Biology Reference
In-Depth Information
Decomposing the Non-Uniform (Non-Affine) Component
The non-uniform part of a deformation differs from the uniform in that it does not leave
the sides of a square parallel. However, like the uniform part, the non-uniform can be
further decomposed into a set of orthogonal components. The decomposition of the non-
uniform deformation is based on the thin-plate spline interpolation function, and produces
components called partial warps. We first describe an intuitive introduction to partial
warps, then a more mathematical one.
AN
INTUITIVE INTRODUCTION TO PARTIALWA
RPS
The non-uniform component describes changes that have a location and spatial extent
on the organism; they are not the same everywhere. They describe spatially graded phe-
nomena such as anteroposterior growth gradients, and more highly localized changes
such as the elongation of the snout relative to the eye. The notion of spatial scale is central
to the analysis, so we need an intuitive notion of spatial scale. In general (but imprecise)
terms, a change at small spatial scale is one confined to a small region of an organism.
To refine that idea, and develop a firmer grasp of the concept, we show several compo-
nents at progressively smaller spatial scales (
Figure 5.5
).
Figure 5.5A
shows a component at large spatial scale that, while broadly distributed, is
not the same everywhere (so it is not uniform). The particular example shown in
Figure 5.5A
is the elongation of the mid-body relative to the more cranial and caudal
regions. A more localized change, confined to the posterior region of the body, is shown
in
Figure 5.5B
a shortening of the region between the dorsal and adipose fins relative to
the dorsal fin and caudal peduncle. Because more distant landmarks are not involved in
the change, it is more localized than the one shown in
Figure 5.5A
. Another localized
change is shown in
Figure 5.5C
, this time confined to the cranial region. This is a shorten-
ing of the postorbital region relative to the regions just anterior and posterior.
The components we have described above and depicted in
Figure 5.5
are partial warps,
but to draw them we had to specify their orientation (we drew them as oriented along the
anteroposterior body axis). That orientation is not actually specified by the partial warps
themselves; rather, it is provided by a two-dimensional vector, the partial warp scores.
There is one two-dimensional vector per partial warp. These scores express the contribu-
tion that each partial warp makes to the total deformation. The scores have an X- and
Y-component, and indicate the direction of the partial warp. The idea of direction or orien-
tation should be familiar from previous chapters. In
Figure 5.6
we show one partial warp
(that depicted in
Figure 5.6B
) multiplied by three different vectors. It may be easiest to see
the directions by looking at the orientation of the vectors at landmarks.
Figure 5.6A
shows
the partial warp oriented horizontally, which in our case corresponds to the X-direction,
so the coefficient of the X-component is large and that of the Y-component is negligible.
In contrast,
Figure 5.6B
shows the vector with a negligible X-component and a large
Y-component.
Figure 5.6C
shows the vector with X-andY-components of equal magnitudes.
We have described partial warps one at a time, but a complete description (and
interpretation) requires combining them all. Taken separately, partial warps are purely
