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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.
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:    Search WWH ::

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