62 ; d] 202 1
6X, A;-B,'
OZ A, BR
02, A, B, . 0. AcnBs
eX X , m x; , E
P ALB. O]A.-B. U@P
[55 -X - Bi. = (n a),
(Ax zm Bx Y oP op
oy. Y, * OA y = Y, * OB i
@ ^A -B, (COP J. A,-B, (OP
AY -X-BY,+X, | (oA, OB)
[A —5.] oP OP
@ c Z0 yz NIS
OP A -B, CP / A,-B,. @P
(AZ -XCBZ,-*X, (S- 2
(A, -B,) OP oP
In these equations, P stands for the parameters of camera- and
inner orientation. All partial derivatives are then plugged into
the law of error propagation to form the wanted 3D model.
3. TESTING THE MATHEMATICAL MODEL IN 2D
In Fig. 1, the points 1 and 2 are cameras position, 3 is an object
point on a building. The coordinates of the 3 point are given
by:
sin p
X,=X,+b*— “cos à
: sin (x — à — )
(5)
Yı=Yırbi- * sin. 0
sin (x-a-B)
3
baseline
Fig.1: 2D model
Applying the law of error propagation (Koch 1988) to the
baseline b and the two angles a and f results in the following
formulas:
OX, Sinp,
Ob Siny
OY, = Sinp * Sina
Ob Siny
OX, _ b Sinf Cosa *Cosy | sina |
oo. Siny Siny
oY | inno * Cos
Lab Sap Sing’ Cosy Cos (6)
Ou Siny Siny
OX inp *
Eh Cox + sup Cosy Cos
op Siny Siny
Y : in
OY, Eh Sine Sinn Cosy + Cosp
op Siny Siny
Then the spatial errors of the 3 point are
2 2 2 2 2
mX," = ES [2] (S * ms [8 | * ms?
oo. p op p Os
(7)
2 2 2 2 2
mY, (o 2) ey «| Dp GR * ms?
Ou p op p Os
A comparison of equations (3) and (6) gives the following
results shown in Fig. 2 with
mP - (mX; + mY3)/2.
4. CONCLUSIONS
As can be seen in Fig. 2, the point errors of the 2D model are
slightly larger than of the 3D model, which indicates that the 3D
model works correctly under the given circumstances. The next
step of this work is to implement the derived formulas in a
visual computer user interface that allows designing
interactively a point network and optimizing it under predefined
restrictions.
References
Koch,K.R., 1988. Parameter Estimation and Hypothesis Testing
in Linear Models, Springer-Verlag Berlin, Heidelberg, p.
121.
Kraus,K., 1993. Photogrammetry, Ferd.Dümmler-Verlag
Bonn, Volume 1, pp.13-14.
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