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analytical solution. Moreover, Lele (1993) also shows that the estima
tor of the mean form
M
can be obtained up to translation, rotation, and
reflection consistently and the covariance parameter
K
*
L
K
L
is
also estimable consistently. However, assuming
D
=
I
imposes restric
tions on the applicability of this model so we next extend the method
to the situation where
D
≠
I
.
3.13.1 Method of moments estimators
We will first consider a model where the perturbation of landmarks
along the D axes are independent and identical to each other, but the
correlations between landmarks are allowed. That is, we consider the
perturbation model where the covariance structure is given by
K
I
D
. The main advantage of this model is that there exists a
noniterative, close form, consistent estimator for the mean form
matrix and the covariance matrix. For notational simplicity, let
*
K
K
L
T
lm
]
.
Let
e
lm
denote the squared Euclidean distance between landmarks
l
and
m
. Thus
e
lm
(
X
l,
1
X
m,
1
)
2
L
[
X
m,
2
)
2
. Under the Gaussian per
turbation model, it can be seen that
(
X
l
,1
(
X
l
,2
X
m
,2
)
2
are
noncentral chisquared random variables with the same scale param
eter
lm
ll
mm
X
m
,1
)
2
and
(
X
l
,2
2
em
and noncentrality parameters
(
m
l
,1
m
m
,1
)
2
m
m
,2
)
2
. Because the two axes are perturbed independently, it
follows that
e
lm
is a sum of two independent noncentral chisquared
random
and
(
m
l
,2
variables
with
common
scale
parameter.
Hence
e
lm
~
lm
X
2
2
(
lm
is the squared Euclidean distance
between landmarks
l
and
m
in the true mean form, that is,
lm
/
lm
)
where
em
.
The first two moments of a noncentral chisquared distribution are
given by (Johnson and Kotz, 1970, Chapter 28):
E
(
e
lm
)
2
lm
lm
and
(
m
l
,1
m
m
,1
)
2
(
m
l
,2
m
m
,2
)
2
and
lm
ll
mm
2
lm
e
lm
=
E
2
(
e
lm
)

var(
e
lm
).
var
(
e
lm
)
4
lm
2
lm
lm
. It thus follows that
To obtain estimators we equate the sample moments with the popula
tion moments and solve the resulting equations for the parameters.
Let
e
lm,i
denote the squared Euclidean distance between land
4
n
å
e
lm
=
1
marks
l
and
m
in the individual “
i
”. Let be the average
e
lm
,
i
n
i
=
1
of the squared Euclidean distance between landmarks
l
and
m
in n
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