Statistical Shape Analysis
Ian Dryden (University of Nottingham) Session I
Dryden and Mardia (1998, chapters 1,2,3,4)
[email protected] ¡ Introduction ¡ http://www.maths.nott.ac.uk/ ild Motivation and applications
¡ Size and shape coordinates
¡ Shape space
3e cycle romand de statistique et probabilites´ ¡ Shape distances. appliques´
Les Diablerets, Switzerland, March 7-10, 2004.
1 2
An object’s shape is invariant under the similarity trans- In a wide variety of applications we wish to study the formations of translation, scaling and rotation. geometrical properties of objects.
We wish to measure, describe and compare the size and shapes of objects
Shape: location, rotation and scale information (simi- larity transformations) can be removed. [Kendall, 1984] Size-and-shape: location, rotation (rigid body trans- formations) can be removed. Two mouse second thoracic vertebra (T2 bone) out- lines with the same shape.
3 4 ¡ Landmark: point of correspondence on each object that matches between and within populations.
Different types: anatomical (biological), mathematical, pseudo, quasi
From Galileo (1638) illustrating the differences in shapes of the bones of small and large animals.
5 6
¡ Bookstein (1991)
Type I landmarks (joins of tissues/bones) Type II landmarks (local properties such as maximal curvatures) Type III landmarks (extremal points or constructed land-
marks) ¡ T2 mouse vertebra with six mathematical landmarks Labelled or un-labelled configurations (line junctions) and 54 pseudo-landmarks.
7 8 1 3
A 2 B 3
2 1 3 1
C 2 D Traditional methods 3
1 2 2 - ratios of distances between landmarks or angles sub- mitted to multivariate analysis 3 3 E - the full geometry usually if often lost F
1 1 2 - collinear points?
Six labelled triangles: A, B have the same size and - interpretation of shape differences in multivariate space? shape; C has the same shape as A, B (but larger size); D has a different shape but its labels can be permuted to give the same shape as A, B, C; triangle E can be reflected to have the same shape as D; triangle F has a different shape from A,B,C,D,E.
9 10
Geometrical shape analysis
Rather than working with quantities derived from or- ganisms one works with the complete geometrical ob- ject itself (up to similarity transformations). ¡ Pioneers: Fred Bookstein and David Kendall
In the spirit of D’Arcy Thompson (1917) who consid- Summaries of the field are given by Bookstein (1991, ered the geometric transformations of one species to Cambridge), Small (1996, Springer), Dryden and Mar- another dia (1998, Wiley), Kendall et al (1999, Wiley), Lele and Richstmeier (2001, Chapman and Hall).
We conside a shape space obtained directly from the landmark coordinates, which retains the geometry of a point configuration at all stages.
11 12 MR brain scan The map of 52 megalithic sites (+) that form the ‘Old Stones of Land’s End’ in Cornwall (from Stoyan et al., 1995).
13 14
0.4 b 12 13 11
0.2 10 n
9 Braincase l 7 8 0.0 6 5 na Face 4 pr -0.2
1 3 2 st -0.4 ba o
-0.4 -0.2 0.0 0.2 0.4 Ape cranium
Handwritten digit 3
15 16
250 ¢ 200 150
• S • S
100 •S S S • •
• S ••S • S 50
(a) (b) 0
0 50 100 150 200 250
Electrophoretic gel matching Face recognition
17 18
Proton density weighted MR image Cortical surface extracted from MR scan
19 20 £
OUR FOCUS: landmarks in ¤ real dimensions
£ ¦
¥
§ ¨ © is a ¤ matrix ( )
Invariance with respect to Euclidean similarity group
¦ ¦
© © ¤ (translation, scale and rotation) =
Size....
¥ ¥
¨
Any positive real valued function such that
¥ 203 Pseudo-landmarks on the cortical surface of the for a positive scalar . brain
21 22
¡ Centroid size:
An alternative size measure is the baseline size, i.e.
" "
¥ ¥ ¥
¥ the length between landmarks 1 and 2:
# & ' ( &
¨ ¨ ! )
# $ % & $ %
0 ¥ ¥ ¥
%
'
% %
¥ ¥
¨ 1
( & # &
# $ %
*
¨ ) where and )
This was used as early as 1907 by Galton for normal-
' ,
izing faces.
¨ +
£
, , -
Other size measures: square root of area, cube root
. /
¥ ¥ ¥
¨ !
- Euclidean norm, of volume
-
£ ¦ £ £ ¦
+ - identity matrix, - vector of ones.
, ,
23 24
% 3 3 3 4 5
2 1 1 1 2 Landmarks: 2
Shape coordinates: )
¡ Bookstein shape coordinates (1984,1986) (For two Fixed coordinate system dimensional data) vs
•
6 6
200 • • 200 • •
100 • 100 • • Local Coordinate system • • 0 0
Im(z) Im(z) • • -100 -100 -200 -200
Are angles appropriate.....?? -200 -100 0 100 200 -200 -100 0 100 200 3 Re(z) Re(z) 1.0 200 • •
2 100 • • 0.5 • • 0 • • Im(z) Im(z) 1 • • • 0.0 1 -100 2 • -200 -0.5 -200 -100 0 100 200 -0.5 0.0 0.5 3 Re(z) Re(z)
1
@ £
' > 3 3 3
2 3 &
8 ¨ 9 : ; 9 < 1 ? ¨ A 1 1 1
Shape: 7
<
9 = ; 9
25 26
3 • 20 20 • 2 3 10 • 10 • 1 •
0 0 2 • 1 -10 -10 -20 -20
-20 -10 0 10 20 -20 -10 0 10 20 (a) (b) 1.0 20 3
10 • 3 0.5 •
0 • • 1 2 • • 0.0
-10 1 2 -20 -0.5 -20 -10 0 10 20 -0.5 0.0 0.5 (c) (d)
In real co-ordinates:
%
$
B C
D D
D
G
E F F F J J F J J
; 9 G ; 9 H I 9 ; 9 H I G ; H I ; H I K L M
H G N
)
$
O C
D D
D
G
E F F J J F J J F
9 G ; 9 H I ; H I ; G ; H I 9 ; 9 H I K L M
H G N
$ P $ & The outline of a microfossil with three landmarks (from
where , and
G G G R S
F F J J
M 9 G ; 9 H I G ; H I
N Q Q Q N G
H Bookstein, 1986).
B O
C C D
D .
; T U U T N
27 28
0.8 W 100• 0.32 0.36 0.40 0.44 • • • • 88• 9.2
87• • • X
88• 88• 9.0 84• • • • • ••• • • • • • 0.7 100• 8.8 slog • • • • • • 92• 65• 75• 84• 8.6 • • • • • • 76• • • 71• • • • • 8.4 74• 84• • •
V • • 8.2 V 0.6 71• • • 0.44 • • • • 72• • • • • • • • • • 67• • • • • •• • • • •• • • • 0.40 64• • • • U •
0.5 • • 65• 0.36 • •
• •
60• Y • • 0.32 • • • • • • 0.75 • • • • • 0.4 • • • • • • • • • • 0.2 0.3 0.4 0.5 0.6 • • • • • • • 0.65 U • • V • • • • • • 0.55 • •
• • 0.45 W 8.2 8.4 8.6 8.8 9.0 9.2 W 0.45 0.55 0.65 0.75 A scatter plot of (U+1/2) for the Bookstein shape vari- ables for some microfossil data. (Bookstein, 1986)
29 30 2
a 4 0.5 1
d E `
5 _ 3
g
f b B 0 V A O B
v
\ ] 0.0 1 2
e F -1
^ 6 -0.5
-2
[ \ ]
-2 -1 0b 1 2
c [ [ B -0.5 0.0 0.5 U Z u
The shape space of triangles, using Bookstein’s co- h
A scatter plot of the Bookstein shape variables for the
3 i
8 ordinates 8 . All triangles could be relabelled T2 mouse data. and reflected to lie in the shaded region.
31 32 ¡ Kendall’s shape sphere (1983) (triangles only) Isosceles triangles 3 Equilateral (North pole) 1 2 θ=0 Unlabelled Right-angled
Kendall’s shape coordinates
φ=2π/3 φ=5π/3 θ=π/2
φ=0 φ=π/3
% 3 3
%
k ¨ l j m ¨ j 1 1 1 j Remove location j Flat triangles
- (Equator) ;
θ=π
1 2
& %
j
@ £
3
3 3
& o p q &
7 n n ¨ ¨ A 1 1 1 1 ;
% Reflected equilateral (South pole) j
A mapping from Kendall’s shape variables to the sphere Simple 1-1 linear correspondence with Booklstein S.V. is
(equ. 2.11 of book)
' s
q
' 3
P P
7
n n
)
A A !
For triangles Kendall’s SV sends baseline to !
, 3 u 3
, r , r
2 ¨ ¨ j ¨
t
s s s
o o o
) ) )
, , ,
s
o q
P P
¨ 7
n n
) )
and ) , so that
%
u
o o
v
2 j ¨
) ) ) .
33 34 w
Kendall’s Bell
3 x
Kendall’s spherical shape shape variables are
then given by the usual polar coordinates
w w w
, x 3 u , x 3 , 3
2 ¨ ¨ j ¨
t t t
w
y y z > y x {
where > is the angle of latitude and t z is the angle of longitude.
35 36
Bookstein coordinates - 3D
¥
3 3
# % # # P #
2 2 2
The Schmidt net for 1/12 sphere Landmarks ¨
-
)
w
}
3 3
& % & & P &
| |
t t t
7 ¨ 7 7 7
3 x > y y 3 > y { z
-
¨ ¨ ! 1
t ~
)
¥ ¥
%
o
@ £
¥
, ' 3 3 3
&
)
¨ ¨ A 1 1
¥ ¥
t C '
%
• 0.0 A B
C C ) • • A BA B C C • C C • A B A B A • AB • B C C ¦
-0.5 A • BC C C AC • B A • • A A AB •AB •AB B where is a rotation matrix C C
• •
¥ ¥ ¥
A B C CA B
3 3 P A • B C C • % A • BA • BA B C C A • B A • B (a function of ) and C C A • B C C A • B ) A • B A • B
-1.0 A C• B A • CB
¥ ¥ t
A C• B A •CB t
' 3 > 3 > 3 3 > 3 > 3 C C %
A C• B A • B A • B A • CB
r - , r - , A C • B A •CB ) A • C B A • C B
-1.5
¥
3 3 >
P % P P
-0.5 0.0 0.5 P
¨ 7 7
7
-
)
@
¥
> >
P P P & &
7 ¨ 7 ¨
where 7 , and for
)
£
3 3
1 1 1 .
37 38 t
Goodall-Mardia QR shape coordinates D
£ ¦
¥ ¥
¨ l ¤ Helmertized landmarks k ( matrix)
SIZE AND SHAPE (JOINTLY)
¥
3 4 3
k ¨ ¤
(a) (b)
is lower triangular
Bookstein 3D coordinates
SHAPE:
¨ r
39 40 Shape coordinates
1. FILTER OUT TRANSLATION: SHAPE SPACE....Kendall (1984) a) Shift centroid to origin b) Take linear orthogonal contrasts, e.g. Helmert con- 1. Remove location (Pre-multiply by Helmert sub-matrix)
trasts
¥ ¥
¨ l c) Shift baseline midpoint to origin k
2. RE-SCALE: @
where th row of the Helmert sub-matrix l is given a) Re-scale to unit centroid size by,
b) Re-scale to unit area
@ @ @
3 3 3 ' 3 > 3 3 > 3 '
& & & &
<
o
1 1 1 1 1 1 ¨
; , c) Re-scale to a standard baseline length =
d) Re-scale to minimize ‘distance’ to a template
@
and the & is repeated times and zero is repeated
@ @ £
3. REMOVE ROTATION: £
' ' 3 3 '
1 1 1
times, ¨ .
, , ,
a) Rotate baseline to horizontal
¥ ¥
¨ l l ¨ k ¨
b) Rotate to minimize ‘distance’ to a template Note (centering matrix) so
-
¥ 1 (centroid size) Bookstein shape coordinates: 1c/2c/3a Kendall shape coordinates: 1b/2c/3a Procrustes shape coordinates: 1a/2d/3b
41 42
¡ Dimensions....
2. Remove size (rescale)
£ ¦
¥ ¥
¤
l k
Original configuration:
¨ ¨ 1
¥ ¥
l
£
' ¤
Centered configuration: ¤
% %
¡ 4
is the PRESHAPE ( F )
; I ;
£
' ' ¤
Preshape: ¤ ,
3. Remove rotation
£
t
' ' ' '
¤ ¤ ¤
Shape: ¤
, , r
¥
4 3
¨ ¤
¥ ¥ ¡ is the SHAPE of . ¡ Shape space is non-Euclidean
43 44
$
SHAPE SPACES Write , for the pseudo-singular value decom-
S
% % $
position where N , and
F F
; I ; I
. Let N
£ £
F ¡ ¡ ¢
H I
o N Q Q Q N
¤ ¤
©
$ %
Assume . [ points in Euclidean dimen-
¤
,
$ ¥
£ ¤
R R R ¨ ¨
E ¡ ¡ ¡ ¢
¢ §
©
%
¢ ¦
H H K
¤
Q Q Q
sions] ¢
R R R R
E ¡ ¡ ¡ ¢
¢ §
¦
¢
H H K
Q Q Q
Le and DG Kendall (1993, Annals of Statistics)
£
t
'
%
¤ ¨
: is a unit radius -sphere. £
¤ ,
Theorem On ª , the Riemannian metric can be expressed
F ¢
as I
t
5
¤ ¨
: is the complex projective space ) .
;
)
D
" ® ¢ " ® ¢ " °
G
« ¬ « «
$
G G G
t
F ¡ ¡
¡
D
z ; I
G G G
¤
¡ ¡
±
: has a singularity set of dimen- D
D
G G
)
¡ ¡ ¡
t
;
H
¢
'
G G H ¯ ¯
sion ¤ and is NOT a homogeneous space.
¤
¢ §
" ® "
®
H
D
%
G G
t
¡
±
D
N
For ¤ the space spaces are topological
¢ ²
H
H
%
spheres. where D are co-ordinates for .
F
±
; I
45 46
t ¨ Planar case: ¤ dimensional data
PLANAR CASE: Procrustes/Riemannian distance
t
5
¨ ¨
)
;
r
3 3 3
%
)
m ¨ j m 1 1 1 j m
Complex configurations j -
Helmertized landmarks
· · ·
3 3
%
%
m ¨ m 1 1 1 m
-
% 3 3 4 5 >
%
m
j k ¨ l j ¨ j 1 1 1 j
;
-
; ·
Now multiplying by 3
j ¸ ¸
with centroids .
# ´
³
t
s 3 s 4 3 µ 4 > 3 z
¨ ©
·
3
j m m
Shape distance ¹ satisfies k
rotates and rescales j . So,
· ·
' '
# $ % # #
*
¸ ¸
j m j m
·
3
m m
¹ j ¨ º º
³ ³
· ·
' '
4 5 > 3
# #
* *
j j ¸ ¸
! m ! m
j
k
) )
· ·
# # m
where m means the complex conjugate of . m
is the set representing the SHAPE of j . This is a
complex line through the origin (but not including it) in NB ¹ is the modulus of the complex correlation
£
'
·
m m
dimensions. The union of all such sets is the j
,
between and .
5
complex projective space )
;
£
t
¡
¨ A ¹ )
: is the great circle distance on .
, r
5
¶ )
NB: ) ; 47 48
3 3
%
m ¨ j m 1 1 1 j m
Complex configurations j -
Bookstein co-ordinates:
'
& %
j m j m
@ £
·
' > 3 3 3
&
8
¨ 1 ? ¨ A 1 1 1
'
%
j j
m m
) Session II Kendall co-ordinates:
¡ Procrustes analysis
@ £
·
& % & 3 3 3
&
¡
n ¨ j j ¨ A 1 1 1
r Tangent coordinates
;
¡
3 3 %
% Shape variability
1 1 1 j ¨ l j m
where j
- ¡
; Shape models ¡
¡ Tangent space inference » Linear relationship:»
¡ Shapes package.
t
%
8
n ¨ ! l
£ ¦ £
t t
% ' '
where l is lower right partition of
l .
£
t
· ·
P P
¨ A ¨ ! A 8
For : n . r
49 50
PROCRUSTES ANALYSIS PLANAR PROCRUSTES ANALYSIS Juvenile (———) Adult (------)
u u % 3 3 u
1 1 1 ¼ Two centred configurations ¨ and
3000
· · ·
% 3 3 5
¼
1 1 1 • ¨ , both in , with •
2000
·
u ½ > ½ ¨
• ¨ ,
, ,
1000 • •
y • • • • •
0
½ u • u • • • • • • [ - transpose of the complex conjugate of ] • • • • • -1000 •
• · Match onto u using complex linear regression -2000 -2000 -1000 0 1000 2000 3000 x
3000
# À
·
u
o p ¾ o ¿ o Á
2000
¨ , •
•
·
3
o Á ¨
1000 • • , y
•
•• • ¥
3
o Á 0 •• ¨ • • •
• • •• • • • • • M
•
-1000
¥
·
3
¨ - ‘design’ matrix
-2000
,
# À
-2000 -1000 0 1000 2000 3000
M
3
p ¾ ¿
x o
¼ ¨ - similarity transformation pa- r ameters Register adult onto juvenile
51 52
Procrustes match = least squares
· · · · ·
Â
½ u ½ Procrustes fit ¨
Minimize the sum of square errors r
0 ¥ ¥
·
½ ½
u 3 u ' u '
Á Á
) ¨ ¨ 1
·
Â
s u '
¨
Procrustes residual vector
M M ·
Full Procrustes fit (superimposition) of on u Minimized objective function
0
· · · ·
s 3 > u ½ u ' u ½ ½ u ½
¨
)
r
· ·
½ ½
u u
(not symmetric unless ¨ )
# À Ä
¥
· Â ·
3
o p ¾ o ¿
¨ ¨
,
à Ã
M Ã Ã Initially standardize to unit centroid size....
where Full Procrustes distance:
·
u
# À
È
%
Æ Ç
¥ ¥ ¥
·
3 u ' ' '
½ ½ u
¿ p ¾
¨
¨ É
, À
;
·
u
à M
M M
%
N Ê N Ë Ì Ì
i.e. N
Ì Ì
· ·
u ½ ½ u
)
Ì Ì
L
'
Ì Ì
¨ Í 1
> 3
o p ¾
· ·
,
Ì Ì
u u Î
¨
½ ½
/ . Å / . Å
Ã
w
Ã
· ·
½ ½
u ' u 3
¨ ¨
%
Ã
· · · ·
½ ½ ½
u u
¿
¨ 1
)
L
r Ã
53 54
d /2
P Æ Ç ρ FULL Procrustes distance - full set of similarity transformations used in matching d
F Æ
PARTIAL Procrustes distance  - matching over trans- lation and rotation ONLY 1 1
For fairly similar shapes they are very similar,
P P
Æ Æ Æ
Ç
o o
¨ ¹ ¹ ¨ ρ/2
as  Â
In this course for simplicity we shall concentrate on FULL Procrustes matching. Section of the pre-shape sphere
55 56
dF · d/2 Procrustes residuals from the match of onto u are
ρ · different from u onto
3000
• • • • 2000
1000 • • y • • •• • 0 •
•• 1/2 •• 1/2 •• • • -1000 •• • •
ρ -2000 -2000 -1000 0 1000 2000 3000
x
w
¿
? 1 ? Ï ¨ 1 A
JUV to ADULT (above): ¨ , .
, , ,
w Ð Ð
à Ã
' >
¿
? 1 ? Ï ¨ 1 Ñ Ò ? ¨ Ó
ADULT to JUV: ¨ , Ã
Section of the SHAPE SPHERE FOR TRIANGLES, Ã
1 A
Æ Ç Æ
, r , , , ¹
illustrating the relationship between , Â and
57 58
CONFIGURATION MODEL
· · Õ
3 3
%
1 1 1 Random sample of Ô configurations from
Female (left) and Male (right) gorilla skulls the perturbation model
# À ×
Ø
·
3 3 3 3
# # # #
o ¿ o Á p
¨ Ö ¨ 1 1 1 Ô
, ,
4 5 #
where Ö - translations
# 4
¿
© - scales
200 200
w
t
> y { z # - rotations
100 100
# 4 5 Á
y y are independent zero mean complex random
0 0 errors Ø is the population mean configuration.
-100 -100
-300 -200 -100 0 100 -300 -200 -100 0 100
Ø Ø x x (a) (b) AIM: to estimate - the shape of
Procrustes mean:
Mean shape? Shape variance/covariance?
Õ
/ . Å È
"
Ø Æ Ø
Ç
·
3
#
¨ 1
)
Ù
# $ % Ã
59 60
· ·
# > #
Consider to be centred: ¨ .
,
-
Ø
(Kent, 1994) Procrustes mean shape is the dom-
Ø
¡
· #
inant eigenvector of à Procrustes fits: match to
Ã
Õ Õ
" "
· · · ·
½ ½ ½
# # # 3
# # #
¨ ¨ j j
r
# $ % # $ %
Ø
· Â · · · ·
½ ½
# # 3 3 3 3
# # # p
¨ ¨ 1 1 1 Ô
· ·
# # # 3 3 3
p
r ,
¨ ¨ 1 1 1 Ô
where the j , are the pre-
r , Ã
shapes.
%
Õ
·
Â
# $ % # * NB Arithmetic mean: Õ has same shape as
Proof We wish to minimize Ø .
Õ Õ
Ã
Ø Ø
· ·
½ # ½
#
" "
Æ Ø
Ç
·
# 3 '
Í
) ¨ ¡
Ø Ø
· ·
,
# Î
½
½
#
$ % # $ %
# Procrustes residuals
Ø Ø Ø Ø
½ ½
'
¨ Ô 1
r
Õ
"
 Â
· ·
,
s # ' ÝÞ 3 3 3 3
# # ßà p
¨ 1 1 1 Ô
Therefore, ¨
,
# $ %
Ô
/ . Å Ú Û
Ø Ø Ø
½
¨ 1
Ü Ü
$ %
Ù Ã
Hence, result follows. 61 62
Procrustes fits (Generalized Procrustes analysis) 0.6 0.6 0.4 0.4
•••• • •••••••• • ••••••• 0.2 •• • •••• •• •••• • • • • • •••• ••••••••••• •••••••• ••••••• 0.2 •• •• •• • •••• ••••••• •• ••••••• •••• ••••••• •••• ••••••• 0.0 • •• ••••••• • • •••••• ••• ••••• •••• ••••• 0.0 ••• ••• • •••• •••••• •• •••• • ••••• ••••• ••••••• ••••••••••• •••••••• •••• • ••• -0.2 • ••••• ••••••• ••• ••••• •••••••••• -0.2 • ••••• -0.4 -0.4 -0.6
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Male Gorillas Female gorillas
63
Other mean shape estimates: ¡
150 Bookstein mean shape
100 h
Take sample mean of Bookstein coordinates 8 50
1.0
ã y ã
0
ã ã
2 2 ã ã ã 2 ã â 2 222222 22ã 2222222•
â 2 2
88 2222222 â
â 88
8â 8 â 8 88â 8 2
88 â 8 8888•888 8 88 8 0.5
-50
ä
ä ä
333ä 33 ä
33ä 33333 3 3333•33
-100 á
v 7• 4•
0.0 å
-150 å
å 5
å 5
5 å å 5 5555555 555å 5 -150 -100 -50 0 50 100 150 5•55 5
x -0.5 6 66 666666 6 66666 6 6•66 666 1 111 66 1111 111111 1 1 1 1•1111 111 -1.0
The male (—-) and female (- - -) full Procrustes mean -1.0 -0.5 0.0 0.5 1.0 shapes registered by GPA. u
Female Gorillas
64 65
[In Book chapter 12]
¡ MDS mean shape (Kent, 1994; Lele 1991) 1.0 Obtain average squared Euclidean distance matrix 0 22 22 2 22222 222222 222• 8 2 22 2 8•88 8 2 2 888888888 888 8
0.5 8 8
8 %
0
'
¨ 33333 let æ (centred inner product matrix) 333333333•
3333333 )
v 7• 4•
0.0
3 3
%
1 1 1 ç è Let ç be the scaled eigenvectors 5
555555 5•555555 5 55 -0.5 6 6
6 6
0
616 666 0
1 6•6 6 6
%
3 3 666 66 3
1 61666 6
¨ ç ç 1 1 1 ç 1 11 66 § 1 1• 1111 1 1 1111 1 11 1 1 1 ) 1 -1.0
-1.0 -0.5 0.0 0.5 1.0 u (invariant under reflections too)
Male Gorillas ¡ IMPORTANT: If shape variations small the mean shape estimates are approximately linearly related. i.e. Multivariate normal based inference will be equiv- alent to first order. (Kent, 1994)
66 67 The partial Procrustes tangent coordinates for a
Tangent coordinates planar shape are giv en by
Ä
# À
½
' 3 4 3
%
q q
+ Ö Ö j Ö
¨ (1)
3 3
%
m ¨ j m 1 1 1 j m ¼
Consider complex landmarks j with
;
/ . Å
w
' ½
Ö j
pre-shape where ¨ . Partial Procrustes tangent Ã
coordinates involve only rotation (and not scaling) to
% 3 3
%
¼ m m
j ¨ j 1 1 1 j ¨ l j l j 1
r match the pre-shapes. ;
Let Ö be a complex pole on the complex pre-shape
½ >
q ¨ sphere usually chosen as an average shape. Note that Ö and so the complex constraint
means we can regard the tangent space as a real
w
£
t
'
© )
Let us rotate the configuration by an angle to be as subspace of ) of dimension . The ma-
;
' ½
%
Ö Ö
close as possible to the pole and then project onto trix + is the matrix for complex projection
;
Ö Ö
the tangent plane at , denoted by . Note that into the space orthogonal to Ö . Below we see a sec-
/ . Å
# À
w
½
' '
¨ Ö j Ö j minimizes ) . tion of the shape sphere showing the tangent plane à coordinates.
68 69
PROCRUSTES TANGENT SPACE ¥
Procrustes tangent co-ordinates of at the pole
§ : v
iθ
¥
' ze
¨ é ¹ § iθ
ze β
t
> { y z where ¹ is the Riemannian distance be- γ r v
¥ F é
tween the shapes of § and , and is the optimal ¥
Procrustes rotation to match to § .
T M
RX
cos ρ
A diagrammatic view of a section of the pre-shape sphere, showing the partial tangent plane coordinates
q and the full Procrustes tangent plane coordinates
# À Ä
Ç
q q
. Note that the inverse projection from to j is
given by
# À %
Ä
½
' 3 4 5
q q o q
j ¨ ) Ö j ) 1 ;
L (2) ,
The rays from the origin in Procrustes tangent space Hence an icon for partial Procrustes tangent coordi-
¥ ê
l ¼ j correspond to minimal geodesics in shape space. nates is given by ¨ . 70 71 -0.54-0.48 0.07 0.10 -0.10 -0.06 -0.18 -0.10 0.14 0.17 0.145 ë 0.170 • • • • • • • • • • • • • • • • • • • • • • • • • •• • •• ••• • • • • • • • ••• • •• ••• • •• • •• • • • •• • ••• ••• • •••• •••• •• • ••• • • ••• • ••••• • ••••• • • •• • •••••• •••• •• •••• • 180 s • ••••• •••• • • •••• • •• ••• • •••• • • • •••• • ••••• ••••• •• ••••• • •• • • • •••• • • • ••• •• • • •• ••• •• • •• • • ••• •• • • • •• • • ••• •••• • ••• • ••• • •• •
• • • • • • • • • • • • 165 ••• •• • • ••• • •• • •• • •• • •• • • • ••• • • • • • • ••• •• • • • • • • • •• •• • •• • •• • •• ••• • • • • •• -0.48 ••••••• • •••••••• •••• •• • • •••• •• • •••••• • • •••••• • •••••••• •••••••• •••••• • •••••• ••••• •• •• ••• •••• •••• • x1 • ••• • •••• • • • • • ••• • •••• • • •••• • • •• • • • • •• • • • • •• • • •• • • •• • •• • • •• • •• • •• • •• • •• • • • • • • ••• • • • •• • -0.54 • • • • • • • • • • • • •• • • • • • • • • • • • •• • • • • • • • • ••••• • • ••••• ••• •• • •• •• • • • •••• ••••• • ••••• • • •••• ••• • • •••• • • ••• • ••• •
0.6 • • • • • • • • • • • • • • • • •• • • • • • • ••• • ••• x2 • •• • •• •• • • •• •••• •• •• •••• • • • ••• • ••• • • •• 0.52 4 •••••• ••••••• • •••• • •••• • • •• ••• • •••• • • •••• • ••••• • •• •• • •••••• •• • •• ••••• • • • • • • • • • • • • • • •• • •• • • • • • •• • •• •• • • •• • • • • • • •• • • • • ••• 0.46 •• •• ••• •• ••• •• • • •• • •• ••• ••• • •• • • • + 0.10 • • • • • • • • • • • • +++ ••••• • • •• •••• ••••• • x3 • •• • • ••• • •• •••••• •••• ••• •• ••• ••• • • • • • ••• • • • • • •• •• • ++ •••••• ••• • •• ••••• • ••• • •••• • •••• • •••••• •••••• • •••• • ••• •• •• ••• • ••• • ++ • • • ••• • • • • • •• • •• •• • •• •• • • • • • • • • • ++ 0.07 + • • • • • • • • • • • • • • •• • • • • •• • • ++ • ••• • • •• •••• • • •••• •• •• • • • • •••• •• • •• • ••• • • • • •••• • • •• •• •• •• • • • •• • •• • ••• 0.4 + ••• •• • •• • • • • • • •• • •• • • • • •• •• • •• • •• •••• ••• ••• •• ••• ••• •• • •••• ••••••• • •••••• • •••• •• • •••• • •••• ••• ••• ••• • • •• •• 0.02 ++ • •• • • •• • •• • • • •• • x4 • ••• • •• • • • • ••• • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -0.02 • ••• • ••• •• • • •• • • • •• • ••• • ••• • ••• • •• • • ••• • • • • • ••• • ••• • •••• • •••••• •••• • •••••• ••• • ••• • • • ••••• • • • • ••• • •• •••• •• • •• -0.06 • • •• • ••••• • • • •••• ••• ••• ••• • • •••• • •••••• •• •• • • • ••• • ••• • • •• • • • ••• •• • ••• •• • • • • • • • • ••• •• • ••• • x5 ••• •• •••• • • •• • • • ••• •• • • • ••• • ••• • • • • • • • • • • • • • •
-0.10 • • • • • • • • • • • • 0.2 ++ + • • • • • • • • • • • • +++++ ++ 0.0 +++++ ++++++ ••• • • • • • ••• ••• • • • • •• • • •• • • • • • • •• • ++++ +++ •••• ••• ••••••• • •••••• • •• •• ••• • •• •• •• • •• •• • x6 •••••• •••••• • • •• ••••• •• • ••• • • •• •• • • ••••• • + • •••••• • ••• • •••••• • ••• • •• •••••• • ••••• •• ••• • • ••••• • • •••• • • •• • • ••• • •• •• • ••• 5 3 •• • ••• • •• • •• •• • ••• • •• • •• •• • •• • • •• • • • •• •• • • •• • • • • • • • • • • •• • • •• -0.05
-0.10 ••• • • •• • ••• • • • • • • • • • • •• • •• • • •• • • •• • • •• • • •• • • •• • ••• •• • •• • ••• ••• •••• • • • ••• • •• ••• • • ••• • y1 •••••• ••••• •••••• •• •• ••• •• • ••••• ••••• •• ••••• • • •• • • •••• •• •••• • • •••••• ••• • • •• •• • • •••• •• • • •• ••• ••• •• • • ••• • • ••• ••• • • ••• •••• •• • •• •• • • • • •••• • •• • •• 0.0
-0.18 • • • • • • • • • • • • •• • • •• • • • • • • • • • • •• • • • • • •
• • • • • • • • • • • • -0.10 •••• • •••• •• • • •••• • •• • ••• • ••• • • •••• • ••• • • • • •••• • • • • •• + + ••••••• • ••••••• •• •••• ••••••• • •••••• • •••••• ••••••• • ••••••• • y2 ••••• •• • ••••• • •• •• •• • ••• •• + + •••• • ••• •• • •• • • • •• • • • • ••• •• • • •• • • •• • • • • •• • +++ +++ • • • • • • • • • • • • • • •• • • •• • • •• ++++ ++++ • • • • • • • • • • • • -0.18 +++ +++ ì •• • • • • • • • • • • • • • • • • • • • • • • + ++ 0.17 • • • • • • • • • • • • ++ + +++ ••••• • • •••• • ••• • ••••• • • ••• •• •••• • ••• •• •• • •• • •••• • •••• • •• • • • •••• •• + ••• • • ••• ••• • •• • •• • • ••• •• •• • •••• • •• ••• y3 • •••• • ••• • ••• • •••••• •••• •• • •••• • ••• • • • • • • •• •• • ••• • •••• ••• •• • •••• ••• •• • • •• •
-0.2 • • • • • • • • • • • •
0.14 • • • • • • • • • • • • 1 2 •• •• • • • •• • • •• •• •• • • • • • • • •••• •••• • • •••• • • •• • • • •• • • ••• • ••• • •••• • ••••• •• • • • ••• • • • •• •••• • • •••• • • •••• ••••• • • •• •• • • •••• ••••• • •• •• • • •••• • •• • ••••• • •• •• •• • • • ••• ••• •• • •• • •• ••• ••• •• • ••• • • • •• 0.44 •••• •••• •••• • • •• ••• • • •• • •• •• •••• • ••• •• • • y4 • •• • ••• •• • ••• •• • • •• • •• • •• • •• • •• • • • • • • • • • •• •
• • • • • • • • • • • • 0.34 • • • • • • • • •• • • • • • • • • • • • • • • • ••••• ••• • • •••• •• •••• • • •••• •• •• ••••• • • ••••• • ••••• ••••• • ••••• •• • ••• 0.170 •• • ••• • •• • • • • • • ••• • • • •• • • • • • •• • • • • •• •• •• • • •• • • •• •• • • • • • y5 • • -0.4 + •• ••• • •• • • • •••• ••• • • • •• • • •• • • • ••• •• ••• •• ••• • ••• • •• • • ••• • ++++ + • ••• • • • • • • • • • • • • •• •• • • • • • • • • •• • • • • • • • •• •• • ++++++ • • • • • • • • • • • • ++ 0.145 •• • • ••• • • ••• ••• • • ••• •••• •• • ••• •• •• • • • • •••• • •• • • •• ••• • •• •• • • • • •• • ••• ••• •• • ••• • • • •• ••• •• •••• • • •••• • •••• • •• • • ••• • ••• •••• • • •• • • • ••• •••• • •• •• • -0.44 ••• •• •• • •• • • • • • • • •• • • • •• • •• •• • • •• • • •• •• • • •• • • •••• •• • •• •••• • y6 ••• • • • • • • • •• •• • • • • •• •• • • • •• • • • •
6 • • • • • • • • • • • • • • ë •• • • •• • • ••
165 180 0.46 0.52 -0.020.02 -0.05 0.0 -0.18 -0.10 0.34 0.44 -0.47 -0.44 -0.47 -0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Pairwise scatter plots for centroid size ( ) and the
3 u Icons for partial Procrustes tangent coordinates for 2 coordinates of icons for the partial Procrustes the T2 vertebral data (Small group). tangent coordinates for the T2 vertebral data (Small group).
72 73
The Euclidean norm of a point q in the partial Pro-
crustes tangent space is equal to the full Procrustes m
distance from the original configuration j correspond-
q
¼ Ö ing to to an icon of the pole l , i.e.
For practical purposes this means that standard mul-
Æ
Ç 3
q tivariate statistical techniques in tangent space will be
m ¼
¨ j l Ö 1 good approximations to non-Euclidean shape meth- ods, provided the data are not too highly dispersed. Important point: This result means that standard mul- tivariate methods in tangent space which involve cal- Full Procrustes tangent coordinates
culating distances to the pole Ö will be equivalent to non-Euclidean shape methods which require the full
¥ An alternative tangent space is obtained by allowing
%
¼ Ö
Procrustes distance to the icon l . Also, if and
>
¿
j
¥ scaling by of the pre-shape in the matching
% q
are close in shape, and q and are the tangent
) Ö ) to the pole . In the above section
plane coordinates, then
Æ Ç Æ
¥ ¥ ¥ ¥ ¥ ¥
' 3 3 3
% % % %
q q
í í ¹ í 1
Â
) ) ) ) (3)
74
¡ O - sample covariance matrix of some tangent co-
Shape variability #
ordinates q ,
Õ
"
¡
, # ' # '
q q q q
O
( (
Overall measure ¼
¨
# $ %
Ô
%
#
q q
*
( Õ
where ¨ .
Õ
%
"
Æ Ç Æ Ø
Ç
·
3
#
é § ¨ Ô 1
)
;
&
O
$ % Ö
# - eigenvectors of : principal components (PCs),
Ã
³ ³ ³
>
%
1 1 è
with eigenvalues 1
)
Æ
Ç Ç î ï ð ñ î
> >
@
é § ¨ 1 ¡
PC score for the p th individual on the th PC is:
@
ò
' 3 3 3 3 3 ó 3
# & #
Æ Ç ï ð ñ î
& q q p
(
¼
¨ Ö ¨ 1 1 1 Ô ¨ 1 1 1
> > >
, ,
é § ¨ 1 ?
¡ PC summary of the data in the tangent space is ¡
PCA in tangent space to shape space è
"
ò
3
# # & &
q q o
(
¨ Ö
& $ %
Ø
·
Â
s '
# #
- PCA of Procrustes residuals ¨
3 3
p
¨ 1 1 1 Ô Ã
# for . q
- PCA of Procrustes tangent coordinates ,
Ø s (project # so to obtain part that is orthogonal to and ¡ Standardized PC scores:
its rotations) Ã
%
³ @
Ø
ô ò
# & # & 3 3 3 3 3 ó
s
# #
p
q
&
)
¨ ¨ 1 1 1 Ô ¨ 1 1 1 1 í
- NB for observations close to we have L
r , , Ã 75 76
Mouse vertebra example: (PC1 = 69%) Mouse vertebra example: Procrustes registration for display
0.6 4 + ++++ ++++ +++
0.4 + ++ 0.6 0.6
0.2 + + +++++++ ++++++ ++ ++++ 4 5 3 • •
0.4 0.4 õ
0.0 5 3 0.2 • • 0.2 • • + ++ ++ +++++ ++++ ++++++ +++++ ++ ++++ 0.0 0.0 + + 1 2 -0.2 1 2 • • • • -0.2 -0.2 -0.4 -0.4 -0.4 ++++++ • • +++++++ 6
6 -0.6 -0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 (a) (b)
77 78 Mouse vertebra example: (PC1 = 69%)
Bookstein registration for display
¡ Important:
If using Bookstein superimposition to calcuate O then 0.8 4 0.8 strong correlations can be induced.....can lead to mis-
0.6 • 0.6 • 5 3
0.4 0.4 leading PCs • • • •
0.2 1 2 0.2 • • • • 0.0 0.0 No problem with Procrustes registration, Kent and Mar- -0.2 • -0.2 • 6
-0.4 -0.4 dia (1997)
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 (a) (b)
79 80
T2 small vertebra outlines PC1: 65% 0.2 4 ++++ +++++ 0.3 0.3 0.3 0.3 0.3 0.3 0.3 +++ •••• •• •••• ••••• ••••• • • ++ •••• •••• ••• •• ••• ••• ••• 0.1 •• 0.1 •• 0.1 •• 0.1 •• 0.1 0.1 •• 0.1 •• + •••••••••• •••••••• •••••• •••••• •••••• •• • ••• •••••• ••• ••••• •• • •••• •• ••• •• ••• •• •••• ••• ••• ••• + + ••• •••• ••• ••• •••• ••• ••••• ••• •••• ••• •••• •• ••••• •• ••• ••• ••• •••• ••• •••• ••• ••• •••• ••• ••• •••• ••• •••• ••• ••••• •••• ••••• •••• ••••• ••• •••••• ••• •••••• •••• ••••••• •••• ••••••• -0.1 ••••• -0.1 ••••• -0.1 ••••• -0.1 •••••• -0.1 •••••• -0.1 •••••• -0.1 •••••• 0.1 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 ö 0.3 -0.3 -0.1 0.1 0.3 3 5 + +++ ++++++++ ++++++ +++ 2 + ++ 0.0 + ++++ ++++++ ++++ ++ +++++ +++++ ++++ 1 0.3 0.3 0.3 0.3 0.3 0.3 0.3 -0.1 •••• •••• •••• ••• ••• ••• •••• •••• •••• •••• •• •• •• •• 0.1 •• 0.1 •• 0.1 •• 0.1 •• 0.1 •• 0.1 •• 0.1 •• •••• •• •••• •• •••••• •••••• •••••• ••• •• ••••• ••• •• ••• •• ••• •• ••• •• ••• •• • ••• •• • •••• •• ••••• ••• •••• ••• ••••• ••• ••••• ••• •••• •• •••• •• •••• •• + •••• ••• •••• ••• ••• ••• ••• ••• ••• •••• ••• •••• ••• ••••• +++ ••• ••••••• ••• •••••• ••• •••••• ••• •••••• ••• ••••• ••• •••• ••• •••• ++ -0.1 ••••• -0.1 •••••• -0.1 •••••• -0.1 •••••• -0.1 ••••••• -0.1 •• •••• -0.1 •• •••• +++++++ •• •• -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 0.3 -0.3 -0.1 0.1 ö 0.3 -0.3 -0.1 0.1 0.3 6 -0.2
-0.2 -0.1 0.0 0.1 0.2
Æ Ç
> >
§ ¨ 1 Ò é PC2: 9%
81 82 0.3 0.3 0.3 0.3
+++ +++++ 0.2 + ++ 0.2 0.2 0.2 ++++++++ ++++ +++++++++ ++++++•+••++•••••• ++++++ ++++++ ++•••••++ +++•••+•+++••• + + + + +•••••+•+ ++••+++•••• + + + + 0.1 +• • 0.1 +•• 0.1 0.1 • ++•••••+••+• •+•+• ++••• + + + + • •• ••+••+•••++•+••••+••+••••• •+++•+•+•+ ++•• ++ + ++ + • •• ••+•+++++ +++++••••• +•+•+•+++•+• •+• ++ + ++ + ••••• • •••+•+++++++ +++++•••• + •++•+•+•+•+•+ +••+ ++ ++ ++ ++ •••+••• ••++++++ ++++•••• +++++++••+•+•+ •+•+• + + + + + + 0.0 • +•+ ++ + 0.0 •++•+• + 0.0 ++ 0.0 ++ •+•+++++++ ++++•••• ••• ••+•+•••••• +•+•+• + + + + ••+••++++ ++++++••••• • • +•++ +•+• ++ ++ ++ ++ ••+++ +++++++••• •••+•+ ++•+•+•+•++ + +++ + +++ •++•+••++ + ++++•••• •+•+ • ••++• ++ + ++ + •+•+•+•+•+ +++•++•+•+•• •+•+ • +••+•+••+•+ ++ +++ ++ +++ -0.1 •+••++ +++++•+••++••++••+•++• -0.1 •+• +• +• +• + + -0.1 + +++ -0.1 + +++ +•+•++•+•++••• +++•+••+••+••++••++ ++++ ++++ -0.2 -0.2 -0.2 -0.2 -0.3 -0.3 -0.3 -0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 (a) (b) (a) (b)
83 84
Pairwise plots:
0.04 0.05 0.06 0.07 0.08 0.09 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 + + + + 620 + + + + + + + + 600 + + + + + + + + + + + + + + + + + + + +
+ + + + 580 + + + + + + + + + + + + + + + + + + +++ + + + + + +
+ + + + + + + + ø 560 ÷ + + + + s + + + + + + + + + + + + 540
520 ø 500
+ + + + 480
+ + + +
0.09 + + + + + + + + + + + + + + + +
0.08 + + + +
+ + + + 0.07 dist + + + + + + + + + + + +
0.06 + + + + + + + + + + + + + + + + + + + +
0.05 + + + + ++ + + + + + + + + + + + + + +
0.04 + + + + + + + ++++ + + + + 0.2 +++ 0.2 + + + + ++ + 2 ++ +++ + + + + ++ ++ + + + +
+ 1 ++ + + + + + + + + + + + ++ + + + + + ++ + + + + + + + + + + + + + ++ ++ ù + + + + + ++ +++ score 1 0 + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + +
+ + + + ++ + + + + ++ + + + + + + + + 0.0 + 0.0 ++ + + + + -1 +++++ ++ ++ ++ + + + + + + + + + + ++ + + +++++ -2 ++++ ++ +++ + 1.5 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++ 1.0 ++++++ +++++++ + + + + 0.5 ++ + + + + + + + + + + + + + + + + + + ++ + + + + + +
0.0 ù -0.2 -0.2 + + + score 2 + + + + +
-0.5 + + + + + + + +
-1.0 + + + + + + + +
-1.5 + + + + + + + + + + + +
+ + + + ÿ -0.2 0.0 0.2 -0.2 0.0 0.2 2 + + + + + + + + + + + + 1 + + + ++ + + + + + + +
+ + + + + + + + + + + + ù + + + + score 3 (a) (b) 0 + + + + ++ +++ + + + + + ++ + + + + + + + + + + + + + + + + + + ++ -1 + + + + + + + +
+ + + + -2
ú þ
û û ü ý ü 480 500 û 520 540 560 580 600 620 -2 -1 0 1 2 -2 -1 0 1 2
Size, shape distance, PC scores 1, 2, 3
85 86 Pairwise plots: Digit 3 data
30 30 30 30 30 30 Size, shape distance, PC1: 50%, PC2: 15%, PC3: 20 20 20 20 20 20 10 10 10 10 10 10 0 0 0 0 0 0
-10 -10 -10 -10 -10 -10 13%, PC4: 8%, PC5: 4%
þ ü 0.2 0.3 0.4 0.5 0.6 0.7 -2 -1 0 1 ü2 -2 -1 0 1 2 -30 -30 -30 -30 -30 -30 • • • • • • • • • • • • ø -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 55
•• • • • • • • • • • • • • • • • • 50 • • • • • • • • • • • • £ • • •• •• • • • • • • 45 s • • • • • • • • • • • •
• • • • •• • • • • • • £ • • •• • • • • •• • • • • • • • • • • • • 40 • • • • • • • • • • •• • • • •• • • • • • • • • • • • • • • •• • • • • • •• •• • • • • • • • • • • • • • • • • • • 35 30 30 30 30 30 30 • • • • • • • • • • • • • • • • • •
20 20 20 20 20 20 • • • • • • 0.7 10 10 10 10 10 10 0.6 0 0 0 0 0 0
0.5 • • • • • • •• • • • • dist -10 -10 -10 -10 -10 -10 • • • • • • 0.4 • • • • • • • • • • • • • • • • • • • • • • • • 0.3 • • • • • • • • • • • • •• • • • • • • • • • • •• • • • • • • • • • • • •• • • • • •• • • • • • • • • -30 -30 -30 -30 -30 -30 0.2 • •• • • • • • ••••• • • •• •• • • •• • • • • • • •• •• •• • • •• • • • • •• • •• • • •• •• • •• • •• • • • • • • • • •
-30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 • • • • • • ¤
• • • • • • 3 ÿ
• • • • • • 2 • • • • • • • • • • • • 1 • • •• •• •• ù • • • • • • • • • • • • • • • • • • • • • •• • score 1 • • • • • •• • •• • • • • •• • • • •• • •• • • • • • • • • • • 0 • • • • • •••• • • •• • • • • • • • • • • • • • •• •• • • • •• • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • •
30 30 30 30 30 30 • • • • • •
• • • • • • -1 • • • • • • 20 20 20 20 20 20
• • • • • • -2 ÿ 10 10 10 10 10 10 2 • • • • • • •• • • • • • • • • • • 0 0 0 0 0 0 • • • • • • •• • • • • • • • • • • 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -10 -10 -10 -10 -10 -10
¥ • • • • • • ù 0 • • • • •• • • • score 2 • •• • • • • • • •• • • •• •• • • • • • • • • • • • • • • • •• • • ••• • • • •• • • • • • •• • • • • • • • • • • • • • • • • • • • • • • •• • •• • •
-1 • • • • • • • • • • • • -30 -30 -30 -30 -30 -30
-30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -2 • • • • • •
• • •• • • • • •• • • ÿ 2 • • • • • • • • • • • •
• • • • • • 1 • • • •• • • • • • • • •• • • • • • • • • • • • • •• •• • • ù score 3 •• • •
• • • • • • • • • • • • • • • • • • 0
¢ ¢ ¢ ¢ ¢ ¢ ••• • • • • ••• • • • • •• • • • • • • • • • • • • • • •• • ••
30 30 30 30 30 30 • • • ••• •• • • • • • • • • •
¡ ¡ ¡ ¡ ¡ ¡ • • • • • •
• • •• • • • • • • • • -1 20 20 20 20 20 20 • • • • • • • • • • • • • • • • • • • • • • • •
10 10 10 10 10 10
ÿ • • • • • • 0 0 0 0 0 0 2 • • • • • • • • • • • • • • • • • •
1 • • • • • • -10 -10 -10 -10 -10 -10 • • • • • •• •• • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 0 • • • • • • • • • • •• • • • • • • • • • •• • • • • • •• • • • score 4 •• • • • • • • • -30 -30 -30 -30 -30 -30 • • • • • • • • • • • • • •• • • • -1 • • • • • • -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 • • • • • • • • • • • • • • • • • -2
• • • • • • ÿ • • • • •• • • • •• • • • • • • 2
• • • • • • • • •• • • § 1 • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • ••• • • • • • • • • • • • • • • • • • • • • •• • 0
30 30 30 30 30 30 • • • • • • • • • • • • • • • • • • • • • • • • score 5 • • •• • ••• • • • • •• ••• •• • • • • • • •• • • 20 20 20 20 20 20 • • • • • • • • • •• • •• • • • • • • • • • • -1
10 10 10 10 10 10 • • • • • • • • • • • • -2
0 0 0 0 0 0
ú þ
¦ ü ý 35 40 45 û 50 55 -2 -1 0 1 2 3 -1 0 1 2 -2 -1 0 1 2 -10 -10 -10 -10 -10 -10 -30 -30 -30 -30 -30 -30
-30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30 -30 -10 0 10 20 30
Æ Ç
t
>
é § ¨ 1 Ñ
87 88
0.4 ++ + 0.4 + + + 0.4 + + + 0.4 + + 0.4 + + 0.4 + + 0.4 + + + + + + + + + + + ++ +++ + + + + + ++ ++ + + + + + + ++ + + + + + + + 0.0 0.0 0.0 + 0.0 + 0.0 0.0 0.0 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
-0.4 + -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4
0.4 ++ 0.4 + + 0.4 + + 0.4 + + 0.4 + + 0.4 + 0.4 + ++ + + + + + + + + + + ++ + + + + + ++ ++ ++ + + +
+ + + + + + + + + ++ + ++ 0.0 + 0.0 + 0.0 + 0.0 + 0.0 + 0.0 + 0.0 + + + + + + + + + + + + + + + + + + + + + + + + + + + +
-0.4 + -0.4 -0.4 -0.4 -0.4 -0.4 -0.4
-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 0.4 0.4 0.4 + + • • • • ++ + + + + • • +• • •• • + • ++++••• ++••• + + + + • • • + + •••++++ ++ •• + + + + + • • + • • ++ • • • • • • + + • + 0.2 + + + 0.2 • ++ 0.2 • + • • • + +• + • •+ • • +•+++ 0.4 + 0.4 + 0.4 + + 0.4 + + 0.4 + + + 0.4 + + + 0.4 + + +• + • + • ••• + + + + + + + + + +• • +•• + • •+ • + ++ + + + + • •+• • + + +• + ++ + ++ + ++ ++ ++ + •• • • + + + • + •+ + + + • • + + + + + • • • • + + +• + + 0.0 0.0 0.0 0.0 0.0 0.0 0.0 + + + + + ++ + +++ 0.0 • + + 0.0 + + 0.0 + + + + • + + + + + + + + + + + • •••++ + ++++••• + + + + + + + + + + + + + ++ + + + + + + •+ -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 • + + + ++ -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 • + + + + + + -0.2 • -0.2 • • + -0.2 + + + + + • • • • + + + •• + • • + • + + ++ + + + •• + + + + + • • +• • + + + + + • + • •• • • • • + • • • -0.4 • -0.4 • • -0.4 • + + + • • 0.4 0.4 0.4 0.4 0.4 0.4 + 0.4 + + + + + + + + + + + + + + + ++ + + + + + + + + +++ + +++ ++ ++ ++ ++ ++ + + + + + + + + + + -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 0.0 0.0 0.0 + 0.0 0.0 0.0 + 0.0 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4
0.4 + 0.4 + + 0.4 + + 0.4 + + 0.4 + + 0.4 + + 0.4 + + ++ + ++ ++ + + + + + + + + + + + + + + ++ +++ +++ ++++ + + + + + + + + + 0.0 + + 0.0 + 0.0 + 0.0 + 0.0 + 0.0 + 0.0 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4
89 90
HIGHER DIMENSIONS
¥ ¥ Ordinary Procrustes analysis (match % to - cen- PERTURBATION MODEL:
tred)....Minimize: )
0 ¥ ¥ ¥ ¥
ð
3 ' ' 3
% %
¿
)
¼
¨ Ö
)
Â
,
) )
Ø
¥
# # # #
¿ o © o #
¼
¨ Ö
Solution: ,
>
Ö ¨ Ø
à Can estimate the shape of by GPA (generalized Pro-
h
i
¼ ¨
crustes analysis): by minimizing Ã
where
¨ h h
¥ ¥ ¥ ¥
Õ
i 3 3 i 4
% %
¼ ¼
¨ ¤
"
Æ Ø
Ç ¥
)
)
3
#
)
¨ ¦
# $ % ¤
with a diagonal ¤ matrix. Furthermore,
. /
¥ ¥
%
¼
3
¿
. /
¨
)
¥ ¥
Ã
%
%
¼ Least squares approach. Iterative algorithm needed
Ã
t
The minimized sum of squares is: for ¤ dimensions
Æ Ç
¥ ¥ ¥ ¥ ¥
% 3 % 3
¨
) )
) ) )
91 92 60 60 60 60 40 40 40 40 • • ••• •• ••• ••• ••• •• 20 • • 20 •• 20 20 ••••••• ••• •• • ••• ••••• •••••• •• ••••• ••• 0 • 0 0 0 •• ••• •••• •••• • • • •• •• ••••••• •••••••••• •• • •••• •••••• ••••• •••• •••• ••• ••• •• -20 • -20 -20 -20 -60 -60 -60 -60
-60 -20 0 20 40 60 -60 -20 0 20 40 60 -60 -20 0 20 40 60 -60 -20 0 20 40 60 (a) (b) (a) (b) 60 60 40 40 •• •• •••• 20 • 20 •••• ••••• ••• •••• 0 0 •••• •••••• • ••• ••••• ••••• ••• •••• •••• -20 -20 -60 -60
-60 -20 0 20 40 60 -60 -20 0 20 40 60 (c) (c)
Male macaques Female macaques
93 0.6 0.6
•+ +• 0.2 0.2 +• +• +• +• +• +• ++• + + •+ +• • • • -0.2 -0.2 -0.6 -0.6
-0.6 -0.2 0.2 0.6 -0.6 -0.2 0.2 0.6 (a) (b) 0.6
+•
0.2 +• +•
•+ +• •+ +• -0.2 -0.6
-0.6 -0.2 0.2 0.6 (c)
Male ( ) Female (+) PC1 (47%) for Males: +/- 9 s.d.
94 95
Hierarchy of shape spaces Original Configuration
remove translation Different approaches to inference: Helmertized/Centred
remove scale remove rotation 1. Marginal/offset distributions 2. Conditional distributions Pre-shape Size-and-shape 3. Directly specified in shape space 4. Distributions in a tangent space remove rotation remove scale
remove reflection 5. Structural models in the tangent space Shape
remove reflection Reflection size-and-shape
Reflection shape
96 97 Preshape distributions (2D)
Shape distributions: offset normal approach
% 3 3
¨ j 1 1 1 j
2D - complex notation: j where
-
> 3 ½ ½
(
¨ j j ¨ j ¨ j
j [ ]
, ,
- -
3 ¡ complex Bingham (Kent, 1994)
3
Û
ô
½
ç j ¨ j j
# À
j ¨ ç j is Hermitian. NB: ç so suitable for 2 shape analysis. 1 2 1 (a) (b) NB: MLE of modal shape is identical to the PROCRUSTES
(least squares) mean Ø
¡ complex Watson (special case of c. Bingham) Mean triangle with independent isotropic zero mean
normal perturbations with variance ) .
Û
Ø Ø
ô
½ ½
ç j ¨ j j
98 99
DIFFUSIONS AND DISTRIBUTIONS Offset normal density (wrt uniform measure) (Mardia
and Dryden, 1989; Dryden and Mardia, 1991, 1992) Diffusion of points in Euclidean shape (WS Kendall):
Û
Æ Æ Æ
¥ ¥
Ø Ø
¥ ¥
t t
# # ' # 3 3 3
p
' 3 ' ' 3
o
¨ æ ¨ 1 1 1 1
t
¹ ¹
,
, ,
)
;
Ø
p
¨ j )
where ) , Ornstein-Uhlenbeck process for Euclidean points
r
Ø Ø Ø
# '
p
*
(
j ¨
) )
× and independent size and shape diffusions [with ran-
&
&
º º
Æ Æ
& '
# $
*
# #
¨
2 ¨
is the Laguerre polynomial. dom time change for shape: size ) ]. Com-
9
r
S
puter algebra package developed through this
Parameters: work.
Ø
ó : 2k-4 mean shape parameters Size and shape, and shape diffusions in (Le). : concentration parameter. Shape density at time : (from previous slide).
100 101 Controls:
Maximum likelihood based inference 0.6
•• ••••••
0.4 •
0.2 • •••••••• •••••• ••• • ••••• ••••••• • •• ••• ••• •••••• ••••• ••• 0.0 • ••• •••• •••••• •••• •••• •• • •• •••• ••••• • • ••••• ••• •••••• • ••• • • ••••• -0.2 -0.4 -0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Schizophrenia patients: 0.6
•• •••• •••• 0.4
0.2 •• ••••• •••••• •••• • ••••• • ••• • •• • • •• ••••• •• •••• • •••
0.0 • • •••• • ••••• •••••• •••• ••••• • •••• •••••• •••••• • •••• • ••••• • •• ••• ••••• -0.2 -0.4 -0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
102 103
Schizophrenia study (Bookstein, 1996; INFERENCE: Multivariate normal model in the tan-
Dryden and Mardia, 1998) gent space (to pooled mean)
£
%
A Ô ¨
¨ landmarks in 2D: Controls and
, ,
Hotelling’s ) test ¨
Ô Schizophrenia patients
,
| |
)
·
% 3 3 3 3
# &
q
)
@
3 3 3 3 3
%
p
1 1 1 Ô ¨ 1 1 1 Ô
Isotropic offset normal model: independent individu- ¨ all mutually indepen-
, , als dent and common covariance) matrices
Inference: maximum likelihood
·
3
q (
( - sample means
3 O - sample covariance matrices 1.0 CS Mahalanobis distance squared:
0.5
0
· ·
' 3
S ' q
C q
( ( ( (
B
¼
¨
) ;
CS x x
0.0
C t
% '
C %
o o O
S S B
Ô Ô Ô Ô where ¨
SC SC r ) C ) SC S -0.5 SC
S C Under l equal mean shapes... S
-1.0
' '
% %
-1.0 -0.5 0.0 0.5 1.0 o
Ô Ô Ô Ô §
0
,
) )
¨
)
t
% % '
o o
Ô Ô Ô Ô §
) )
ï ï
t
Õ Õ
%
> > >
A 1 ¨ 1 ?
LR test ) .
,
< ; ;
) ) =
Monte Carlo permutation test, p-value: 0.038. N §
under l . [ = dimension of the shape space] S 104 105 Pairwise plots:
Gorilla (female/male): Size, shape distance, PC scores in direction of mean difference
0.03 0.05 0.07 -2 -1 0 1 2 -3 -2 -1 0 1 2 £ t mm m m m mm m m m m mm mm m m m m m m m m mm m 300
mmmm m mm m mmmmmm m mmmmmm mmmmmmm mmm mmm mmmmmm m
Ñ ¤ ¨ ¨ m m m m mm mmm m mmm m m m m m mmm m landmarks in dimensions mmmmm m mmmm mmm mmmm mmmmm mmm mm m mmmmm 280 m m mm m m m m m m m m
s m m m mmm m mm m mm mm m mm m
$ $ $
260
$
$
$
$
$ $
$ $ $
$ $ $ f f f f f $ f f f f f f f f ff ffff f f f f
f ffffff ff f fffff f f fffff f fff f fff f ff f ff f fff f ff fff$ fff f $ ff f f fff f f f $ ff f ff f ff ff f f f ff f ff f ff ffff f ff ff f f ff f f f f ff ff f f ffff f 240 f ff f f ff f fffff f ff ff fff f ffff f f ffff f f f ffff ff f m f m m f f m f m f m
f m f m m f m f$ f m m f
ff mm f f mm m mf f m f f m f f mm mm f f $ 0.07 ff mm f fm m m m f f mm f f f mf m m mf f
f m f m m $ f m f f m m f !
t f m f m m f m f f m m f
$
$ $
f dist $ f f f f f $
f f mm f m m m f m f $ mm f f f fmm mmf f %
f m f m m f m f $ f m m f
% > 3 $ 0.05 f fff mmm f fff mm m mmm ff f f fm mf mff ffmfmfm m mm fmm$ ff f
f f mm f f m m mm f $ f mfm f mff m m mff
$ $
f m m f$ f mm mm f f m f f mfm f mmff $ $
ff mm f f m m m f m f mm f f$ f mf m mf f
¨ A Ô ¨ Ô ff ff mmm f ff mmmm m f fmmm f f m mmmf f f f f f mff mmm mfmmmf ff ff ff mmm m f fff mmm mm f mff mmmf f ff mff f f mmm mmmff f f 0.03 m m m m m m m m m m m m 4
) m m m m m m mmmm mmmm m m mm m m mm mmm mm mmmm m mmmm m m mmm mmm m m m m mm mm mm mmm 2 mmmmm mmmm m m m mmmmm mmmm m mmmmm mmmmm
m m m m m$ m mmmm mmmmm mmm mm m mmmmm mm mm m mmmmmm
m m m m $ m m f m f m m f m f mf m f 0
f m mf score 9 m f m f f m m $ f
$ f f f$ f f f
f f f f $ f f
$ $
$
f f f f f f f f f f f $ f $
f ffff ff ff f f ff ff f f ff $ f f f ff f fff f $ f f ff f $ f f f f f f f ff ffffff fffffffff f ff ff ff ff f f f f ff fffff f f f ff fffff ff fffff f f -2 f f f ff f f ff f f f f ff f$ f f
fff f f f f f fff f f f f f f f f fff f
£
t t
f f f f f f f f f $ f ff
& $ $
f f $ f f f f
$ $ '
2
$ $ f f f f f f f $ f f f f f
f f f $ f f f
ff ff m mff f f f f ff ff f m mf f ff f f f fff f f m mf f f f f f
1 ¨ ¨ § ff fff m ff f f mf ff f f f m f f ffmf f ff ff f m m f ff f f % ff fff m f m f f f ff f ffff f m fmf f f ff fffff mf fmf ff f 0 , m m m m m m
m mmmm mmm m m mmm score 11 mmmm mmm mmmm$
$ $ ff f f mmmm mmffmmf mf m f$ ff mmmmm m m mmmm fff f mfmf ff mm m mmmfm fmf f -1 f m m f f m m f f m m f f m mmm m fmm mm f mmmm mm m mmmmf f mmmm m mmmm mf mmm m m m m mm mm m m m m m
-2 m mm mm m m mm m mm m mm m m mm $
f f f f f f 3 f f f f f f
2 $
ff m f f m f f m fmf f f m m f $ f
$ $
$ $ f f f f f f $ fff f f f f ff fff f fff f ff f ff f f f ff f fff f ff$ f f
fff f f f ff f f ff f f f f f f 1 $
f ff f f f f f f f f $ f f f f f f f fff f f f f $ f f f ff fff f ff ff m f m f ff m m f f $ f fm mf f
0 t " f f mmmm mm f m f m f f mm m mmmf mf $ score 2 f f m mm mm m f m f m m mfm $ f m m m m f f mm mm f ff mmmmm mmmmmffm m f f f f mmmm mmm mmf m f f f mfmf mm m mmmfmf f mmm mm mf mmm m f mm m m mmmm f m fmmm m m mmm f -1 m mmm mmm m m mm m m m m mmm m mm
-2
1 Ò ¨ mmm mm m mm m mm mmm m m m The test statistic is f m f m f m m f f m m f m m m m m m
2 mmmm mm mmm m mmmm mmmm m m m m m mmm m
mm mm m mmm m m m mm m mm
m m m m m m
m m mm mm m m mm $ mm 1 m mm m m m mmm m m mm mmm
f m f m f m m f m f m $ f $
f f m fm f ff$ m fm f m ff mff
$ $
% v # > > > >
m m m m m m
0 f f f f f f
mm m m m m m $ m m m m
f f f f $ f f $ f f f f $ f f ff ff m mm m f mmmff fm f f fmm mm m mmmf m f $ ff m mf mfmf f m mmmff f f
fff fff m fmfmff f f f f fff fff mm m mf f f ff f f mmf fff f f score 12 $ mmf f fff f f
1 Ò ¨ 1
f f f f f f fff f f f f ff f f f $
-1 f m fm f m f m f m f ff m f m f f f m m m f f$ m f f mf f
and $
f ff f ff f ff ff f f f f $ ff f , -2 ff f f f f f f f f f
) f f f f f f
-3 f f f f f f
$ $ N ff f f f f f f f f f f 2
fff m ff f m f $ f f m m f f f fm f f f ff m
$
$
$ $ $
f f f f $ f f
1 $
f fff f fff f ffff f f f $ ff f fff f f f fff f $ $
f f m f f m f f m m f f f m f $ f f m % fff ff mmm f ffmffmm f ff ff mfm m m mf mf f f f $ f mmm f ffff f ffmmff f m 0 f ffffff m m mm fmmmfmmf mfmf f ff ffffff fmmmmm m mmmfmfmmfmff f f m mmmmffmmf f$ fff f ff ff fffmmmfmmmm f mmmmm mmmmm f f mmmmmm m mm m mf mmmmm f mf mmmm score 1 f f m m mf m f ff mm m mf f m m f f fmf m -1 m mmm m mm mm mm mmm mm m m m m m m m m m m m -2 m m m m m m mm m m
m m m m m m -3 ' 240260280300 -2 0 ' 2 4 -2 -1 0 1 2 3 -3 -2 -1 0 1 2
106 107
Goodall’s F test:
( + If then
Comparing several groups: ANOVA
Õ Õ
Æ Ø Ø
Ç
% 3
Õ Õ
)
)
< ;
=
)
< <
Õ Õ
à Ã
¨
Æ Ø Æ * Ø
< ) ¥ )
= Ç
Ç Balanced analysis of variance with independent ran-
3 3
# % #
o
# $ % # $ %
* *
@
¥ ¥
) )
<
=
Õ
3 3 3 3 3
& % &
)
1 1 ¨ 1 1 1 Ô -
dom samples 1 from
à Ã
,
ï ï
-
Õ Õ
Under l :
S - Ô
Ô groups, each of size .
F
)
< ; I
=
N
Ø Ø &
¡ Schizophrenia data: Let be the group full Procrustes means and is the Ã
overà all pooled full Procrustes mean shape. A suitable
£
t
A ¤ ¨
¨ landmarks in dimensions test statistic is
,
Õ .
Æ Ø Ø
Ç
3
#
# $ %
*
3
% )
'
Õ
.
Ô ¨ Ô ¨
Õ
¨ Ô Ô Ô - 1
, ,
Æ Ø
¥
Ç
à Ã
)
,
' & # 3 &
& $ % # $ %
* *
-
Ô
)
,
£
t t t
'
Ã
§ ¨ ¨
Under l equal mean shapes:
S
! !
> >
ï ï
¨ 1 Ñ 1 Ñ í 1
, and
Õ . Õ . Õ
% %
, , ,
) ) )
N + ,
F F
; I ; I N
Permutation test: p-value = 0.04 and reject l for large values of the statistic.
S
¡ Hotelling’s ) test p-value = 0.66
108 109
By analogy with analysis of variance we can write
Õ
"
Complex Watson inference: "
Ø Ø
# 3 u & 3
o
¹ j ¹ ¨
) )
# $ % & $ %
à Ã
à Ã
Ø
Õ
3 3 5 3
%
1 1 1 j
Two independent random samples j from
Õ
/
u % 3 3 u 5 3
" "
1 1 1
Ø /
and from . We wish to test be-
# 3 u & 3
o o
) ¹ j ) ¹ æ
# $ % & $ %
tween
à Ã
/
Ø / Ø /
Ø Ø
% 3
¨ l ¨ Ó
l where is the overall MLE of if the two groups are
S
Ã
# 0
Ã
Ø Ø Ø
t
æ
y 1 { z
> pooled, and is analogous to the between sum of
where ¨ , (i.e. rep-
resents the shape corresponding to the modal pre- squares. Since,
Õ
Ø
" "
Ø Ø
shape . For large it follows that
,
# 3 u & 3
o
Õ
v %
) ¹ j ) ¹ í )
t
Õ
# $ % & $ %
F F
)
à Ã
; I ; I
à Ã
" "
Ø /
# 3 u & 3 ,
o
Õ
v
) ¹ j ) ¹ í )
t
# $ % & $ %
F F
)
I I
; it follows that
Õ
" "
Ø Ø
3 u & 3 '
and we also have #
o
æ ¨ ¹ j ¹
) )
Õ
# $ % & $ %
à Ã
à Ã
" "
Ø /
3 u 3 ,
# &
o
Õ
v
¹ j ¹ í
) ) )
t
Õ
# $ % & $ %
F F
) )
; I ; I
à Ã
" "
Ø /
3 ' u 3 ,
# &
v
¹ j ¹ í 1
) ) )
t
# $ % & $ %
)
;
à Ã
110 111
Bayesian approach to inference
2
Õ
z 3 % 3 3
7 1 1 1 7 ¨
º
2 2
3
Õ
% 3 3 3 z 3
7 1 1 1 7
1
4
2 2 2
3
l
Õ
3 3 3 z 3
Therefore, under we have %
S
7 1 1 1 7
t
'
o
Ô ¤ æ
Õ
¨
Ø /
Ø
# 3 u & 3
o
# $ % & $ %
* *
)
#
¹ j ¹
) )
e.g. Data j complex Watson( , known )
Ø
à Ã
Prior complex Bingham ( known)
Õ
v v
í
F F
) ) )
; ; I ; I
N
and so we reject l for large values of . Using S
Taylor series expansions for large concentrations)
% % %
Ø /
3 3
Ø Ø
3
o
Õ Õ
æ í Ô ¤ ) ¹
z % 3 3 z % 3 3
; ; ;
j 1 1 1 j ( j 1 1 1 j
à Ã
Õ
º
6 7
Û 5
"
Ø Ø Ø
and so for large the test statistic is equivalent to Ø
½ ½ ½
# 8 9
o #
)
( j j
:
# $ %
the two sample test statistic of Goodall (1991).
Û
Ø Ø
½
3
o
¨
Conjugate prior
MAP: dominant eigenvector of o
r 112 EDMA (Euclidean distance matrix analysis) [Lele, 91+,
Stoyan, 91]
£ ¦ £
¥
: form distance matrix ( matrix of pairs of
inter-landmark distances (ILDs))
Ø
Estimate : population form distance matrix
Ø / @ £
3 u 3 3 3 3 3
& & & &
2 + ¨ 1 1 1
) . Then
,
)
=
0
' u ' u 3
o
2 ; 2 < ; < ¨
) ) ) ) ) ) )
< ; <
; (4)
r
)
= Ø Ø / /
' '
o
¨ ; < ; <
) ) ) < ; .
Moment estimator
The smoothed Procrustes mean of the T2 Small data:
v
/ .
=
0 0
³ ³
' >
> >
©
¨ ) ) ) 1
¨ ¨ 1
< ; < ; <
(Top left) , (top right) , (bottom left) ;
,
³ ³
> > >
1 ¨
¨ , (bottom right) . , , Removes bias.
Estimate of mean reflection size-and-shape
? ? =
0
3 3
; < ; <
§ ¨
) Ã 113 114
EDMA-II (Lele and Cole, 1995)
EDMA-I test (Lele and Richtsmeier, 1991)
A
Ù and estimates of average form distance matrix Ã
Form ratio distance matrix forà each group
* *
0 ¥ ¥
3
# & # & # &
¨ 1 (5) r Scale by group size measure
Test statistic:
/
Ø / Ø /
0 0
& 3 # & 3 3
# = Largest entry in arithmetic difference of scaled
¨ @ @
& # &
# (6)
r
à à Ã
à matrices
N N Use bootstrap procedures. More powerful than EDMA-I
115 116
Rao and Suryawanshi (1996)
B
¥
: form log-distance matrix
shape log-distance matrix is
B B B
¥ ¥
½
' ( 3
¼
¨
, ,
%
t
" "
B B
¥
;
(
# &
¨ 1
£ £
'
# $ %
& $ # %
, Average reflection shape
Û
B
Ø
0
½
§ 1
Average form log-distance matrix is Ã
Õ
Å
"
B
Ø Æ
, 3 3
# %
¨
# $ % C
)
Ô
Ã
Æ
% 3 #
where is the distance between landmarks
)
¥
% #
and for the p th object . )
Average reflection size-and-shape
Û
B
Ø
0
§ 1 Ã 117
For small variations estimates of mean shape or size- and-shape are all very similar...(Kent, 1994)
¡ SIZE-AND-SHAPE Distance based (+): Invariance under translation and rotation (not scale) Landmarks not necessarily needed (eg. maximum breadth) Perturbation model:
Consistent estimation under general normal models
Ø
¥
# # # 3 3 3 3
o © o # p
¼
¨ Ö ¨ 1 1 1 Ô , Distance based (-): ,
Invariant under reflections ¡ ALLOMETRY
Visualization not straightforward The relationship of shape given size
A choice of metric for averaging needs to be made
118 119 Microfossils: Bookstein’s (1996) Microfossils
0.32 0.36 0.40 0.44 • • • • 9.2
• • X • • • • ••• 9.0 • • • • • 8.8 slog • • • • • • • • • • • • 8.6 • • • • • • • • 8.4 • • 8.2 • • 0.44 • • • • • • • • • • • • • • • • • •• • • • •• • • • 0.40 • • • U • • •
0.36 • •
• • • • 0.32 • • • • 0.75
0.8 • • • • • • • • • • • • • • • • • 100• • • • • 0.65 88• • • • • 87• • • V 88• 88• • •
84• • • Y • • 0.55 0.7 100• • • • • 92• 65• 75• 84• 0.45 8.2 8.4 8.6 8.8 9.0 D 9.2 0.45 0.55 0.65 0.75 76• 71•
74• 84• V V 0.6 71•
72• 67• Regression:
64•
Å Å
h
0.5
1 3 i 1
% %
¿ o ¿
65• o
¨ ¨
C C ) 60• ) Tthe fitted values (with standard errors) are
0.4
t t
1 % > > > % ' > > > >
0.2 0.3 0.4 0.5 0.6 ¿
1 Ò Ò 1 ¨ 1 1
U ¨ , ,
t t
Ã
1 ' > > > >
Ã
¿
1 Ò 1 A ¨ 1 1
¨ and
,
) )
Ã
Ã
h h
t
Å
i i
o
¨ ¨
8 8
versus i r
, Significant linear relationship between and . C 120 121
T2 Small mouse vertebrae data
-0.10 -0.05 0.0 0.05 -0.02 0.0 0.02 • • •
• • • 185 • • •
• • • 180 • • • • • • •• •• • • • • • • • • • • • • • 175 s • • • • • • • • • •• • • • • • • •
170 Shapes package in R: • • • • • • • •• • • • 165 • • • •• •• • • • • • • • • • • • 0.05 • •• • • • • • • • • • • • • • • • • •• •• 0.0 • • • • •• • PC1 • • http://www.cran-r-project.org • • • • •• • • • • • • -0.05 • • •
-0.10 • • • • • • • • • • • • • • • • • •
• • • • • • 0.02 ••• • • • • • • • • • • • Library of shape analysis routines. • • • • • •
• • • 0.0 • • • • • • • • • • • PC2 • • • • • • • -0.02 • • • • • • -0.04 • • • • • • • • • • • • Also see: • • • • • • 0.02 • • • • • • • • • • • • • • • • • •
0.0 • • • PC3 •• • • • ••• •• • • •• • • • • • • •• • • • • • -0.02
• • • • • 165 170 175 180 185 -0.04 -0.02 0.0 0.02 http://www.maths.nott.ac.uk/ ild/shapes
NB: Approx. linear relationship between PC 1 and centroid size.
122 123 DEFORMATIONS AND THIN-PLATE SPLINES
The thin-plate spline is the most natural interpolant in two dimensions because it minimizes the amount of bending in transforming between two configurations, which can also be considered a roughness penalty.
The theory of which was developed by Duchon (1976)
¦ t
and Meinguet (1979). Consider the landmarks
,
@ £
3 3 3
Session III: &
1 1 1
¨ , on the first figure mapped exactly
,
£
u 3 3 3
#
p
1 1 1
into ¨ , on the second figure, i.e. there
,
£ t
¡ Deformations are interpolation constraints,
¡
@ £
t
& & 3 s 3 3 3 3 3
Shape in images u
; ;
¨ E ¨ ¨ 1 1 1
(7)
, ,
¡
@ £
Temporal shape
& % & 3 & 3 3 3 3
¨ E E ¼ ¨ 1 1 1
and we write E
,
¡
) Shape Regression for the two dimensional deformation. Let
¡ Discussion
*
3
% u % u u
¼ ¼
¨ ¨
1 1 1 1 1 1
) )
* £ ¦
t so that and are both matrices.
A pair of thin-plate splines (PTPS) is given by the
bivariate function
% 3
¼
E ¨ E E
)
ô ò
3
o o
¼
¨ (8)
124 125
£
# & # ' &
¦ £ ¦
t
ò
¨
' % 3 3 ' 3
3 where and is the -vector of
,
¨ 1 1 1 ¼
where is ,
ones. The matrix F
and
,
J
I
Å
,
3 > 3
G
) H
> >
Í
¨ C ¨ K
(9) ¼
> 3 >
,
¨ 1
> >
¼
is symmetric positive definite and so the inverse ex-
£ ¦
t " t
ô
o
The parameters of the mapping are ists, provided the inverse of exists. Hence,
¦ £ ¦ £
t t t t
F F F F
3
and . There are interpola- %
,
I J I J * I J * I J
; ,
tion constraints in Equation (7), and we introduce six %
GH GH GH GH
ô
3
> > > >
¨ ¨
K K K K
¼ ¼ ;
more constraints in order for the bending energy in ,
> > > >
¼
Equation (14) below to be defined: ¼
%
say. Writing the partition of as
;
> 3 >
¼ ¼
¨ 1
¨ (10)
% % %
,
%
)
3
%
¨ L
The pair of thin-plate splines which satisfy the con- ;
) ) M
straints of Equation (10) are called natural thin-plate )
% %
£ ¦ £
where is , it follows that
splines. Equations (7) and (10) can be re-written in
% %
*
F F F
matrix form
¨
I J I J * I J
ô
%
*
¼
% 3 3
,
¿ ¿
L ¨ ¨
) (12)
GH GH GH
ô
3
> > >
)
¼
¨
K K K
¼ ¼
M Ã
(11) Ã
,
%
> > >
¼ ¼
giving the parameter values for the mapping. If
;
£ '
exists, then we have and so the rank of the bending energy matrix is A .
% % % % N N % N % N %
' 3
¼ ¼
¨
; ; ; ; ;
% N % N % N % %
It can be proved that the transformation of Equation
3
¼ ¼ ¼
¨ ¨
) )
; ;
; (13)
N % N %
(8) minimizes the total bending energy of all possible
' 3
¼
¨
) )
*
; ;
N interpolating functions mapping from to , where
3
where ¨ , using for example Rao (1973,p39).
, the total bending energy is given by
P
%
¿ ¿
¨
Using Equations (12) and (13) we see that and E
) Ã
are generalized least squares estimators, andÃ
& & &
) ) )
"
E E E
)
t
) ) )
u
o o
> % 3 '
¿ ¿
2 1
¼
¨ ) ) 1
u u
& $ % Q Q R S
T T T
2 2
)
) )
=
à Ã
Mardia et al. (1991) gave the expressions for the case
T T T
T (15)
when is singular. A simple proof is given by Kent and Mardia (1994a).
The minimized total bending energy is given by,
£ ¦ £ O
The matrix æ is called the bending energy
% %
. / . /
* *
P
¼ ¼
¨ ¨ 1
matrix where E %
% (16)
O ¨ 1 æ (14)
In calculating a deformation grid we do not want to There are three constraints on the bending energy see any more bending locally than is necessary and matrix
also do not want to see bending where there are no
> 3 >
¼ ¼
æ ¨ æ ¨
O O
, data.
Early transformation grids modelling six stages through life (from Medawar, 1944).
Early transformation grids of human profiles
126 127 TRANSFORMATION GRIDS
Following from the original ideas of D’Arcy Thompson A regular square grid is drawn over the first figure and (1917) we can produce similar transformation grids, at each point where two lines on the grid meet # the
using a pair of thin-plate splines for the deformation corresponding position in the second figure is calcu- *
from configuration matrices to . lated using a pair of thin-plate splines transformation
u 3 3 3
# #
p
E ¨ 1 1 1 Ô U Ô U
¨ , where is the number , of junctions or crossing points on the grid. The junc- tion points are joined with lines in the same order as in the first figure, to give a deformed grid over the sec- ond figure. The pair of thin-plate splines can be used to produce a transformation grid, say from a regular square grid on the first figure to a deformed grid on the second figure. The resulting interpolant produces transformation grids that ‘bend’ as little as possible. We can think of each square in the deformation as be- ing deformed into a quadrilateral (with four shape pa- rameters). The PTPS minimizes the local variation of these small quadrilaterals with respect to their neigh-
(a) (b) bours.
128
Consider describing the square to kite transformation which
was considered by Bookstein (1989) and Mardia and Goodall
£
t
¤ ¨
(1993). Given ¨ points in dimensions the ma-
* It is found that
F
trices andF are given by
> > > >
J J
I I
> > >
3
J J
¼
1 Ò ? L
¨ (17)
J J
' > > > > ' > > > >
,
t
1 Ñ A 1 Ñ A 1 Ñ A 1 Ñ A
G G
M
' > ' >
, , , ,
*
1 ?
G G
3
, ,
t
¨ GH ¨ GH 1
> ' > '
K K
1 ? ô
> 3
, ,
¨ ¨ +
t
> >
)
1 ?
, ,
¨ and so the pair of thin-plate splines is given by E
We have here F
% 3
J
I
E E ¼
, where
>
¾
J
)
J
G
J
>
3
%
¾
G
E ¨
(18)
3
,
G
v
¨
GH
>
¾
&
"
K
t
> > ' ' &
o
E ¨ 1 Ñ A 1
>
¾
, ,
& $ %
)
t " ! t
>
¾
! ¨ 1 A ¨ ¨
where ¨ and
,
t "
t Note that Equation (18) is as expected, because there is no
1 Ò Ò
. In this case, the bendingF energy matrix is
change in the direction. The affine part of the defor-
,
I J
' '
J mation is the identity transformation.
J
, , , ,
G
% %
' '
G
> >
, , , ,
æ ¨ ¨ 1 Ñ A G 1
O H
' '
,
K
, , , ,
' '
, , , ,
129 2 2
1 • 1 •
0 • • 0 • •
-1 • -1 • -2 -2
-2 -1 0 1 2 -2 -1 0 1 2
2 2
1 • • 1 • •
0 0 • -1 • • -1 • -2 -2
-2 -1 0 1 2 -2 -1 0 1 2
on the first figure at a different orientation, then the de- formed grid does appear to be different, even though the Transformation grids for the square (left column) to kite transformation is the same. This effect is seen in the Figure
(right column) (after Bookstein, 1989). In the second row
m
where both figures have been rotated clockwise by ? in m the same figures as in the first row have been rotated by ? the second row. and the deformed grid does look different, even though the transformation is the same.
We consider Thompson-like grids for this example (above). A regular square grid is placed on the first figure and de- formed into the curved grid on the kite figure. We see that the top and bottom most points are moved downwards with respect to the other two points. If the regular grid is drawn 0.6 0.6
0.4 • 0.4 •
0.2 • •• 0.2 • •• • • 0.0 • 0.0 • • • • • • • • • • • • -0.2 • -0.2 • • 13000 13000 13000 -0.4 -0.4 12000 12000 12000
-0.6 -0.6 • • 11000 11000 11000 • • • • •
10000 10000 • • 10000 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 •••• • •• • • • • •• • • • (a) (b) 9000 9000 • • 9000 •• •• •
8000 8000 8000 • •
6000 7000 8000 9000 10000 11000 12000 6000 7000 8000 9000 10000 11000 12000 6000 7000 8000 9000 10000 11000 12000
A thin-plate spline transformation grid between the con- 13000 13000 trol mean shape estimate and the schizophrenia mean 12000 12000 11000 11000 • • • • shape estimate. 10000 10000 • • • •• 9000 • 9000 • • • • •• •• • • • •• • 8000 8000
6000 7000 8000 9000 10000 11000 12000 6000 7000 8000 9000 10000 11000 12000 (left) We see a square grid drawn on the estimate of
mean shape for the Control group in the schizophrenia
¦
t !
> >
U
¨ A ¨ Ñ Ò
study. Here there are Ô junctions A series of grids showing the shape changes in the
£ A
and there are ¨ landmarks. (right) we see the , skull of some sooty mangabey monkeys schizophrenia mean shape estimate and the grid of new points obtained from the PTPS transformation. It is quite clear that there is a shape change in the centre of the brain, around landmarks 1, 9 and 13. 130 131 Here we have labelled the eigenvalues and eigenvec-
PRINCIPAL AND PARTIAL WARPS ³ tors in this order (with % as the smallest eigenvalue corresponding to the first principal warp) to follow Book- Bookstein (1989, 1991)’s principal and partial warps stein’s (1996b) labelling of the order of the warps. The are useful for decomposing the thin-plate spline trans- principal warps do not depend on the second figure * . formations into a series of large scale and small scale The principal warps will be used to construct an or- components. thogonal basis for re-expressing the thin-plate spline
transformations. The principal warp deformations are
Consider the pair of thin-plate splines transformation univariate functions of two dimensional , and so could
£
u 4
4 be displayed as surfaces above the plane or as con-
© © )
from ) to , which interpolates the
* £ ¦
t tour maps. Alternatively one could plot the transfor-
points to ( ) matrices. An eigen-decomposition
ô ô
u % & 3 &
o
£ ¦ £
¼
mation grids from to ¨
æ
O @
of the bending energy matrix of Equation (14) )
ô ô
%
³ ³
³ for each , for particular values of and . Note that
% y y y
P
1 1 )
has non-zero eigenvalues 1 )
; the principal warps are orthonormal.
% 3 3 3
P
Ö 1 1 1 Ö
with corresponding eigenvectors Ö . The
)
;
% 3 3 3
P
£
Ö Ö 1 1 1 Ö
eigenvectors are called the princi- ' A
) The partial warps are defined as the set of
;
@ £
3 3 3 '
pal warp eigenvectors and the eigenvalues are called &
¨ 1 1 1 A
bivariate functions é , where
,
* ³ * ³ the bending energies. The functions,
ò
& & & & & &
&
¼ ¼ ¼
é ¨ Ö ¨ Ö Ö 1
@ £
ò
& 3 3 3 ' 3
&
@ *
¼
¨ Ö ¨ 1 1 1 A ,
The th partial warp scores for (from ) are de-
ò
' % 3 3 '
¨ 1 1 1
are the principal warps, where fined as
* @ £
¼
& % 3 ó & & 3 3 3 ' 3
. ó
¼ ¼
¨ Ö ¨ 1 1 1 A
, ) 132
and so there are two scores for each partial warp. 0.6 0.6 0.6 0.4 0.4 0.4
0.2 • • 0.2 • • 0.2 • •
0.0 • 0.0 • 0.0 •• •
• •• • • • •
Since -0.2 •• -0.2 • • -0.2 • •
P
-0.4 -0.4 -0.4
"
-0.6 -0.6 -0.6
;
ò
& 3
¼
¨ é
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 V 0.2 0.4 0.6
$ % & we see that the non-affine part of the pair of thin-plate 0.6 0.6
splines transformation can be decomposed into the 0.4 0.4 @ 0.2 • • 0.2 • • sum of the partial warps. The th partial warp cor- •
0.0 • 0.0 • • • • • • • responds largely to the movement of the landmarks -0.2 • -0.2 • •
-0.4 -0.4 @
which are the most highly weighted in the th prin- -0.6 -0.6 @ -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 cipal warp. The th partial warp scores indicate the
contribution of the @ th principal warp to the deforma- *
tion from the source to the target , in each of the The five principal warps for the the pooled mean shape Cartesian axes. of the gorillas
133
0.6 0.6 0.6
0.4 0.4 0.4
0.2 • • 0.2 • • 0.2 • •
• 0.0 • 0.0 • 0.0 • •• • • •• -0.2 • • -0.2 • • -0.2 • • 0.6 0.6 • • -0.4 -0.4 -0.4 0.4 0.4 -0.6 -0.6 -0.6
• ö 0.2 • 0.2 • • -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 •
• 0.0 •• 0.0 •• • • • • •• -0.2 -0.2 0.6 0.6 0.6 -0.4 -0.4 0.4 0.4 0.4 -0.6 -0.6 0.2 • • 0.2 • • 0.2 • •
0.0 • 0.0 • 0.0 • -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 •• •• ••
-0.2 • -0.2 -0.2 (a) (b) • • • •• • • • -0.4 -0.4 -0.4 -0.6 -0.6 -0.6
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 ö 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
A thin-plate spline transformation grid between a fe- male and a male gorilla skull midline. Affine and partial warps for Gorilla (Female to male mean shapes)
134 135
RELATIVE WARPS
Principal component analysis with non-Euclidean met- rics 0.06 0.06 0.06 f f f f f mf f
0.04 f f 0.04 f f 0.04
Æ m ð f ff f f f m mmm m m è
f f ff m 3 f f ff mm m
0.02 0.02 0.02 m f
m f f ff m m m f
f m m © f f f f f f f m f f mm Define the pseudo-metric space as the real f m m f f mm f fff m mmf m m mmf mm m m m m f m m f f f m mf m f m 0.0 m f 0.0 f f 0.0 f f m m f f fm f f f m fmm m m m fmmm m f f ff f ó m m m mm m f m f f m m m m f f
-0.02 -0.02 -0.02 ff m f m f f -vectors with pseudo-metric given by m m mm m f f f f m m mm -0.04 -0.04 -0.04
-0.06 -0.06 m -0.06
ð
m m Æ
% 3 % ' % ' 3
-0.06 -0.04 -0.02 0.0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0.0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0.0 0.02 0.04 0.06
¼
2 2 ¨ W 2 2 2 2
;
) ) )
% ó 2
where 2 and are -vectors, and is a gener-
;
) 0.06 0.06 0.06
0.04 0.04 0.04 alized inverse (or inverse if it exists) of the positive m mm f m mm f fm 0.02 mff mm m 0.02 ff 0.02 m m f m f f f mf mf f f f fff mm m mmff mff f m m ff mm f f ff f mm mmf fmff fmf mf f f mmm m semi-definite matrix . f f f f ff ff f m f fmff 0.0 m 0.0 f mf 0.0 m f m mf m mf fm mm mmm mffmmf mf f f f f f fff m mm mfm fm f f f f m f mf m m mm f f m m m m ff m f f m f f m -0.02 -0.02 -0.02 mf -0.04 -0.04 -0.04
-0.06 -0.06 -0.06 The Moore–Penrose inverse is a suitable choice of -0.06 -0.04 -0.02 0.0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0.0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0.0 0.02 0.04 0.06
generalized inverse. If is the population covariance
Æ
ð
3
% %
2 2 2
matrix of 2 and , then is the Maha-
) )
Affine scores and the partial warp scores for female lanobis distance. The norm of a vector 2 in the metric
(f) and male (m) gorilla skulls. space is
%
Æ ð
ð
3 >
¼
2 ¨ 2 ¨ 2 2 1
)
; L
We could carry out statistical inference in the metric
+
space rather than in the usual Euclidean ¨ 136 137 space (after Bookstein, 1995, 1996b; Kent, personal Appendix). The first few PCs in the metric space (with
communication; Mardia, 1977, 1995). A simple way loadings given by the eigenvectors of
Æ
ð
è
% %
4 3 u 4
©
to proceed is to transform from 2 to
%
) )
; L ; L
9
2
) in Euclidean space. For example, consider
; L
ó ) can be useful for interpretation, emphasizing a dif-
principal component analysis (PCA) of Ô centred -
Õ
3 3
% ferent aspect of the sample variability than the usual
1 1 1 2
vectors 2 in the metric space. Transform-
%
u #
# PCA in Euclidean space. If our analysis is carried out
¨ 2
ing to ) the principal component (PC)
; L loadings are the eigenvectors of in the pseudo-metric space, then we say the our anal-
Õ ysis has been carried out with respect to .
"
,
u # u
#
¼
¨ 1
J
# $ %
Ô If a random sample of shapes is available, then one
@
ó & 3 ¨
Denote the eigenvectors ( -vectors) of as Ö may wish to examine the structure of the within group
J
J
3 3 ó ó y '
1 1 Ô
1 (assuming ), with corresponding variability in the tangent space to shape space. We
, ,
³ @
& 3 3 3 ó
1 1 1
eigenvalues ¨ . The principal com- have already seen PCA with respect to the Euclidean
,
J @
ponent scores for the th PC on the p th individual are metric, but an alternative is the method of relative warps.
%
Relative warps are PCs with respect to the bending
s # & u # # 3 3 3 ó
& & p
¼ ¼
¨ Ö ¨ Ö 2 ¨ 1 1 1 1
)
; L
,
J J
energy or inverse bending energy metrics in the shape
So, the (unnormalized) PC loadings on the original tangent space.
%
&
Ö
data are ) which are the eigenvectors of
; L
%
Õ
% %
J
#
# $ % #
*
Õ
¨ 2 2 ¼
) )
, where (us- Ô
L ; L
; Consider a random sample of shapes represented
9 9
Õ
3 3
%
q q
1 1
ing standard linear algebra, e.g. Mardia et al., 1979, by Procrustes tangent coordinates 1 (each is
£ Ø
t t
³ ³
'
3 3
%
#
1 1 æ
a -vector), where the pole is chosen to be an with 1 the eigenvalues of with corre-
)
)
;
% 3 3
average pre-shape such as from the full Procrustes #
1 1 1 Ö
sponding eigenvectors Ö . The eigenvec-
)
;
3 3
mean. The sample covariance matrix in the tangent %
#
ç 1 1 1 ç
tors are called the relative warps. The
O )
plane is denoted by and the sample covariance ;
# relative warp scores are
¨ +
matrix of the centred tangent coordinates 2
£ ¦ £
t t
) X
0
# 3 3 3
@ £
q p t "
¸
l ¼ ¨ 1 1 1 Ô
3 3 3 ' 3 3 3
# & & #
is denoted by . p
¼
,
¨ ç æ 2 ¨ 1 1 1 ¨ 1 1 1 Ô 1
)
;
L
, ,
In our examples we have used the covariance matrix )
of the Procrustes fit coordinates. The bending energy O
matrix is calculated for the average shape æ and then Important remark: The relative warps and the rel-
¨ + æ O
the tensor product is taken to give æ , ative warp scores are useful tools for describing the
£ ¦ £ £
t t t "
) )
X '
which is a matrix of rank . We write non-linear shape variation in a dataset. In particular
@
æ æ
; for a generalized inverse of (e.g. the Moore– )
) the effect of the th relative warp can be viewed by Penrose generalized inverse).
plotting
0 %
Ø Z
ô
3
&
&
) We consider PCA in the tangent space with respect )
¼
l æ ç Y
L L
to a power of the bending energy matrix, in particular ) 0
for various values of ô , where æ
with respect to O .
#
0 0
0 0
"
) ³
;
æ ¸ æ
) )
) )
¼
; ;
¨ Ö ; Ö 1 Let the non-zero eigenvalues of æ
;
L L
L L
;
) )
$ %
% 3 3 % 3 3
#
)
;
1 1 1 Y ç 1 1 1 ç
be Y with corresponding eigenvectors
) ) ;
and ;
#
0
0
"
)
³
; The procedure for PCA with respect to the bending
3
)
¼
;
æ ¨ Ö Ö
)
;
; L
;
;
L
1
o
$ %
)
¨
; energy requires and emphasizes large scale ,
variability. PCA with respect to the inverse bend-
1 '
ing energy requires ¨ and emphasizes small
,
1 >
S
æ ¨ +
scale variability. If ¨ , then we take
)
£ ¦ £
t t as the identity matrix and the procedure) is exactly the same as PCA of the Procrustes tangent 0.6 0.6
0.4 0.4
1 > coordinates. Bookstein (1996b) has called the ¨ • • 0.2 • 0.2 •
• case PCA with respect to the Procrustes metric. 0.0 • • • 0.0 •• • • • • •
-0.2 • -0.2 -0.4 -0.4 -0.6 -0.6 0.06 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
0.04 m f f (a) (b) f 0.02 f m m f ff ff mfffff ff f mm mm mf f f f 0.0 m m f m m m m mmmm f fmfff mf mm m f 1
m m Relative warps: ¨ , -0.02 m
-0.06-0.06 -0.02 0.0 0.020.040.06
m
0.15 m m m 0.10 f m mm f f m f f 0.05 mm m m f f mmf f f f
0.0 m fm f f f f mm mf ff f mm m m fm f f f mm f f f -0.05 mm f f 0.6 0.6 m 0.4 0.4
+ -0.15 + + -0.15 -0.050.0 0.050.100.15 0.2 + 0.2 + +
0.0 ++ 0.0 ++ + + + +
-0.2 -0.2 + + -0.4 -0.4 0.6 0.6 -0.6 -0.6 0.4 • 0.4 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.2 • 0.2 • ••
(a) (b) 0.0 • • • 0.0 • • • • • -0.2 • -0.2 • • -0.4 -0.4 -0.6 -0.6
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 (a) (b) Deformation grids for the two uniform/affine vectors
for the gorilla data.
1 '
Relative warps: ¨ ,
138 139 f m
0.05 f f m m f f f ffffffffff f m m m f f m fm 0.0 m mf f mm mm ff mm m mm f f m mm f f f m mf m
-0.05
[ ¡ -0.05 [ 0.0 0.05 Prior distribution for configuration
Geometrical object description = SHAPE + REGISTRATION 0.6 0.6
0.4 • 0.4 where REGISTRATION = • • 0.2 0.2 • LOCATION, ROTATION and SCALE
0.0 • • • 0.0 • • • • • • •• -0.2 • -0.2 ¡ Use training data to estimate any parameters -0.4 -0.4 -0.6 -0.6
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
1 >
Relative warps: ¨
140 141
¡ Deformable templates: Grenander and colleagues
¡ Point distribution models (PDM) Cootes, Taylor, et al. CASE STUDY:
¡ Bayesian approach Object recognition: face images
- Prior model for object shape and registration using [example from Mardia, McCulloch, Dryden and John- SHAPE ANALYSIS son, 1997]
- Likelihood for features measuring goodness-of-fit (fea- ture density)
Bayes Theorem Posterior inference
142 143
LANDMARKS or FEATURES
¡
3 u
2 Grey level image +
¡ Scale space features (e.g. Val Johnson, Duke; Stephen Pizer et al, UNC Chapel Hill, USA)
Convolution of image with isotropic bivariate Gaussian
kernel at a succession of ‘scales’ ( )
3 u 3 u
2 2
3 3
) )
o o
¨
u
J J
9 9
2
T T
) )
3 u
T T
2 ¨
Æ Æ
<
= =
' % 3 u ' , %
F ] ]
+ 2
; I
t
<
= =
= \
z
Q
) )
)
Use 2D FFT
¡ ‘Medialness’ : Laplacian of blurred scale space im- age
144
¡ Feature density (likelihood)
£ !
Pilot study - Face Identification: Choose ¨ land-
marks on the medialness image at scales 8, 11, 13
ñ a a ñ b b
×
/ Å ^ Å Ú . /
_
× × × ×
3
<
F
@ (
I
= `
# $ % º
¡ Johnson et al. (1997) motivate this as mimicking a human observer.
¡ Features are treated as independent ¡ High medialness at feature high density
¡ Treat non-feature grey levels as independent, uni- formly distributed (like a human observer ignoring those
pixels).
¡ Parameters # need to be specified
145 146 Registration parameters
¡ Location Original raw face data
Ø
3
)
9 9
9
Ø
3
)
J J J
200 200 ¡ 150 150 Rotation • 1 2 • 1 22 1 2 1 2 11 22
111 5 222 d
d • e y y 6 6 555 77
100 5 100 5
• • 6 5555 7 7 w 6 7
6 7 6666 5 77 7
c 3 7 6 4
c 63 89 7
g h ) •• • • f 9 3 3 98 4 g 44 18 h 3 9898889 4 3 4 3 3 9899 44
50 50 3 3 89 44 ¡ Isotropic scale 0 0 0 50 100 150 200 0 50 100 150 200
x x
3
¿
)
Ë
Ë
3 3 3 3 c 3 c 3 3
Hyperparameters estimated
J J
9 9 Ë from training data (10 faces) Ë
147 148
Bookstein registered data ¡ Least squares Procrustes approach
• • • • • •••• •• 0.4 • •• •••• • • • 0.0
0.2 • • • ••• • ••• • •• • -0.5 •• • •• • • • • • • • • • • • • i • • • • • v • • •• • • •• • • 0.0 • •• • • • • • • -1.0 • •
•• -0.2 •• • • • • ••••• • ••• • • • •• •• •••• • • • • • •••• • • • • •• • • • • • • • • •• ••• • • • • • • • • • • • ••• • -0.4 -1.5
-0.5 0.0 0.5 -0.4 -0.2 0.0 0.2 0.4 u
149 150 ¡ First five PCs (explaining 54.4, 29.9, 6.0, 3.7, 2.7% ¡ Vector plot from mean to 3 S.D.s for first three PCs of variability in shape).
• • • • • • • • • • • • • • 0.4 0.4 0.4 0.4 0.4 0.4 0.4 • • • • • • • • • • • • • • • • •
• • • • 0.4 0.4 0.0 • • 0.0 • • 0.0 0.0 0.0 0.0 0.0 • • • • • • • • • • • • • • • • • • • •• • • •• • -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 0.2 0.2 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 • • • • • •
0.0 0.0
• • • • • • • • • • • 0.4 • • 0.4 • 0.4 0.4 0.4 0.4 0.4 • • • • • • • • • • • • • • • • • • • • • • • • • • • 0.0 0.0 0.0 0.0 0.0 0.0 0.0
• -0.4 -0.4 • • • • • • • • • • • • • • • • • • • • • -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4
-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.2 0.4 -0.4 0.0 0.2 0.4
• • • • • • • • • • • • • • 0.4 0.4 0.4 0.4 0.4 0.4 0.4 • • • • • • •
• • • • • • • • • • • • • • 0.0 0.0 0.0 0.0 0.0 0.0 0.0 • • • • • • • • • • • • • • • • • • • • • • • -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4
-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 • • • • 0.4 0.4 0.2 0.2 • • • • • • • • • • • • • • 0.4 0.4 0.4 0.4 0.4 0.4 0.4 • • • • • • • • • • • • •
• • • • • • • • • • • • • • 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 • • • • • • • • • • • • • • • • • • • • • • -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 • • • • • •
-0.4 -0.4
• • • • • • • • • • • • • • 0.4 0.4 0.4 0.4 0.4 0.4 0.4 -0.4 0.0 0.2 0.4 -0.4 0.0 0.2 0.4 • • • • • • •
• • • • • • • • • • • • • • 0.0 0.0 0.0 0.0 0.0 0.0 0.0 • • • • • • • • • • • • • • • • • • • • • -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4
-0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4 -0.4 0.0 0.4
151 152
¡ .... FACE PRIOR
Assume registration and shape independent ¡ Draw samples from posterior using MCMC
Multivariate normal prior model (configuration density): ¡ Object recognition: maximize posterior to obtain most
likely configuration given the image
^ Å Ú . /
Ø Ø w
ô ô
z z 3 3 3 3 % 3 3
¿
¨ 1 1 1 è
J
¡
9
m
b b m
m
a m
a Straightforward Metropolis-Hastings algorithm
j p
j k
j n
} j k
× s ×
=
a b =
=
=
* r
¸
<
~
=
) l q
) l
) l o
) l
;
= = = = <
=
\ \ \ \
( q o Proposal distribution: independent normal centred on current observation, with varying variance (linearly de-
Bayes theorem Posterior density: creasing over 5 iterations, then jumping back up)
^ Å Ú . / / Å
z
@ Update each parameter one at a time
º
^ Å Ú . / / Å ^ Å Ú . /
3
z
( @ º
153 154 Results: MCMC output for face 2 (in training set): Pos- Translations, scale, rotation, PC1, PC2 terior, Prior, Likelihood 136 132 chainsx[i, ]
128
u u 0 t 500 1000 1500 2000 2500 1150 1:nsim 1100
1050 x 84
chainsp x 82
1000 x 80
chainsx[i, ] w
78
t u u 950 0 500 1000 1500 2000 2500
1:nsim v
900
u u 0 t 500 1000 1500 2000 2500 1:nsim 98
94 v chainsx[i, ] 90
-2
u u 0 t 500 1000 1500 2000 2500 1:nsim -4 -6 0.04 chainspr 0.0 -8 chainsx[i, ]
-0.04
u u 0 t 500 1000 1500 2000 2500 1:nsim
-10
u u 0 t 500 1000 1500 2000 2500 1:nsim 0.0 1200 chainsx[i, ]
-0.8
u u 0 t 500 1000 1500 2000 2500 1150 1:nsim 1100 1050 chainsli -0.5 1000 chainsx[i, ]
-1.5
t u u 950 0 500 1000 1500 2000 2500 1:nsim
900
u u 0 t 500 1000 1500 2000 2500 1:nsim
155 156
PC3,...,PC8 MAP estimate overlaid on scale 8 2.0 1.0 0.0
chainsx[i + 6, ]
u u 0 t 500 1000 1500 2000 2500
1:nsim 250 -1.0 -2.5
chainsx[i + 6, ]
t u u
0 500 1000 1500 2000 2500 200 1:nsim 1.5 0.0 150 chainsx[i + 6, ]
-1.5
u u 0 t 500 1000 1500 2000 2500 S S 1:nsim
• • y 2.0 S 1.0 100 S • S -0.5
chainsx[i + 6, ]
u u t • • 0 500 1000 1500 2000 2500 1:nsim S S S • •• • 2 50 1 0 -1
chainsx[i + 6, ]
u u 0 t 500 1000 1500 2000 2500 1:nsim 0 2 1
0 0 50 100 150 200 250 -1
chainsx[i + 6, ]
u u 0 t 500 1000 1500 2000 2500 1:nsim
157 158 Shape distance to training set.... Procrustes distance
¹ and Mahalanobis
z Procrustes IMAGE REGISTRATION
7• 0.20 5• 3• 6• 0.15 1• 9• 8• dist 0.10 4• 10• 2• 0.05 0.0 2 4 6 8 10 image no.
| Mahalanobis 6
5 1• 3• 7• 8•
4 4• 5• 6• 9• 3
{ 10• dist 2• 2 1 0
2 4 6 8 10 image no.
159 160
IMAGE AVERAGING
Õ
% 3 3
1 1 1 ç
Consider a random sample of images ç con-
¥ ¥
Õ
% 3 3
1 1 taining landmark configuration 1 , from a pop-
ulation mean image ç with a population mean config-
Ø Ø
uration . We wish to estimate and ç up to arbitrary Euclidean similarity transformations. The shape of Ø
can be estimated by the full Procrustes mean of the (a) (b)
¥ ¥
Õ
½
% 3 3
#
1 1 E landmark configurations 1 . Let be the
def ormation obtained from the estimated mean shape
Ø
to the p th configuration. The average image has
theà grey level at pixel location given by
Õ
"
½
(
,
#
#
¨ ç E 1
ç (19)
# $ % Ô
(c)
161 162 (a) (b)
(a) (b)
(c) (d)
(c) (e)
Images of five first thoracic (T1) mouse vertebrae.
163 164
SHAPE TEMPORAL MODELS
Stochastic modelling of size and shape of molecules over time: HIGH DIMENSIONAL.
¡ Practical aim: to estimate entropy. Use tangent space modelling. 2 0 PC score 1 −3 0 1000 2000 3000 4000
time 2 0 PC score 2 −3 0 1000 2000 3000 4000
time An average T1 vertebra image obtained from five ver- 2 0 PC score 3 −3 tebrae images. 0 1000 2000 3000 4000
time 2 0 PC score 4 −2
0 1000 2000 3000 4000
time
165 166 REGRESSION
The minimal geodesic in shape space between the
* *
¥ ¥
ò
> { 3
¹
shapes of and where ¨ [Rie- S mannian distance] is given by:
¡ Temporal correlation models for the principal com-
ponent (PC) scores of size and shape. [AR(2)]
*
¥
¡
ò ò ò ò ~ ò ò
, ' 3 > y y
o
¨ é
Non-separable model - different temporal covariance
ò
-
S S } structure for each PC but constant eigenvectors over S
time.
*
¥
¡ é
where é is symmetric (i.e. is the optimal
- -
Improved entropy estimator based on MLE, interval -
* ¥ estimators. Procrustes rotation of on ).
¡ Properties of estimators under general correlation structures, including long-range dependence.
¡ Temporal shape modelling directly in shape space.
Practical regression models: tangent space regres-
sion through origin ¶ fitting geodesics in shape space.
167
SMOOTHING SPLINES
Smoothing spline fitting through ‘unrolling’ and ‘un-
wrapping’ the shape space .
)
On the Procrustes tangent space at time , the shape S
space is rolled without slipping or twisting along the Spline fitting in : unrolling the spline to the tangent
)
space at is the corresponding cubic spline fitted to continuous piecewise geodesic curve in . The piece- S wise linear path in the tangent space is the) unrolled the unwrapped data. path. Le (2002, Bull.London Math.Soc.), Kume et al. (2003). A point off the curve is unwrapped onto the tangent Piecewise linear spline piecewise geodesic curve
plane.
in . ) k Σ 2 (t) β α 2 Y
α
1
* α (t) α 2 1 *
β
k
Σ Y* 2 (t 0 )
168 169
NONPARAMETRIC INFERENCE Ø ¡ The full Procrustes mean is a consistent estima- tor of ‘extrinsic mean shape’ (PÃ atrangenaru and Bhat-
tacharya, 2003)
Ø DISCUSSION ¡
Central limit theorem for and a limiting ) distribu-
Ã
tion for a pivotal test statistic confidence regions. ¡ At all stages geometrical information always avail- ¡ Bootstrap confidence interval for mean shape based able on a pivotal statistic - NEEDS CARE in a non-Euclidean
space. ¡
¡ Statistical shape analysis of wide use in many disci-
Coverage accuracy of bootstrap confidence region
plines.
Ô
) .
;
¡ £
¡ Great scope for further application in image analy- Bootstrap sample hypothesis test (not necessary sis, e.g. medical imaging. to have equal covariance matrices in each group).
¡ Need to simulate from the null hypothesis of equal ¡ mean shapes, and so the individual samples are moved Non-landmark - curve - data along a geodesic to the pooled mean without chang- ing the inter-sample shape distances.
¡ Simulation studies indicate accurate observed sig- nificance levels and good power.
170 171
Selected References to papers: DG Kendall (1984,Bull.Lond.Math.Soc), Bookstein (1986,Sta- tistical Science), WS Kendall (1988,Adv.Appl.Probab.), DG Kendall (1989,Statistical Science), Mardia and Dry- den (1989, Adv.Appl. Probab; 1989, Biometrika), Dry- den and Mardia (1991, Adv.Appl,Probab) Dryden and Mardia (1992, Biometrika) Goodall and Mardia (1993, Annals of Statistics), Le and DG Kendall (1993, An- nals of Statistics), Kent (1994, JRSS B), Le (1994, J.Appl.Probab.), Kent and Mardia (1997, JRSS B), Dry- den, Faghihi and Taylor (1997, JRSS B), WS Kendall (1998, Adv.Appl.Probab.), Mardia and Dryden (1999, JRSS B), Kent and Mardia (2001, Biometrika), Le (2002, Bull.London.Math.Society), Albert, Le and Small (2003, Biometrika).
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