ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
  
  
drawing 
Figure 2: Relation between scene, image and drawing. 
Vertical line Li, passing through point Ui, maps to 1? — 
uf. Horizontal line L» has the same vanishing point as its 
projection 1. 
Minimizing the Euclidean distance of e and 0 
ele — p'A'Ap under Ip| — 1, 
leads to the well known algebraic solution 
A'Ap — Ap, 
namely the right eigenvalue of A. 
The statistically optimal solution instead of the Euclidean 
distance minimizes the Mahalanobis distance of e and 0 
el le =p ATH IA, under ID| —4. 
Here the covariance matrix of the residuals e — e(y, p) 
can be obtained from error propagation 
Oe(y, 
Xs BLISS B (y py im SU PL. pep. 
dy 
(3) 
It can be shown (see (Fórstner, 2001)) that the solution can 
be obtained iteratively from 
~(v ^ (v) — v LU 
AD TC) A) p^*D — Ap 
using 
Sv) 
£2) = (86.5) ES, BE.) 
where we need the fitted values y of the observations: 
zs T T: 
yg; = pr Du B; B;X,,Bi B; Yi 
and where the Jacobian B — Bm“, p^-D) has to be 
evaluated at the fitted values of the previous iteration (v — 
1), 
Selfdiagnosis and performance characteristics: The re- 
sult can be evaluated based on the optimal value of the Ma- 
halanobis distance 
026 ló-—plA'Yy. Ap ~ xl, 
where the Jacobian A = A(y) and X; needs to be evalu- 
ated at the fitted values. 
In case the assumed model holds, €) is x2, distributed with 
R degrees of freedom, where R is the redundancy of the 
system: 
R=G—(U—1) 
which is the number of independent constraints minus the 
number of unknown parameters, here 11. 
Assuming the test has not been rejected, we may use the 
estimated covariance matrix of the estimated parameters 
Ea QA XA)! with 52 = ; (4) 
to evaluate the obtained accuracy of the parameters. The 
calculation ofthe pseudo inverse can make use of the known 
nullspace p ofthe matrix. In case the geometric constraints 
hold, one can conclude that the observations have a stan- 
dard deviation which is larger by a factor of 62. 
Further, we could use the covariance matrix Se to predict 
the reprojection error: 
A A A aT 
35s — (la € X )Ess(l3 9 X")! - PXxxP (5) 
using (2), Ÿ = PX, and taking the uncertainty of both P 
and X into account. 
Or one could determine the covariance matrix 3 22 of the 
P - 
(A[h) = 
A ~—1~ 
estimated projection center Z = —H Hh from 
(H| — HZ) (for the derivation cf. appendix): 
A 
>= 
A71 AT i x AT 
22 -H (4 813) Mgg (28 I3) H 
This uncertainty may be compared with some specification 
coming from the application. 
3 SCENE CONSTRAINTS 
In the following we derive constraints between measurable 
quantities in the image on one hand and the map and the 
unknown projection matrix on the other hand. 
3.1 Vertical Lines 
We first turn our attention to vertical structures in the scene. 
Man made objects are very rich of such features, one may 
think of buildings, all kinds of technical installations, etc. 
Observations l;,; — 1,...,7 of vertical lines L; in the im- 
age reflect as a point coordinate in the drawing, say x¢ = 
(x7, y*);. Two 3D-points U; and V; on the same verti- 
cal line, therefore, only differ by their third component (cf. 
fig. 2). Thus, the vertical line is fully characterized by two 
points 
U; = (X, Zu DE and VieLY, Z1, 6) 
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