v—Ax-l
M UM where (1)
C, 2-6, P
l is the vector of observations, v is the vector of residuals, x is
the vector of unknown parametrs, A is the design matrix, C, is
the covariance matrix of observations, and P is the weight
matrix. The observations are assumed to be normally
distributed random variables.
If design matrix A is of full rank, the least-square estimate for
x is:
£=(A"PA) A"PI (2)
and the corresponding covariance matrix is:
C soi[A PA)" i0. (3)
It is possible to evaluate the measurement accuracy before
actual measurements. The design matrix A is totally defined by
the geometry of the network and by the information of the
observations between the points. The real observations do not
effect to the structure of the design matrix A. The weight
matrix P is made based on the accuracies of the observations.
In photogrammetry, the weight matrix is usually P-o "I.
(Fraser,1989). That means that image observations are equally
weighted. The cofactor matrix Q., (in equation 3) is the
function of the network geometry and the observation accuracy
only. All information about the precision of unknown
parameters is therefore included in the variance-covariance
matrix C, (3), which is get by multiplying the cofactor matrix
Q. by the variance of the unit weight.
In this way, the precision of the network can be evaluated
before the actual measurements. To evaluate the precision of
the points in network, only their approximate values has to be
known. These approximation values are needed for the
linearization of the condition equations.
At first the simulated image coordinates are calculated from
the measurement model information (camera parameters,
camera calibration data, measuring points coordinates). After
that the cofactor matrix and the variance-covariance-matrix are
created. Various precision measures can be calculated from the
variance-covariance matrix of the unknown parameters. These
are, for example, the variances of intersected 3D-coordinates,
error ellipses, and error ellipsoids. These values are visualized
in 3D object space. The shape of the error figures depends on
the network structure (geometry). The variance of the unit
weight is only a scaler. The variance of the unit weight could
be improved, for example, by taking multiple exposures. The
selection of a priori o^ can based on earlier measurements or
experiments.
4.1 Limiting error propagation
So far, the calculation of precision values is based on limiting
error propagation (Brown, 1980; Fraser, 1989). Only the
precision of the image coordinate measurement is taken into
436
account in calculating accuracy estimates for the XYZ
coordinates. The assumption is made, that projective and
additional parameters are precisely known. This means that
possible errors of the calibration are not taken into account.
The simulation model could be expanded so, that the effect of
these is also taken into account when calculating precision
values. In this case
Using limiting error propagation, the variance-covariance-
matrix of unknown object coordinates can be obtained as a
simple inverse of a 3 x 3 matrix (Fraser, 1989):
OC) =(A] PA,) (4)
zi
i
The error propagation could be expanded to total error
propagation, when also the effect of calibration errors are taken
into account.
4.2. The variances of the unknown parameters
The variances of unknown object coordinates are get from the
diagonal elements of the variance-covariance matrix C, . The
standards errors of the unknown parameters are square roots of
these diagonal elements. In this way the standard errors (sX,
sY, and sZ) for unknown object coordinates are calculated.
However, the errors are not always biggest in the direction of
coordinate axes. The example of visualization of mean errors
was already shown in figure 3.
Instead of the separate mean errors for X, Y, and Z, one
precision measure can be used. One possible measure is the
mean radial spherical error (MRSE) (Mikhail, 1976, p. 34). In
three-dimensional case this measure is visualized with a
sphere, which radius is equal to the MRSE value. It gives an
approximation about the precision of the point. An example of
the visualization of MRSE values was given in figure 4. More
exact way to express the precision of the point is to use error
ellipsoids.
4.3. The error ellipses and the error ellipsoids
The variance-covariance matrix includes the variances and the
covariances of the unknown parameters. The error ellipses and
the error ellipsoids are defined by calculating the eigenvalues
and eigenvectors from the variance-covariance matrix or its
submatrix. The probability of the point to lie inside the
standard error ellipse (k=1) is 0.394. In case of the standard
error ellipsoid the probability is 0.199. (Cooper, 1987).
The standard error ellipses in three different planes (XY, XZ,
YZ) are visualized with the MMD tool. Standard error
ellipsoids are also calculated and visualized.
5. CONCLUSIONS
The design of measurements using three-dimensional CAD-
models enables the more detailed measurement plan. Using
network simulation, it is possible to consider the reachable
measurement accuracies in advance.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
The M
measur
process
values «
system.
tool.
The to
measure
be calil
used to
well-sel
every wl
point «
calculat
The me:
part fro
tool us
environi
into acc
planning
values
measuri
the des
simulati
work is
REFER
Ailisto,
three-dir
submilli:
pp. 2603
Brown,
Photogra
Tehcnice
Melbour
Cooper,
Collins ]
Sons & I
Cowan, (
vision t
Analysis
Fraser, |
Topograr
Photogra
chief), A
Sensing,
Luhmann
von A
Zeitschrit
pp. 103-1
Mason,
Photograi
Photograr
