Morphometric Data - An Invariant Approach to Statistical Analysis of Shapes - page 52

For a three-dimensional object, the translation vector t t 1

is a 1 by 3 vector.
One can combine the two operations and obtain the landmark
coordinate matrix of a translated and rotated object by MR ( 0 - )

t 2 t 3

1 t . Figure 2.7 provides the pictorial representation of these
three operations. Figure 2.7 provides the original landmark
coordinates for the triangle, a rotation ( R ) of the original, a
translation ( t ) of the original, and the combination of rotation
and translation ( R + t ) of the original.

2.8 Statistical model and inference
for the measurement error study
In the measurement error study described in the previous part of this
chapter, we fixed a single skull to the digitizer and collected the coor-
dinates of the same set of landmarks multiple times. Let M i denote the
true but unknown landmark coordinate matrix for this skull in this
fixed orientation. Each data collection episode produced a landmark
coordinate matrix, say M i , where “i” denotes the i-th data collection
episode. Due to measurement error, we will not get exactly the same
landmark coordinate values during every data collection episode. The
variability among the M i 's denotes the measurement error. We need a
statistical model to study and quantify this variability.

2.8.1 Preliminaries of the matrix normal distribution

First notice that the M i 's are related to the true landmark coordinate
matrix M . This relationship may be written as M i

E i . That is, M i
is obtained by adding an error matrix E i to the true coordinate matrix
M . We assume that this error matrix has certain properties. These
assumptions are detailed below.
Assumption 1 : There is no systematic bias introduced when locating
a landmark in space. That is, the measurement errors at any particu-
lar landmark along any particular direction cancel each other out and
are on an average zero.
Assumption 2 : The errors introduced at any particular landmark
and along any particular direction are distributed according to a
Normal distribution. Though normally distributed, these errors, how-
ever, may be correlated with each other.

M