Biology Reference
In-Depth Information
That is a component of the deformation, not a component of an ontogeny. Only by looking
at the total deformation can we say where change occurs.
To summarize our intuitive presentation of spatial scale, we repeat our major points.
First, any non-uniform deformation can be decomposed into a series of components (partial
warps) at progressively smaller spatial scales. Each component describes a pattern of rela-
tive landmark displacements, based on the spacing and location of landmarks in the refer-
ence form. Each partial warp is multiplied by a two-dimensional vector (the partial warp
scores) that measures the contribution made by the partial warp (in each direction) to the
total deformation. We now present a more technical introduction to the thin-plate spline.
AN
ALGEBRAIC INTRODUCTION TO PARTIALWA
RPS
Algebraically, partial warps are obtained by eigenanalysis of the bending energy matrix.
Eigenanalysis may be familiar from a quite different context, for example, principal com-
ponents analysis, where it is used to extract eigenvectors (PCs) of the variance
covariance
matrix of measurements. The exact same mathematics is involved in calculating the partial
warps; the difference lies in the matrix being analyzed. Rather than extracting eigenvectors
of a variance
covariance matrix, we instead extract them from the bending-energy matrix.
(We will discuss eigenanalysis further, in context of principal components analysis in
Chapter 6; here we focus on the derivation of the bending energy matrix.)
The idea behind the thin-plate spline is that it will approximate the observed deformation
by a linear combination of a function that is the smoothest available and that fully describes
the observed deformation. The function satisfying that pair of requirements has the form:
Z
R
2
ln R
2
ð
X
;
Y
Þ 52
U
ð
R
Þ 52
(5.1)
where R is the distance between a pair of landmarks in the reference configuration (scaled
to unit centroid size). This particular function satisfies the biharmonic equation:
!
2
d
2
dx
2
1
d
2
dy
2
2
U
Δ
5
U
ð
R
Þ ~ δ
ð
0
;
0
Þ
(5.2)
where
δ
(0,0)
is the generalized function, or delta function, which is defined to be zero
everywhere except at X
5
0, Y
5
0, with the seemingly odd requirement that:
ð
ðδ
ð
0
;
0
Þ
dx dy
Þ 5
1
(5.3)
The delta function is oddly behaved, but mathematically tremendously useful, as it has
useful normalization properties. It is sometimes called a functional, rather than a function.
U is said to be the fundamental solution of the biharmonic equation, which is the equa-
tion for the shape of a thin steel plate lifted to a height Z (X, Y) above the (X, Y)-plane.
This is because the bending energy (BE) of the steel plate at a point (X, Y) is given by:
2
2
2
d
2
U
dx
2
d
2
U
dx dy
d
2
U
dy
2
1
2
1
52
BE
ð
X
;
Y
Þ
(5.4)
