is a K
K symmetric matrix.
STEP 8: Calculate the eigenvalues and eigenvectors of B ( M ) .
Let us denote its eigenvalues arranged in a decreasing order
by 1 ,
and the corresponding eigenvectors by
h 1 , h 2 ,…, h n .
STEP 9: 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
STEP 10: Calculate the form matrix (the matrix of all pair-
wise distances) for the above landmark coordinate matrix.
This is an improved estimator of the mean form matrix.
Note: The M obtained in STEP 3 can be used to graphically repre-
sent the mean form of the sample. It should be remembered that this
estimator is a representation of the mean form M only up to rotation,
reflection, and translation. In practice, to obtain a graphical represen-
tation that is biologically meaningful, one may need to reflect the
above estimator by multiplying one or more of the axes by -1.
Having obtained the estimator of the mean form matrix M described
above, it is fairly simple to obtain the estimator of the covariance
matrix, K * . The following steps provide the details.
b) Algorithm for estimation of K *
STEP 1: Calculate the centered matrices X 1 c , X 2 c ,…, X n c corre-
sponding to the observations X 1 , X 2 ,…, X n .