ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002 
  
points in a system of equations, we obtain: 
Qa; uJ b; ul Ci Ul 
  
  
Xl, 5X, XV, P. 
0. —wXI wX] Ps 
wy XI 0 —upX] (11) 
Yau] 
Tov] e; 
Tex le ne 
SX. 
the matrix being of size (27 + J + 2K) x 12. The solution 
which minimizes 57, |e;|? + 57, Je;|? + 37, |ex[? is the 
right eigenvector. 
Note that almost any combination of scene constraints may 
be used in a way that the overall number of constraint is 
equal to or greater than 11. However, at least two point ob- 
servations in different heights have to be provided in order 
to fix the vertical origin and the vertical scaling. 
The solution may be changed by arbitrarily weighting the 
individual constraints, therefore, it definitely is not opti- 
mal. 
4.2 Uncertainty of Measurements 
For achieving an optimal solution we may want to take the 
uncertainty of the measured entities into account. These 
are the points x, wd, yd and X, from the drawing and 
the points xj, from the image. We, therefore, have to make 
assumptions about their uncertainty, or — better — derive it 
empirically. 
In our investigation we adopt the following assumptions: 
Points in the drawing: We assume the points in the draw- 
ing to be measured independently with equal accuracy in 
both coordinates: Therefore, we assume their homogeneous 
coordiante vector to be normally distributed with 
d 
x Nn a Y 2.2), where a a = 07 
qd 
© © = 
© — © 
© OÖ © 
In case we know the height of a point from the drawing, we 
assume it to be defined with the same accuracy. Therefore, 
the uncertainty of the 3D-points taken from the drawing 
are characterized by: 
x = N(pxa, Yxaxad), 
1000 1000 
0100 0100 
Sxaxa = 0x4 | 9010 | OF = 0% 0000 
0000 0000 
With o,d = coxa. The second covariance matrix is taken 
for points with fixed heights. 
The points in the image are measured with a different ac- 
curacy. We assume the same simple structure of the distri- 
bution: 
x~Np.. Xow), where Bp = 0° . (12) 
OO = 
OS =O 
Cc CO 
2D-lines: Original measurements in the image as well as 
in the drawing consist of points only, points at infinity and 
lines are treated as derived entities. 
Following standard error propagation methods, we obtain 
the uncertainty of an image (or a drawing) linel = x xy = 
Sy = —S,X by 
öl ONT: gl 2 
> = Y M —e zn 
i: Ay ~¥¥ £3 i ox (=) 
T T 
= $,3,9,-T 8,148, 
which with (12) yields 
2 0 ES y 
> = 02 0 2 —D2 = V2 
-21—91 —22— ya yi^ t 01° + ya” + 12? 
assuming stochastic independence of x and y. 
Point at inifinity: The point X, has covariance matrix 
ad 20, 
S Xo: E 202, 0 0 . 
2x2 2x2 
4.3 Jacobians for Optimal Estimation 
We now provide the Jacobians A;(y;) and B;(y;, p) for 
the individual constraints. 
4.3.1 Observed Vertical Lines We have the vector y! = 
(L , x1) ofthe measured entities, relevant for vertical lines. 
Therefore, we obtain from (7) 
de; Il! GUT 
ar y ATL DU = gs i i 
A; (yi) A;(li, qd; ) op ( M ® vi ) . 
  
The other Jacobian can be obtained by 
Ae; Oe; e, OU, 
B.(y;.p) - Bil af, p) = ok = (250 O05 TU: 
(yi, p) (Li; 27, p) dy, (5 eo) 
U7P' ITP(l 0.) 
TpT T T 
VIP" ITP(l 0) 
  
  
using the Jacobian OU;/Ox? from (6). Note that we here 
only consider 1; and the coordinates X — z? and Y = 
y? on the drawing as uncertain. The heights of the 3D 
points, Z; and Zs, may be fixed arbitrarily and, therefore, 
are treated as deterministic values. 
4.3.2 
yj 7 
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