Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 66

tion on the table introduces a different translation and rotation specif-
ic to that transparency.
Rotation refers to change in orientation characterized as movement
around an axis ( Figure 2.7 ). Imagine placing a pin through the origi-
nal pile of transparencies. This pin can be used as an axis about which
the black transparencies spin. Upon rotation, the relative locations of
the points on any single transparency remain the same, but the orien-
tation of the perturbed observations (the triangles) changes by rotation
around an axis. Mathematically, rotation of an object corresponds to
multiplication of the landmark coordinate matrix by an orthogonal
matrix (see Chapter 2, Part 2 ).
Translation, on the other hand, corresponds to a black transparen-
cy remaining stable in terms of rotations around axes but sliding in
any direction along the plane defined by the tabletop ( Figure 2.7 ). As
with rotation, the relative locations of points are maintained under
translation. Mathematically, translation corresponds to adding a
matrix of identical rows to a matrix (see Chapter 2, Part 2 ).
With these definitions of rotation and translation in hand, we can
now present the data using the full perturbation model which represents
the observations after they have been arbitrarily translated and rotated.
A single observation, X i , incorporating the full perturbation model is
represented mathematically by: X i

1 t i where E i is the ran-
dom perturbation of the mean form M , R i is the orthogonal matrix
corresponding to rotation of X i , and t i is the translation matrix. We
emphasize that each observation, X i , may be rotated and translated dif-
ferently. In the context of the transparency experiment, X i corresponds
to the landmark coordinate matrix representing the i -th transparency
after it has been dropped onto the floor, picked back up, and put onto the
table. The original landmark coordinate matrix for any black trans-
parency (before disturbing it) is given by M

( M E i ) R i

E i . Once the matrix has
been arbitrarily rotated and translated, it is written as ( M 1 t i .

Let us now deal with modeling the perturbation pattern that was
used to generate the black transparencies from the red one.
Perturbation patterns are quantified by the covariance structure of the
random variables, E i . These covariance matrices are best conveyed by
a graphic representation of the variability implied by different covari-
ance structures. Let us begin by accepting, for instructional purposes
and mathematical convenience, the assumption of Gaussian (Normal)
perturbations across all landmarks on a two-dimensional, three-land-
mark object. Complex objects with more landmarks in three
dimensions can be modeled using similar covariance matrices.

E i ) R i