Biology Reference
In-Depth Information
Three of the eigenvalues of the bending energy matrix are zero, corresponding to the
components with no bending (with X- and Y-coefficients, these eigenvectors account for
the six uniform components of the deformation). The remaining K
2
3 eigenvectors are the
explicitly localized components of a deformation. These eigenvectors are called the partial
warps; the vector multipliers of the partial warps are called the partial warp scores (follow-
ing
Slice et al., 1996
). They are “partial” because they describe part of a deformation. We
should note that
Bookstein (1991)
called the eigenvectors of the bending energy matrix
principal warps, analogous to principal components. By “partial warp”, he meant the vec-
tor multiple of a principal warp. Slice and colleagues use the term principal warp to refer to
a partial warp interpreted as a bent surface of the thin-plate spline, and because the latter
terminology has become standard, we use it here.
As evident in the definition of
L
2
K
;
only one matrix of landmarks enters into the calcula-
tion of bending energy; the coordinates of the form usually called the reference or starting
form. Thus, the eigenvectors that give us a coordinate system for shape analyses are a
function of one single form. This may be highly counterintuitive, because more familiar
eigenvectors, such as principal components, are functions of an observed variance
covariance matrix. They are functions of variation (or differences) among observed forms.
That is not the case for the eigenvectors of the bending energy matrix. The eigenvalues of
the bending energy are the bending energies that would be required to modify a given
shape by a single unit of shape difference at each spatial scale. Thus, the partial warps are
not themselves features of shape change, they are simply a coordinate system or basis for
the space in which we analyze shape change.
The “A” coefficients in
Equation 5.10
describe the uniform deformation of the shape.
There are six of these coefficients, which is enough to describe the six components of the
uniform deformation of shape. However, we know that the reference and the target do
not differ by rotation, rescaling or translation, because those differences were removed by
the superimposition process. Consequently, we do not need six parameters to describe the
uniform component of
the deformation, only the two components derived in the
Appendix.
By convention, partial (or principal) warps are numbered from the lowest to highest
bending energy; the one with the highest number corresponds to the one with greatest bend-
ing energy. The two uniform components are sometimes called the zeroth principal warp.
Thinking of the uniform components in those terms is useful because it emphasizes that the
uniform components cannot be viewed separately from the non-uniform ones. Including the
uniform terms also completes the tally of shape variables. The K
2
3 partial warps contribute
2K
2
6 scores; adding the two uniform scores brings the count up to 2K
2
4.
DECOMPOSING THE DEFORMATION OF THREE-DIMENSIONAL
DATA
As in the two-dimensional case, the difference between three-dimensional configura-
tions of landmarks can be described as a deformation of one shape (reference) into the
other (target). This deformation can be decomposed into uniform and non-uniform parts
(or affine and non-affine). The non-uniform part can be further decomposed into 3(K
2
4)
