Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 105

which is the squared Euclidean distance matrix correspond-
ing to the above mean form matrix.
STEP 2: Calculate the corresponding centered inner product

matrix by

where is a K K symmetric matrix.

STEP 3: Calculate the eigenvalues and eigenvectors of B ( M ) .
Let us denote its eigenvalues arranged in a decreasing order
by and the corresponding eigenvectors by 1 , 2 ,…, K

h 1 , h 2 ,…, h K .
STEP 4: If the original data are from two-dimensional objects,
the estimate of the mean form matrix is given by

and if the original data are from three-dimensional objects
the estimate of the mean form matrix is given by

STEP 5: Calculate the form matrix (the matrix of all pairwise
distances) for the above landmark coordinate matrix. This is
an improved estimator of the mean form matrix.

The matrix M obtained in Step 3 can be used to graphically repre-
sent the mean form of the sample. It should be remembered, however,
that this estimator is a representation of the mean form M only up to
rotation, reflection, and translation. In practice, to obtain a graphical
representation that is biologically meaningful, one may need to reflect
the above estimator by multiplying one or more of the axes by -1.