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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
non-iterative, 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
non-central chi-squared random variables with the same scale param-
eter lm ll mm
X m ,1 ) 2 and ( X l ,2
2
em and non-centrality 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 non-central chi-squared
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 non-central chi-squared 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       Search WWH ::

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