e evalu- 
ed with 
y of the 
inus the 
use the 
jeters 
(4) 
rs. The 
e known 
straints 
a stan- 
predict 
(5) 
both P 
; of the 
(Hh) = 
fication 
surable 
and the 
> scene. 
ne may 
S, etc. 
the im- 
y QD 
> verti- 
ent (cf. 
by two 
6) 
ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002 
  
where X; — gi and .Y; = yd. The two Z-values Z4 and 
Z» may be arbitrarily chosen, however, for reasons of nu- 
merical stability they should be chosen within the range of 
the heights, e. g. the minimum and maximum height in the 
scene. 
In general, for the projection of a 3D point on the im- 
age, say x; = PX, which lies on a line 1; in the image, 
17x; E] TPX; — 0 should hold. Hence, two points on a 
vertical line provide us with two constraints on the projec- 
tion matrix. We have 
ITPU. vod pu 1! e UT ! 
ÜPV, ) = es =0 or Ev lp-o- 0. 
(7) 
Thus, the two residuals e; of the constraints should vanish. 
Introducing a third point constraint on the same vertical 
line would not add any information since the line is al- 
ready fully defined by two points. When adding a third 
constraint, the rank of the matrix on the left hand side of 
(7b) would not exceed two. 
3.2 Horizontal Lines 
Besides vertical lines, horizontal lines are frequently ob- 
servable features. In this case, observed lines in the image 
correspond to lines in the drawing. 
Let the line in the drawing be defined by two measured 
points x4 and yd. The point at infinity of that line is (y?" — 
zit. 0). As the point at infinity of a line does not depend 
on its position we obtain the point at infinity of the 3D-line 
as 
yf a} 5j 
2x1 
0 
Actually R; and S; are the coordinate differences of the 
two points defining the direction of the line in the drawing. 
The observed line 1; should pass through the image of 
X oj, namely PX ;, known as the vanishing point. Hence, 
we obtain 
PX; =e; 20 or (L@Xo)'p=e; +0, @) 
only constraining the first two columns of P. They do not 
have any influence on the third and fourth column of P. 
In fact, by providing a line in the drawing, we only define 
the direction of this line. We are neither constraining the 
line’s height nor can we use the horizontal position of the 
line. Therefore, we also may use contours of horizontal 
cylinders with a given direction. 
3.3 Observed Points 
The third type of scene constraint we consider in this paper 
is the observation of points in the image which are marked 
as points in the drawing, which is the classical setup of es- 
timating the projection matrix from points. Let the obser- 
vation of a point feature in the image xx = (uk, vk, Wk)" 
be corresponding to a point element in the drawing X; = 
(Ug, Vie, Wi, Tie). Both points are linked by projection 
such that x, = PX}, holds. Thus, we obtain the constraints 
XE X PX 
fro) 
or Su. P Xr = —Spx, Xk = fx 
or (S. 0G XDpso f, 0. 
Only two of these constraints are independent. As the mea- 
sured points are finite, the first two constraints are guaran- 
teed to be independent. Thus, we use the two constraints 
S,, PX, = —Spx,Xx = ey + 0 
m | (9) 
or Sur CO XLpoe 29 
T 
eof Ww ur 0 —wk Uk 
where wu; is the i-th unit vector. 
3.4 Other Constraints 
In (Bondyfalat et al., 2001) additional constraints are ex- 
poited and used to constrain the fundamental matrix F and 
the projection matrix: Observing parallel lines in a given 
plane, leads to constraints which are linear in the elements 
of F but quadratic in the elements of P. In our scheme 
we, therefore, cannot include them in the same manner. 
However, parallel lines which are in the horizontal plane 
are likely to be contained in the drawing and parallel lines 
not being horizontal or vertical are not very likely to be 
present at the object. The same argument holds for observ- 
ing two orthogonal lines in a given plane. Not using these 
two types of constraints, therefore, is practically accept- 
able and gives the way for a direct and an optimal solution 
of P. 
4 ESTIMATION OFP 
We propose to use a two step procedure to estimate P sim- 
ilar to (Matei and Meer, 1997). In the first step we solve 
directly for P using the classical algebraic solution. In the 
second step we take the uncertainties of all measurements 
into account in order to obtain a statistically optimal solu- 
tion. 
4.1 Direct Solution 
In order to obtain a first estimate for P we integrate all 
constraints into a system of equations and solve in a least 
squares sense. 
Writing the constraints for ¢ = 1, ... observations of ver- 
tical lines, j = 1,...J horizontal lines, and k = 1,... K