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Tans-
othe-
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which
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ing a
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0.01m < 0, < 1m
0.005m « oc, « 0.1
A classification of accuracy leads to the relation o, » o, »
05; therefore these figures will be improved if data of cartog-
raphy, photogrammetry, and surveying is merged with each
other.
2 Block adjustment with
independent models
In order to apply the chaining procedure for checking the
metric quality of map digitizing the underlying adjustment
model is reviewed. The important criteria of block adjust-
ment with independent models (K. Schwidesfky/F. Acker-
mann, 1976) are extented to:
e the computing units can be isolated map regions, whole
maps and image pairs
e the functional relation between the model/object space
is a spatial similarity transform
e the block unit is constrained by means of control
points, additional check points, and in case of pho-
togrammetric images the perspective centres
Fig. 1 gives the well-known individual position of the dif-
ferent independent models.
Fig. 1: Connection of independent models to a block unit
(from K. Schwidesfky/F. Ackermann, p. 206)
The spatial similarity transform can be derived by differ-
ential or purely geometric considerations (K.R. Koch, 1987).
Let be B the matrix of coefficients providing for three trans-
lations, three rotations and a change in scale (K.R. Koch/D.
Fritsch, 1981)
1 0 0 1 0
0 1 0 0 1
0 0 1 0 0
575
In this matrix the coordinates X;,Y;, Z; can be approxi-
mate values of the object coordinates of point P;; in case of a
two-dimensional transform this matrix shrinks to two trans-
lations, one rotation and the scale change simply by deletion
of the third row and the third column, and the forth and
fifth row.
Thus, for every model j the following observation equa-
tions are valid
Xi fo fvi rv 0 -Z Y X\jan
Y]-vej«Ivi-Ful-Fz-w vlla
2f Nw Nl (Vois LT x v zin
| i \dA
In short we can write (2) as
E(l;) = bj+vi; =x: — Byp;, D()-ePB? (3)
in which /;; is the observation vector for point P; in model
j and vj its corresponding residual vector, z; is the vector
of unknown object coordinates of point P, Bj; contains the
coefficients of the similarity transform und p; is the vector
of the seven unknown datum parameters of model j. The
operators E and D characterise expectation and dispersion
respectively.
Control points can be considered twice: on the one hand
non-random coordinates constrain (2) in form of the linear
equation system
He =0 (4)
and on the other hand random coordinates deliver additional
observation equations
U VU U U OUU 0
V}|+|wv|=|V ; D( V ) = a 0 Ovy
W » vy J. W f. W [; 0 0 OWW
1 i V %
(5)
The parameter estimation by means of least-squares is done
using (3) - known as Gauss Markov model — in which the
residuals are minimized according to
loll} = min (6
subject toH z — 0
if non-random control points exist.
3 Hypothesis testing
Testing the parameters and residuals of the parameter esti-
mation we have to decide on the distribution of the resid-
uals. This means the chain of hypothesis tests we propose
is highly dependend on these results. For that reason two
main approaches have to be outlined which may depend on
the Normal Distribution and symmetric distributions respec-
tively.
3.1 Hypothesis tests based on Normal
Distribution
In order to verify the results of the parameter estimation
let us first start with checking the residuals. Therefore the
following initial test has to be solved:
