Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 120

STEP 5:The estimate of the coordinates of the mean form is given by:

This can be used to calculate the form matrix by calculating the dis-
tances between the three landmarks.
The estimate of the covariance matrix K *

K L T is given by L

and the estimate of the eigenvalues of

D is given by

Generalization of this algorithm to three-dimensional objects and
objects with more than 3 landmarks is notationally complicated but
straightforward. We do not provide the details here.

3.14.2 Generating observations using the estimated mean form and
the covariance matrix

I D in
detail. This is the model that is extensively used throughout this mono-
graph. The following algorithm is described for a two-dimensional
object. Generalization to three-dimensional object is straightforward.
Let M K -1 be a submatrix consisting of the first ( K -1) rows of M , the
mean form coordinate matrix obtained in Step of Algorithm 1. Let K -1 *
denote the ( K -1) ( K -1) submatrix of K * . This matrix consists of the
first ( K -1) rows and ( K -1) columns of K * . In general, K -1 * should be a
positive definite matrix. The non-positive definiteness of K -1 * implies
that the sample size is much too small to fruitfully conduct any statis-
tical analysis, see the discussion in Part 1 of this chapter.

We will describe the algorithm for the covariance model K

Algorithm for generating matrix normal random variates

STEP 1: Obtain the Cholesky decomposition of K -1 * . That is,
obtain an upper triangular matrix C such that K -1 *

CC T .
STEP 2: Generate 2 ( K - 1 ) random numbers from N (0,1) distri-