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object is three. When the object is three dimensional, the first K diag-

onal elements of
V
quantify variability in the X direction, the next
K

elements quantify the variability in the Y direction, and the last
K

elements quantify the variability in the Z direction for landmarks 1

through
K
. In the measurement error study we concentrate on the

diagonal elements (variances) and ignore the off-diagonal elements

(covariances). The matrix
V
can be used to identify those landmarks

that are highly error prone.

In practice, the matrix
V
is unknown, and we have to use the obser-

vations from
N
data collection episodes to estimate it. Let
M
1
,
M
2
,...
M
N

denote the observations (the landmark coordinate matrices) from
N

data collection episodes.

We start with calculating the mean of these observations. Let

denote the sample mean matrix. Notice that

M
is a

K
by 2 or a
K
by 3 matrix, depending on the dimension of the object.

Let
A
be an
m

n
matrix. Then
A
<i>
denotes the
i
-th column of
A
.For

example,
M
i
<2>
denotes the second column of the observation
M
i
. With

this notation, for a two-dimensional object, the variance-covariance

matrix
V
is obtained using the following formula. The first equation

provides the variance-covariance matix in the X direction, the second

equation provides the variance-covariance matrix in the Y direction,

and the third one provides the covariances between X and Y directions.

N

1

V

(

MMMM

1

1

)(

1

1

)

T

X

i

i

N

i

1

N

1

2

2

2

2

T

V

(

MMMM

)(

)

Y

i

i

N

1

i

N

1

1

1

2

2

T

V

(

MMMM

)(

)

XY

,

i

i

N

i

1

Extension to three-dimensional objects is straightforward. The

additional terms needed are given by:

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