Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 142

empirical results and scientific conclusions. The alternative approach
is to choose not to adopt a coordinate system. In this section, we pre-
sent an approach that enables the comparison of forms independent of
any and all coordinate systems. This is the approach that we favor. The
following section clarifies the significance of comparing forms using a
coordinate system-free method. We begin by presenting the concept of
“orbit” as it relates to the study of form difference. Understanding this
concept will reinforce our insistence on using only that information
that we know unambiguously when comparing forms.

4.7.1 Orbits of equivalent forms and description of form difference

Let M be a landmark coordinate matrix of dimension K

D . Following
our definition, the form of an object as represented by this collection of
landmark coordinates is that characteristic which remains invariant
under the group of transformations consisting of rotation, translation,
and reflection. Now, think of the collection of all K D matrices that

can be obtained by any rotation, reflection, and translation of M . That
collection of matrices is called an orbit described by M under the group
of transformations consisting of rotation, reflection, and translation
( Figure 4.6 ) . All matrices within any single orbit represent exactly the
same form because they differ only on the basis of translation, rotation,
or reflection. An invariant descriptor of form change describes the dif-
ference in the orbits that the forms occupy.
As an aid to understanding the idea of an orbit, consider a topo-
graphic map of a mountainous area. Each contour on a topographic map
corresponds to a surface of constant (equal) elevation. No matter where
a point lies in longitude or latitude, elevation remains the same as long
as the point stays on the defined contour. In other words, as long as a
point remains on a single contour line, elevation is invariant with
respect to latitude and longitude. Orbits in the space of all form matri-
ces as defined above are similar to contours on a topographic map.
Suppose that a person is hiking on the surface described by the
topographic map. Suppose further that this hiker carries a sensor that
sends out a signal that identifies only his elevation exactly. If we are
trying to locate this hiker but only have the signal from his sensor, we
can place that hiker onto a contour defined by a particular elevation,
but not to any particular location on that contour. This is analogous to
the limitations of our knowledge when dealing with landmark coordi-
nate data. All that we can know with certainty is the orbit to which the