Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 104

n
å

s 2 ( e m ) = 1

( e lm , i - e lm ) 2 individuals and let be the variance of the

n

i = 1

squared Euclidean distance between landmarks l and m in n individuals.

2

e lm = ( e lm

- s 2 ( e lm )) 0.5 for l , m = 1,2, ¼ K .

Then,

The estimator of the mean form matrix, which consists of the
Euclidean distances between all pairs of landmarks, is thus given by
To generalize these results to three-dimensional objects, notice that

æ ö

e lm
j lm

e lm ~ lm

( m l ,3

c 2

where

( m l ,1 m m ,1 ) 2 ( m l ,2 m m ,2 ) 2 m m ,3 ) 2

ç ÷ lm

è ø

em . The first two moments for this distribution
are given by: E ( e lm ) 3 lm lm and var ( e lm ) 6 lm 2

and lm ll mm

2

4 lm lm . It thus

follows that

In this case, we calculate for all pairs of

landmarks l

1,2,.…, K ; m 1,2,…, K. The estimator of the mean form

matrix is obtained by:

This matrix may not necessarily correspond to a configuration of K
in two- or three-dimensional Euclidean space. This estimator can be
improved by constraining it to be a form matrix corresponding to a
two- or three-dimensional (as the case may be) object as described
below.

STEP 1: Construct the matrix