Biology Reference
In-Depth Information
The semilandmark method based on bending energy produces smooth differences
between curves on different specimens, the distance minimizing method produces speci-
mens with the minimum possible distance between them. There is no clear consensus at
this point about which approach is preferable. However, it does appear that the distance
minimizing approach will often be more statistically conservative. If we want to show that
the mean shape of two groups is different, then the distance minimizing alignment would
produce specimens at the minimum possible distance, effectively biasing a statistical test
against finding any differences. So, if there is enough difference remaining in the two
groups to reject the null hypothesis despite the possible bias of the alignment procedure,
one can be reasonably sure the differences in mean shape are not an artifact of the align-
ment procedure. Any possible bias in the distance biased alignment is certainly against the
result we want to show. Bending energy alignment does produce smoother looking differ-
ences, but also appears to increase the variance within the data (
Sheets et al., 2004, 2006
).
APPENDIX
Calculating the Shear and Compression/Dilation Terms
Here we present the mathematical derivation of formulae for calculating the uniform
components of a deformation that changes shape. Unlike the formulae for computing the
non-uniform part of a shape change, which have been stable over the last decade, the for-
mulae for computing the uniform part have changed repeatedly. Over the last several
years, the uniform component has been computed using the formulae presented by
Bookstein (1996)
. The ones based on the Procrustes distance are the ones we present here.
We begin with a conceptual framework for Bookstein's derivation of the current formulae;
then follow that with the full mathematical details.
Conceptual Framework
The goal of this derivation is to find a unit vector that describes the direction of defor-
mation at each landmark due to shearing or compression/dilation, followed by a
Procrustes generalized least squares (GLS) superimposition of the deformed shape back
onto the original (undeformed) one. This represents what we measure in data: a deforma-
tion followed by a superimposition operation. Thus, both mappings must be taken into
account. When we are done, we will have a set of unit vectors that describe the deforma-
tion under shearing or under compression/dilation. We can then take the dot product of
the observed deformation with the unit vectors to obtain the component of the observed
deformation lying along the shear or compression/dilation vectors. These are what we
have been calling the explicit uniform components of the deformation.
Notice that we are taking a verbal description of the situation, turning the verbal state-
ment into two mathematical operations or mappings (shear or compression/dilation, fol-
lowed by the superimposition), then using those mappings to determine the direction of
the vectors describing the deformation. That allows us to calculate components of any
deformation along those desired directions. What might not be obvious yet is that vectors
describing the uniform deformations depend on only one form
the one that we are
