The Study of Growth - An Invariant Approach to Statistical Analysis of Shapes - page 239

Hypothetical Form Matrix

0.00000
2.37584 0.00000

1.97359 1.01250 0.00000

2.27668 3.94614 2.87906 0.00000

3.88223 4.50303 4.06614 2.82918 0.00000

2.98274 3.88200 3.55668 2.47735 1.24550 0.00000

If we want to plot the relative landmark locations of the hypothetical
form in three-dimensional space, we need the coordinates for the land-
mark locations of the hypothetical form. To re-write the hypothetical
form matrix as a landmark coordinate matrix, the form matrix is sub-
jected to spectral decomposition (see also Chapter 3 , Part 2 ). To do this
we first square the Euclidean distance matrix corresponding to the
hypothetical form matrix. Following the notation given in Part 2 of
Chapter 3 , this is written as:

2 2

0

K
KM

12 1 K

2

0

12

HFM M ()

M M M M

2 2

K

0

1 K 2 K

The squared Euclidean distance matrix corresponding to the hypo-
thetical form in our example is given as:

0.00000
5.6446 0.00000

3.89505 1.0251 0.00000

5.18327 15.5720 8.2889 0.00000

15.0717 20.2773 16.5335 8.00426 0.00000

8.89674 15.0699 12.64997 6.13726 1.55127 0.00000

Next, we need to calculate the corresponding centered inner product-
matrix using

1
2

HEMM H T BM () [ ( ] = −