1. The statistical analysis of surface data
Keith Worsley, McGill
Jonathan Taylor, Stanford
Robert Adler, Technion

2. Isotropic Gaussian random field in 2DZ ( s ) N ; 1 i a n o t r p c G u d m e l , 2
< w h = V @ P x S E C \ : g z + A 3 v F W H M 4
Intrinsic volumes or
Minkowski functionals EC
densities
Z(s)
white noise
filter = *
FWHM

3. Volumes of tubes: Getting the P-value of Gaussian fields directly (Siegmund, Sun, 1989, 1993)x i m a t e h G u s n l d b y K - L v f c j ( ) w Z N ; 1 : X = U , Â . C g T 2 P S V H W 9 3 k
& D + E

4. Jonathan Taylor’s Gaussian Kinematic Formula (2003) for functions of non-isotropic Gaussian fields
< D . Z ( ) = 1 ; : n b i d m o h G a u r l T f , g  F c X x R z j E C \

5. Beautiful symmetry: L i p s c h t z - K l n g u r v a e ( S ) E C d e\ X ) = D d L R L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ( R )
Steiner-Weyl Tube Formula (1930)
Taylor Kinematic Formula (2003)
Put a tube of radius r about the search region λS and rejection region R:
Z2~N(0,1) r R r
Tube(λS,r)
Tube(R,r) = S d @ Z s λS
Z1~N(0,1)
Find volume or probability, expand as a power series in r, pull off coefficients: V ( T u b e S ; r ) = D X d 2 + 1 L P ( T u b e R ; r ) = 1 X d 2 !

6. SurfStat F i t l n e a r m x d ® c s o Y » N ( X ¯ ; V µ ) b y R M L u1 N ( X p ; V q ) b y R M L u g h , v w : = 2 z j f
< \
" # - + / T S P E C 3

7. Cluster extent, rather than peak height, for inference (Friston, 1994)
Choose a lower level, e.g. t=3.11 (P=0.001)
Find clusters i.e. connected components of excursion set
Measure cluster extent
by resels
Distribution:
fit a quadratic to the
peak:
Distribution of maximum cluster extent:
Bonferroni on N = #clusters ~ E(EC). Z
D=1 L D ( c l u s t e r )
extent t
Peak
height L D ( c l u s t e r ) Y Â k s

8. MS lesions and cortical thickness (Charil et al., 2007)
Idea: MS lesions interrupt neuronal signals, causing thinning in down-stream cortex
Data: n = 425 mild MS patients
5.5 5
4.5 4
Average cortical thickness (mm)
3.5 3
2.5
Correlation = ,
T = (423 df) 2
1.5 10 20 30 40 50 60 70 80
Total lesion volume (cc)

9. Thresholding? Correlation random field
Correlation between 2 fields at 2 different locations,
searched over all pairs of locations, one in S, one in T:
MS data: P=0.05, ν=424, c=0.325, T=6.48 P m a x s 2 S ; t T C ( ) c E f : g = d i X j L h 1 ! + b k l

10. References
Adler, R.J. and Taylor, J.E. (2007). Random fields and geometry. Springer.
Adler, R.J., Taylor, J.E. and Worsley, K.J. (2008). Random fields, geometry, and applications. In preparation.