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this method cannot give reliable information concerning how two
forms differ from one another. The mathematical details of this non-
identifiability are provided in Part 2 o f this chapter (see also Lele and
McCulloch, 2000).
4.5 Transformational grids and deformation-based
approaches
Deformation approaches, sometimes including the use of transforma-
tional grids, provide an alternate way to study form difference. These
methods present the difference between forms as the changes required
to transform one object into the other. Differences are expressed as mag-
nitudes along specified directions and are depicted by the way in which
a uniform grid placed over the original object changes or deforms during
the transformation from the initial to the target object. However, as with
the superimposition approach, the loss of a common coordinate system
has profound effects for the description of form change.
In this discussion of deformation approaches, we refer to one of the
objects as the initial object and to the other as the target object.
Results of the analysis will be expressed as those changes that are
required to transform the initial object into the target object. To
explain the deformation approach, consider a lump of bread dough.
Flatten the top of the dough ball and draw a square grid directly onto
it. Now, plot the coordinates for the first object (e.g., the three land-
marks of the red transparency) directly onto the grid that is drawn on
the dough. Next, use your hands or a rolling pin to push, pull, or roll
the dough in any manner you please. The relative positions of the land-
marks have changed, as well as the appearance of the grid. The
distorted grid is an excellent descriptor of the form change. It “local-
izes” and summarizes the form change/difference relative to the
original, uniform grid.
Now take a blank, circular transparency and place it over the rolled
dough. Trace the changed landmark positions onto this transparency
using a green marker. This landmark configuration becomes the
deformed or target object in this experiment. Refer to it as the green
transparency. To simulate the loss of the common coordinate system,
do two things: 1) roll the flattened dough with the grid markings back
into a ball, and 2) shuffle the red and the green transparencies.
We need to determine if it is possible to recreate the (distorted) grid
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