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

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

-5 -4 -3 -2 -1 0 1 2 3 4 5

-100

-80

-60

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-20

0

20

40

60

80

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

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???

-3

-2

-1

0

1

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1

2

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4

-2 0 2

-2

0

2

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

0 0.5 1 1.5 2 2.5 3 3.5 4-2

0

2

4

6

8

10

Eule

r ch

ara

cteri

stic

, 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)

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0

0.5

1

1.5

2

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-2

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2

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1

0 0.5 1 1.5 20

50

100

150

0 0.5 10

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0.4

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

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

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)

4 6 8 10

2

4

6

8

10

12

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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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1

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0.06

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0.14

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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

-3

-2

-1

0

1

2

3

0.06

0.08

0.1

0.12

0.14

4 6 8 102

4

6

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12

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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

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)

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 …

0 10 20 30 40 50 60 70 80

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Average lesion volume

Ave

rag

e co

rtic

al t

hic

kne

ssn=425 subjects, correlation = -0.568

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)

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) )

0

0.5

1

1.5

2

2.5x 10

5

corr

elat

ion

Same hemisphere

0 50 100 150-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0

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0.8

1

distance (mm)

corr

elat

ion

0 50 100 150-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0

0.5

1

1.5

2

2.5

x 105

corr

elat

ion

Different hemisphere

0 50 100 150-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0

0.2

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0.6

0.8

1

distance (mm)

corr

elat

ion

0 50 100 150-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

threshold

thresholdthreshold

threshold

Histogram

‘Conditional’ histogram: scaled to same max at each distance