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sions of the mean. We have demonstrated that because of the nuisance
parameters of translation, rotation, and reflection, only the form matrix
corresponding to the coordinates of the mean form can be estimated.
In most research situations, we must estimate the mean template
from a sample of available observations. We will denote the estimated
form matrix corresponding to the mean of the first group by FM ( Â ) and
the estimated form matrix corresponding to the mean of the second
group by FM ( ˆ B ) . The reader can think of an estimated mean form as a
form matrix representing the average of the linear distances from all
forms in the sample, although the exact estimation procedure is a bit
more complex (see Part 2 of this chapter). The mean form difference
matrix provides the estimation of the difference between mean forms.
Details of the estimation procedures are given in Part 2 of this chapter.
Estimating the mean form difference matrix
The estimated mean form difference matrix is given by
Estimating the scaling factor
Although there are a number of possible estimates that can be used as
a scaling factor, in this example we use the geometric mean of the
distances. In this case, the estimated
scaling factor is given by
FM ij ( Â )} I/L .
S ( Â )
{
Estimating the mean shape matrix
The estimated mean shape matrix is given by .
We emphasize that the definition of a shape matrix is dependent upon
the choice of the scaling factor.
Estimating the mean shape difference matrix
The estimated mean shape difference matrix is given by
Using ideas of the form space, we have provided ways to estimate
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