
/√(4
log2
)
Edge length × λ
L0(²) = 1, L0(¡ ) = 1, L0(N) = 1L1(¡ ) = edge length, L1(N) = 1
2perimeterL2(N) = area
L0(S) =P
² L0(²) ¡P
¡ L0(¡ ) +P
N L0(N)L1(S) =
P¡ L1(¡ ) ¡
PN L1(N)
L2(S) =P
N L2(N)
How to ¯nd Lipschitz-K illing curvature L d(S)
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Non-isotropic data?
Z~N(0,1)
Can we warp the data to isotropy?i.e. multiply edge lengths by λ?
Locally yes, but we may need extra dimensions.
Nash Embedding Theorem:dimensions ≤ D + D(D+1)/2
D=2: dimensions ≤ 5
¸ = Sdµ
@Z@s
¶
=
p4log2
FWHMs2
s1
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Lipschitz-Killing curvature of simplices
Lipschitz-Killing curvature of union of simplices
L0(²) = 1, L0(¡ ) = 1, L0(N) = 1L1(¡ ) = edge length, L1(N) = 1
2perimeterL2(N) = area
L0(S) =P
² L0(²) ¡P
¡ L0(¡ ) +P
N L0(N)L1(S) =
P¡ L1(¡ ) ¡
PN L1(N)
L2(S) =P
N L2(N)
¸ = Sdµ
@Z@s
¶
=
p4log2
FWHM
Warping to isotropy not needed – only warp the triangles
Z~N(0,1)
Edge length × λ
FW
HM
/√(4
log2
) Z~N(0,1)s2
s1
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Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Zn
We need independent & identically distributed random fieldse.g. residuals from a linear model
Replace coordinates of the simplices in S⊂RealD by(Z1,…,Zn) / ||(Z1,…,Zn)|| in Realn
…
Unbiased!
Unbiased!
Lipschitz-Killing curvature of simplices
Lipschitz-Killing curvature of union of simplices
L0(²) = 1, L0(¡ ) = 1, L0(N) = 1L1(¡ ) = edge length, L1(N) = 1
2perimeterL2(N) = area
L0(S) =P
² L0(²) ¡P
¡ L0(¡ ) +P
N L0(N)L1(S) =
P¡ L1(¡ ) ¡
PN L1(N)
L2(S) =P
N L2(N)
Estimating Lipschitz-K illing curvature L d(S)
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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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Average lesion volume
Ave
rag
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rtic
al t
hic
kne
ssn=425 subjects, correlation = -0.568
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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: P=0.05, c=0.325, T=7.07Cao & Worsley, Annals of Applied Probability (1999)
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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) )
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Histogram
‘Conditional’ histogram: scaled to same max at each distance