The geometry of random
fields in astrophysics
and brain mapping
Keith Worsley, Farzan Rohani, McGill
Nicholas Chamandy, McGill and Google
Jonathan Taylor, Stanford and Université de Montréal
Jin Cao, Lucent
Arnaud Charil, Montreal Neurological Institute
Frédéric Gosselin, Université de Montréal
Philippe Schyns, Fraser Smith, Glasgow
Astrophysics
Sloan Digital Sky
Survey,
release
6, Aug. ‘07
Sloan
Digital Skydata
Survey,
FWHM=19.8335
2000
Euler Characteristic (EC)
1500
1000
500
"Meat ball"
topology
"Bubble"
topology
0
-500
-1000
"Sponge"
topology
-1500
Observed
Expected
-2000
-5
-4
-3
-2
-1
0
1
Gaussian threshold
2
3
4
5
fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, …
First scan of fMRI data
Highly significant effect, T=6.59
1000
hot
rest
warm
890
880
870
500
0
100
200
300
No significant effect, T=-0.74
820
hot
rest
warm
0
800
T statistic for hot - warm effect
5
0
-5
T = (hot – warm effect) / S.d.
~ t110 if no effect
0
100
0
100
200
Drift
300
810
800
790
200
Time, seconds
300
Linear model regressors
Alternating hot and warm stimuli separated by rest (9 seconds each).
2
1
0
-1
0
50
100
150
200
250
300
350
Hemodynamic response function: difference of two gamma densities
0.4
0.2
0
-0.2
0
50
Regressors = stimuli * HRF, sampled every 3 seconds
2
1
0
-1
0
50
100
150
200
Time, seconds
250
300
350
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??? Too conservative …

False discovery rate??? Not appropriate …
Detecting sparse cone alternatives

Test statistics are usually functions of Gaussian fields, e.g. T or F statistics

Let’s take a challenging example: a random field of chi-bar statistics for
detecting a sparse cone alternative.

Application: detecting fMRI activation in the presence of unknown latency of
the hemodynamic response function (HRF)

Linear model with two regressors of the HRF shifted by +/-2 seconds

Fit by non-negative least-squares; equivalent to a cone alternative
Hemodynamic response function
2
0.4-2
+2
-2
1
0.2
+2
0
0
-0.2
Stimulus and regressors
0
10
20
Time,  (seconds)
-1
10
20
30
40
50
Time,  (seconds)
60
70
Â
¹=
Chi-bar
max Z1 cos µ + Z2 sin µ
0·µ·¼=2
Example test statistic:
Z1~N(0,1)
Z2~N(0,1)
s2
3
2
1
0
-1
-2
Excursion sets,
Xt = fs : Â
¹ ¸ tg s1
-3
Threshold
t 4
Rejection regions,
Z2
2
3
Search
Region,
S
2
Rt = fZ : Â
¹ ¸ tg
Cone
alternative
0
Z1
Null
1
-2
-2
0
2
Euler characteristic heuristic again
Search Region, S
Excursion sets, Xt
EC= #blobs - # holes
= 1
7
6
5
2
1
1
Euler characteristic, EC
10
Heuristic :
¸ t)
P(max Â(s)
¹
Observed
8
0
s2S
6
¼ E(EC) = 0:05
Expected
4
) t = 3:75
2
0
-2
0
0.5
EXACT!
1
1.5
2
E(EC(S \ Xt )) =
X
D
d=0
2.5
3
L (S)½ (t)
d
d
3.5
4
Threshold, t
E(EC(S \ Xt )) =
X
D
L (S)½ (t)
d
d
µ
@Z
¸ = Sd
@s
d=0
Lipschitz-Killing curvature Ld (S)
Steiner-Weyl Tube Formula (1930)
• Put a tube of radius r about the search
region λS
14
r
12
10
Tube(λS,r)
8
λS
6
EC density ½d (t)
Morse Theory
µ Approachµ(1995)
1
@2Z
¡
E
½d (t) =
1fZ ¸tg det
¸d
@s@s0
¯
¶ µ
¶
¯ @Z
@Z
¯
P
=
0
=0
¯ @s
@s
µ ¡
¶
For Z a Gaussian
random
field
½d (t) =
4
2
2
4
6
8 10 12 14
• Find volume, expand as a power series
in r, pull off coefficients:
jTube(¸S; r)j =
X
D
d=0
¼d
L
¡d (S)r d
D
¡(d=2 + 1)
¶
p1 @
2¼ @t
d
P(Z ¸ t)
For Z a chi-bar random field???
¶
Lipschitz-Killing
curvature Ld (S)
of a triangle
r
Tube(λS,r)
λS
µ
¶
@Z
¸ = Sd
p @s
4 log 2
=
FWHM
Steiner-Weyl Volume of Tubes Formula (1930)
Area(Tube(¸S; r)) =
X
D
¼ d=2
L
¡d (S)r d
D
¡(d=2 + 1)
d=0
= L2 (S) + 2L1 (S)r + ¼ L0 (S)r2
= Area(¸S) + Perimeter(¸S)r + EC(¸S)¼r2
L (S) = EC(¸S)
0
L (S) = 1 Perimeter(¸S)
1
2
L (S) = Area(¸S)
2
Lipschitz-Killing curvatures are just “intrinisic volumes” or “Minkowski functionals”
in the (Riemannian) metric of the variance of the derivative of the process
Lipschitz-Killing curvature Ld (S) of any set
S
S
S
¸ = Sd
Edge length × λ
12
10
8
6
4
2
.
.. . .
.
. . .
.. . .
.. . .
.
. . . .
. . . .
. . . .
.. . .
. . .
... .
.
4
..
.
.
.
.
.
.
.
.
.
.
.
6
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
..
. .
. ...
. ..
. . .
. .
. . .....
. . .
. ....
..
..
10
µ
@Z
@s
¶
of triangles
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = edge length, L (N) = 1 perimeter
1
2
L1 (N) = area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of triangles
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
Non-isotropic data? Use Riemannian metric
µ
¶ of Var(∇Z)
¸ = Sd
Z~N(0,1)
s2
3
@Z
@s
2
1
0.14
0.12
0
-1
-2
Edge length × λ
12
10
8
6
4
2
..
.
.
.
. .
. .
.
.. .
.
.. .
.
.
.
.
.
.
.
. . .
.
. . .
.. .
.
.
. .
...
.
. .
.
.
.
.
.
.
.
.
.
.
.
4
6
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
.
..
. .
. ...
. ..
. . .
. .
. . .....
. . .
. ....
...
10
s1
0.1
0.08
0.06
-3
of triangles
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = edge length, L (N) = 1 perimeter
1
2
L1 (N) = area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of triangles
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
Estimating Lipschitz-Killing curvature Ld (S)
We need independent & identically distributed random fields
e.g. residuals from a linear model
Z1
Z2
Z3
Z4
Replace coordinates of
the triangles in S ½ <2
by normalised residuals
Z
jjZ jj ;
Z = (Z1 ; : : : ; Zn ) 2 <n :
Taylor & Worsley, JASA (2007)
Z5
Z6
Z7
Z8
Z9 … Zn
of triangles
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = edge length, L (N) = 1 perimeter
1
2
L1 (N) = area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of triangles
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
Beautiful symmetry:
E(EC(S \ Xt )) =
X
D
µ
L (S)½ (t)
d
d
@Z
¸ = Sd
@s
d=0
Lipschitz-Killing curvature Ld (S)
Steiner-Weyl Tube Formula (1930)
¶
EC density ½d (t)
Taylor Gaussian Tube Formula (2003)
• Put a tube of radius r about the search region λS and rejection region Rt:
14
Z2~N(0,1)
r
12
10
Tube(λS,r)
8
Tube(Rt,r)
Rt
r
λS
6
t-r
t
Z1~N(0,1)
4
2
2
4
6
8 10 12 14
• Find volume or probability, expand as a power series in r, pull off1
coefficients:
jTube(¸S; r)j =
X
D
d=0
¼d
L
¡d (S)r d
D
¡(d=2 + 1)
P(Tube(Rt ; r)) =
X (2¼)d=2
d!
d=0
½d (t)rd
EC density ½d (t)
of the Â
¹ statistic
Z2~N(0,1)
Tube(Rt,r)
r
Rejection region
Rt
t-r
t
Z1~N(0,1)
Taylor’s Gaussian Tube Formula
(2003)
1
X
(2¼)d=2
½d (t)rd
d!
P (Z1 ; Z2 2 Tube(Rt ; r)) =
d=0
½0 (t) =
Z
= ½0 (t) + (2¼)1=2 ½1 (t)r + (2¼)½2 (t)r2 =2 + ¢ ¢ ¢
Z 1
=
(2¼)¡1=2 e¡z2 =2 dz + e¡(t¡r)2 =2 =4
t¡r
1
(2¼)¡1=2 e¡z2 =2 dz + e¡t2 =2 =4
t
½1 (t) = (2¼)¡1 e¡t2 =2 + (2¼)¡1=2 e¡t2 =2 t=4
½2 (t) = (2¼)¡3=2 e¡t2 =2 t + (2¼)¡1 e¡t2 =2 (t2 ¡ 1)=8
..
.
Taylor & Worsley, Annals of Statistics, submitted (2007)
General cone alternatives
»
Let Z = (Z1 ; : : : ; Zn )
N(¹; In£n ¾ 2 ). We wish to test
H0 : ¹ = 0 vs: H1 : ¹ 2 Cone = fau : a ¸ 0; u 2 U ½ Unit Spheren¡1 g:
¹ = max 2 (u0 Z)2 :
If ¾ is unknown the LR test statistic is equivalent to B
u U jjZ jj2
» Beta( j ; n¡j ) then
Lin & Lindsay, Takemura & Kuriki (1997): let B
j;n¡j
¹ ¸ t) =
P(B
X
n
2
2
wj P(Bj;n¡j ¸ t):
j=1
Taylor & Worsley (2007): let ½F (t) be the EC density of the random ¯eld F
d
½B¹ (t)
d
=
X
n
wj ½Bj;n¡j (t):
d
j=1
Applying the EC heuristic:
µ
¶ X
µ
¶
n
¸t ¼
¹
P max B(s)
wj P max Bj;n¡j (s) ¸ t :
s2S
j=1
s2S
Proof, n=3:
Gaussian
random field in 3D
»
If Z(s) N(0; 1) ¡is an¢ isotropic Gaussian random ¯eld, s 2 <3 ,
with ¸2 I3£3 = V @Z ,
@s
E(EC(S \ fs : Z(s) ¸ tg))
L (S)
0
L (S)
1
Lipschitz-Killing
curvatures of S
L (S)
2
L (S)
3
Z
1
1
e¡z2 =2 dz
(2¼)1=2
t
1 ¡2
+ ¸ 2Diameter(S) £
e t =2
2¼
1
+ ¸2 1 Area(S) £
te¡t2 =2
2
(2¼)3=2
1
£
3
+ ¸ Volume(S)
(t2 ¡ 1)e¡t2 =2 :
(2¼)2
= EC(S) £
If Z(s) is white noise convolved with an
isotropic Gaussian ¯lter of Full Width
at
p
Half Maximum FWHM then ¸ = 4 log 2 .
filter
FWHM
FWHM
½0 (t)
½1 (t)
½2 (t)
½3 (t)
EC densities
of Z
The»accuracy of the EC heuristic
If Z(s) N(0; 1) ¡is an¢ isotropic Gaussian random ¯eld, s 2 <3 ,
with ¸2 I3£3 = V @Z , and for some ® > 1,
@s
µ
¶
P max Z(s) ¸ t = E(EC(S \ fs : Z(s) ¸ tg)) + O(e¡®t2 =2 )
s2S
Z 1
1
L (S) = EC(S) £
e¡z2 =2 dz
0
(2¼)1=2
t
L (S) + ¸ 2Diameter(S) £ 1 e¡t2 =2
1
2¼
Lipschitz-Killing
1
L (S) + ¸2 1 Area(S) £
¡t2 =2
te
curvatures of S
2
2
(2¼)3=2
L (S)
1
£
3
+
¸
Volume(S)
(t2 ¡ 1)e¡t2 =2
3
(2¼)2
½0 (t)
½1 (t)
½2 (t)
½3 (t)
+ O(e¡®t2 =2 ):
The expected EC gives all the polynomial
terms in the expansion for the P-value.
EC densities
of Z
What is ‘bubbles’?
Nature (2005)
Subject is shown one of 40
faces chosen at random …
Happy
Sad
Fearful
Neutral
… but face is only revealed
through random ‘bubbles’

First trial: “Sad” expression
Sad
75 random
Smoothed by a
bubble centres Gaussian ‘bubble’
What the
subject sees
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0


Subject is asked the expression:
Response:
“Neutral”
Incorrect
Your turn …

Trial 2
Subject response:
“Fearful”
CORRECT
Your turn …

Trial 3
Subject response:
“Happy”
INCORRECT
(Fearful)
Your turn …

Trial 4
Subject response:
“Happy”
CORRECT
Your turn …

Trial 5
Subject response:
“Fearful”
CORRECT
Your turn …

Trial 6
Subject response:
“Sad”
CORRECT
Your turn …

Trial 7
Subject response:
“Happy”
CORRECT
Your turn …

Trial 8
Subject response:
“Neutral”
CORRECT
Your turn …

Trial 9
Subject response:
“Happy”
CORRECT
Your turn …

Trial 3000
Subject response:
“Happy”
INCORRECT
(Fearful)
Bubbles analysis

1
E.g. Fearful (3000/4=750 trials):
+
2
+
3
+
Trial
4 + 5
+
6
+
7 + … + 750
1
= Sum
300
0.5
200
0
100
250
200
150
100
50
Correct
trials
Proportion of correct bubbles
=(sum correct bubbles)
/(sum all bubbles)
0.75
Thresholded at
proportion of
0.7
correct trials=0.68,
0.65
scaled to [0,1]
1
Use this
as a
0.5
bubble
mask
0
Results

Mask average face
Happy

Sad
Fearful
But are these features real or just noise?
 Need statistics …
Neutral
Statistical analysis
Correlate bubbles with response (correct = 1, incorrect =
0), separately for each expression
Equivalent to 2-sample Z-statistic for correct vs. incorrect
bubbles, e.g. Fearful:


Trial 1
2
3
4
5
6
7 …
750
1
0.5
0
1
1
Response
0
1
Z~N(0,1)
statistic
4
2
0
-2
0
1
1 …
1
0.75

Very similar to the proportion of correct bubbles:
0.7
0.65
Results

Thresholded at Z=1.64 (P=0.05)
Happy
Average face
Sad
Fearful
Neutral
Z~N(0,1)
statistic
4.58
4.09
3.6
3.11
2.62
2.13
1.64

Multiple comparisons correction?
 Need random field theory …
Euler Characteristic = #blobs - #holes
Excursion set {Z > threshold} for neutral face
EC = 0
30
Euler Characteristic
20
0
-7
-11
13
14
9
1
0
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z > t).
Observed
Expected
10
0
-10
-20
-4
-3
-2
-1
0
Threshold
1
2
• Exact expression for
E(EC) for all thresholds,
• E(EC) ~ P(max Z > t)
3
4accurate.
is extremely
The»result
If Z(s) N(0; 1) ¡is an¢ isotropic Gaussian random ¯eld, s 2 <3 ,
with ¸2 I3£3 = V @Z ,
@s
µ
¶
P max Z(s) ¸ t ¼ E(EC(S \ fs : Z(s) ¸ tg))
s2S
Z 1
1
£
L (S)
= EC(S)
e¡z2 =2 dz
0
(2¼)1=2
t
L (S)
£ 1 e¡t2 =2
+
¸
2Diameter(S)
1
2¼
Lipschitz-Killing
1
L (S)
¡t2 =2
2 1 Area(S) £
+
¸
te
curvatures of S
2
2
(2¼)3=2
L (S)
1
£
3
+
¸
Volume(S)
(t2 ¡ 1)e¡t2 =2 :
3
(2¼)2
If Z(s) is white noise convolved with an
isotropic Gaussian ¯lter of Full Width
at
p
Half Maximum FWHM then ¸ = 4 log 2 .
filter
FWHM
FWHM
½0 (t)
½1 (t)
½2 (t)
½3 (t)
EC densities
of Z
Results, corrected for search

Random field theory threshold: Z=3.92 (P=0.05)
Happy
Average face
Sad
Fearful
Neutral
Z~N(0,1)
statistic
4.58
4.47
4.36
4.25
4.14
4.03
3.92

Bonferroni threshold: Z=4.87 (P=0.05) – nothing
Bubbles task in fMRI scanner

Correlate bubbles with BOLD at every voxel:
Trial
1
2
3
4
5
6
7 …
3000
1
0.5
0
fMRI
10000
0

Calculate Z for each pair (bubble pixel, fMRI voxel) – a
5D “image” of Z statistics …
Thresholding? Cross correlation random field
Correlation between 2 fields at 2 different locations,
searchedµover all pairs of locations,
one in S, one in T:
¶

P
max C(s; t) ¸ c
s2S;t2T
=
¼ E(EC fs 2 S; t 2 T : C(s; t) ¸ cg)
dim(S)
X dim(T
X)
i=0
2n¡2¡h (i ¡ 1)!j!
½ij (c) =
¼h=2+1
L (S)L (T )½ (c)
i
j
ij
j=0
b(hX
¡1)=2c
(¡1)k ch¡1¡2k (1 ¡ c2 )(n¡1¡h)=2+k
k=0
¡( n¡i
2
X
k X
k
l=0 m=0
+ l)¡( n¡j + m)
2
¡
¡
¡
¡
l!m!(k l m)!(n 1 h + l + m + k)!(i ¡ 1 ¡ k ¡ l + m)!(j ¡ k ¡ m + l)!
Cao & Worsley, Annals of Applied Probability (1999)

Bubbles data: P=0.05, n=3000, c=0.113, T=6.22
MS lesions and cortical thickness

Idea: MS lesions interrupt neuronal signals, causing thinning in downstream 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 …

Threshold: P=0.05, c=0.300, T=6.48
n=425 subjects, correlation = -0.568
5.5
Average cortical thickness
5
4.5
4
3.5
3
2.5
2
1.5
0
10
20
30
40
50
60
Average lesion volume
70
80
Discussion: modeling







The random response is Y=1 (correct) or 0 (incorrect), or Y=fMRI
The regressors are Xj=bubble mask at pixel j, j=1 … 240x380=91200 (!)
Logistic regression or ordinary regression:
 logit(E(Y)) or E(Y) = b0+X1b1+…+X91200b91200
But there are only n=3000 observations (trials) …
Instead, since regressors are independent, fit them one at a time:
 logit(E(Y)) or E(Y) = b0+Xjbj
However the regressors (bubbles) are random with a simple known distribution, so
turn the problem around and condition on Y:
 E(Xj) = c0+Ycj
 Equivalent to conditional logistic regression (Cox, 1962) which gives exact
inference for b1 conditional on sufficient statistics for b0
 Cox also suggested using saddle-point approximations to improve accuracy of
inference …
Interactions? logit(E(Y)) or E(Y)=b0+X1b1+…+X91200b91200+X1X2b1,2+ …
Three methods so far
The set-up:
S is a subset of a D-dimensional lattice (e.g. pixels);
Z(s) ~ N(0,1) at most points s in S;
Z(s) ~ N(μ(s),1), μ(s)>0 at a sparse set of points;
Z(s1), Z(s2) are spatially correlated.
To control the false positive rate to ≤α we want a good
approximation to α = P(maxS Z(s) ≥ t):



Bonferroni (1936)
Random field theory (1970’s)
Discrete local maxima (2005, 2007)
Simulations (99999)
0.1
0.09
0.08
Random field theory
Bonferroni
0.07
P value
0.06
0.05
Discrete local maxima
0.04
2
0.03
0
0.02
-2
0.01
0
Z(s)
0
1
2
3
4
5
6
7
8
9
10
FWHM (Full Width at Half Maximum) of smoothing filter
Discrete local maxima

Bonferroni applied to events:
{Z(s) ≥ t and Z(s) is a discrete local maximum} i.e.
{Z(s) ≥ t and neighbour Z’s ≤ Z(s)}

Conservative
Z(s2)

If Z(s) is stationary, with
Cor(Z(s1),Z(s2)) = ρ(s1-s2),
all we need is
P{Z(s) ≥ t and neighbour Z’s ≤ Z(s)}
a (2D+1)-variate integral
≤
Z(s-1) ≤ Z(s) ≥ Z(s1)
≥
Z(s-2)
“Markovian” trick

If ρ is “separable”: s=(x,y),
ρ((x,y)) = ρ((x,0)) × ρ((0,y))
 e.g. Gaussian spatial correlation function:
ρ((x,y)) = exp(-½(x2+y2)/w2)
Then Z(s) has a “Markovian” property:
conditional on central Z(s), Z’s on
different axes are independent:
Z(s±1) ┴ Z(s±2) | Z(s)

Z(s2)
≤
Z(s-1) ≤ Z(s) ≥ Z(s1)
≥
So condition on Z(s)=z, find
Z(s-2)
P{neighbour Z’s ≤ z | Z(s)=z} = ПdP{Z(s±d) ≤ z | Z(s)=z}
then take expectations over Z(s)=z
 Cuts the (2D+1)-variate integral down to a bivariate integral

The result only involves the correlation ½d between adjacent voxels along
each lattice axis d, d = 1; : : : ; D. First let the Gaussian density and uncorrected
P values be
Z 1
p
Á(z) = exp(¡z 2 =2)= 2¼; ©(z) =
Á(u)du;
z
respectively. Then de¯ne
1
¡
Q(½; z) = 1 2©(hz) +
¼
where
Z
®
exp(¡ 1 h2 z 2 = sin2 µ)dµ;
2
0
r ¡
1 ½
h=
:
1+½
³p
´
¡1
® = sin
(1 ¡ ½2 )=2 ;
Then the P-value of the maximum is bounded by
µ
¶
Z
P max Z(s) ¸ t · jS j
s2S
t
1 Y
D
Q(½d ; z) Á(z)dz;
d=1
where jS j is the number of voxels s in the search region S. For a voxel on
the boundary of the search region with just one neighbour in axis direction d,
replace Q(½; z) by 1 ¡ ©(hz), and by 1 if it has no neighbours.
Comparison

Bonferroni (1936)
 Conservative
 Accurate if spatial correlation is low
 Simple

Discrete local maxima (2005, 2007)
 Conservative
 Accurate for all ranges of spatial correlation
 A bit messy
 Only easy for stationary separable Gaussian data on rectilinear lattices
 Even if not separable, always seems to be conservative (but no proof!)

Random field theory (1970’s)
 Approximation based on assuming S is continuous
 Accurate if spatial correlation is high
 Elegant
 Easily extended to non-Gaussian, non-isotropic random fields