Biology Reference
In-Depth Information
Because compression/dilation and shear alter shape whereas translation, rotation and
scaling do not, it is common to talk about the two that alter shape without mentioning the
ones that do not. All of them need to be accounted for, so we will refer to compression/
dilation and shear as the explicit uniform deformations or explicit uniform terms because they
are the ones explicitly tracked. We will refer to the others as the implicit uniform deforma-
tions or implicit uniform terms. They are implicit because they can be mathematically deter-
mined from the superimposition method used, the explicit uniform components, and the
non-uniform components of a deformation
they are the translation, rotation and scaling
that must have been carried out. Both explicit and implicit uniform terms are needed, in
addition to the non-uniform terms, to draw the deformation correctly.
Each deformation has an inverse. Applying the inverse of a deformation is equivalent
to traveling backwards along the path that was taken until we arrive back at the starting
point. We can think of the deformation in terms of a 2K-dimensional vector (i.e. two
dimensions per landmark). There would be a vector at each landmark indicating the direc-
tion in which that particular landmark will be mapped under the deformation (although
there are only 2K
4 independent dimensions). In the inverse of the deformation, the
directions of the arrows would be reversed. The inverse of a translation is the same magni-
tude of translation in the opposite direction (negative X instead of positive X). Similarly,
we can represent rotation as an angular displacement so its inverse is a negative angular
displacement (counterclockwise instead of clockwise). Scaling is slightly different because
it involves multiplication (whereas translations and rotations could be treated as addi-
tions). Scaling is multiplication by a factor F; its inverse is multiplication by the inverse of
F (1/F). Unfortunately, the algebraic descriptions of the last two deformations and their
inverses are not quite as simple (as we will see below). Graphically, we can see that the
inverse of compression/dilation involves a reversal of which axis is compressed and
which is dilated, and that the inverse of a shear is a shear of the same amount along the
same axis in the opposite direction.
Several different approaches exist to calculating orthogonal axes to represent the two
uniform components of shape difference found in a data set. The first approach, as
developed by
Bookstein (1996)
is to determine the pattern of shape change at each land-
mark of the reference after the two operations of shear and compression/dilation, fol-
lowed by partial Procrustes superimposition. A derivation of this approach is shown in
the Appendix. The other approach is to use the thin-plate spline method described
below to partition the shape variation in the data into the affine (uniform) and non-
affine components (
Rohlf and Slice, 1990; Rohlf and Bookstein, 2003
). The affine portion
can then be subjected to a singular value decomposition (akin to a principal components
analysis) to develop two orthogonal basis vectors (in two dimensions) which span the
space of possible affine changes in shape. The uniform shape terms found via singular
value decomposition will be linear combinations of shear and compression/dilation.
The approach based on singular value decomposition is readily adapted to three-
dimensional data as well. The choice of whether to use the affine terms as per
Bookstein (1996)
or based on singular value decomposition is not particularly important,
they are simply slightly different basis sets of the uniform terms. Since the terms should
not be interpreted independently of one another in any case, the choice between the
two is not noticeable in the final statistical analysis.
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