Biology Reference

In-Depth Information

Think about the bread dough that you rolled out in the last experi-

ment, and you will realize that the deformation resulting from rolling

the dough may not necessarily be affine. That is, shear may have actu-

ally occurred, but because we have only three landmarks the best we

can infer under the maxim of parsimony is that the deformation is

affine. The mathematical reason for this situation is very similar to

that found in classical regression. If only two data points are available,

a straight line is the only curve that can be fit to the data. No matter

how nonlinear the true relationship between the covariate and the

response variable might be, the regression model can only provide a

linear relationship because there are only two data points. A more

informed idea about the underlying nonlinear relationship requires

additional data points. For example, if the underlying relationship is

quadratic, a minimum of three data points is required. Similar to this

situation, in the study of deformation of objects, unless one has a sub-

stantial number of landmarks, the type of deformations that may be

inferred is limited and is related to the number of landmarks available

for study. This means that the deformation representing the most par-

simonious solution changes with the number of landmarks. It can be

shown that addition or subtraction of just one landmark changes the

estimate of form change. This indicates that the original estimate can-

not equal the true form change.

Lack of a common coordinate system

Suppose that the underlying deformation is affine and that three land-

marks will, in fact, provide adequate information for determining this

deformation. Can the information within the red and green trans-

parencies that have been disturbed from their original positions

provide this deformation? The same difficulty faced in the superimpo-

sition case presents itself here: we do not know and can never know the

configuration of landmarks (the red and green transparencies) in their

original version that includes information on their relative position in

space. All that we have are the red and green transparencies which are

rotated and translated versions of the original landmark configura-

tions. Since we do not know the relationship of the original

configurations to these translated and rotated versions, we can never

know “the” coordinate system within which to conduct the deformation

study.

To clarify, let's go back to the lump of bread dough and assume that

the underlying deformation caused by the rolling of the dough is, in

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