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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
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
and the corresponding eigenvectors by
1 ,
2 ,…,
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.
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