Biology Reference

In-Depth Information

The following expressions provide the first two moments of the ran-

dom matrix
Y
(Alam and Mitra, 1990).

D

å

2

j

j

+
s
mn
d
lp
j
+
s
mp
d
l
j
}

Cov
(
Y
lm
,
Y
np
)
=

{
s
ln
s
mp
+
s
lp
s
mn
+
s
ln
d
mp

+
s
lp
d
mn

j
=
1

Without loss of generality, one can assume
1

1
. Notice that there

1)/2
unknowns. We choose as many moments and

equate the sample moments with the corresponding population

moments. Solving the resultant equations, one can numerically obtain

the estimates of the parameters
(

are
(
KD

1)

K
(
K

*
K
,

D
)
. Given

these estimates, we obtain the mean form of the object (up to transla-

tion, rotation and reflection). From the definition of
j
's, it is clear that

they are symmetric matrices with rank 1. One can write
j

2
,…,

D
,

1
,

2
,…,

d
j
d
j
T
,

where
d
j
is a scaled eigenvector. If the observations are coming from

two-dimensional objects, the estimate of the mean form up to transla-

é

0

0

ù

ˆ
M
=

tion, rotation, and reflection is given by

ê

ú
˙
,

l
2

d
1

d
2

ë

û

whereas for three-dimensional objects, the estimate of the mean form

up to translation, rotation and reflection is given by

é

0

0

0

ù

M
=

ê

ú
˙
.

d
1

ˆ

d
2

ˆ

d
3

l
2

l
3

ë

û

Estimation of the variability around landmarks

In many important biological problems, it is of interest to know the

variability around each of the landmarks individually. So far the

results described above suggest that it is possible to estimate
L

K
L
T

but not
K
itself. However, if one is willing to assume a somewhat sim-

pler structure for
K
and
D
, then it is possible to estimate the

variability around each of the landmarks. We assume that the land-

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