CIPA 2003 XIX th International Symposium, 30 September - 04 October, 2003, Antalya, Turkey 
In Eq. (1), radial symmetric lens distortion Ar has been ignored. 
It can be introduced as follows: 
Xj -(xj — x 0 )(kjr 2 + k 2 r 4 )-x F - 
(2) 
-[yi —(Yi -y o )(kir 2 + k 2 r 4 )-y F ]t = 0 
In this case, all point observations are adjusted in one step, with 
common unknowns the two distortion coefficients (kj, k 2 ). If it 
cannot be assumed that the principal point, about which distor 
tion is assumed to be radial-symmetric, is near the image centre, 
then a first solution without distortion is required to provide an 
adequate estimate for its location as regards distortion. Finally, 
let it be noted that the mean standard error of unit weight (cr 0 ) 
from the adjustments for the three vanishing points is a measure 
for the precision of the adjustment. 
2.2 Use of line elements 
In this case, the basic mathematical model, given in Petsa & Pa- 
tias (1994), could be summarised as follows. Let G be an object 
line and £ the central projection plane of corresponding image 
line g. Further, let n be the normal vector of e and 8 = [L M N] 
the direction vector of the object line. If R denotes the matrix of 
image rotations, then 
n'RS = 0 (3) 
is the condition for the orthogonality of the two vectors n and 8. 
This is equivalent to the parallelism of space line G (and, hence, 
of all object lines of the same direction) with projection plane e 
(and, hence, all projection planes of object lines of direction 8). 
The vector n is a function of the three interior orientation para 
meters and image line elements. Regarding the latter, two alter 
native formulations are possible, giving rise to two different ap 
proaches. 
In the first case, it is line parameters that are considered as ob 
servations to be adjusted; in the other case, all individual point 
measurements on all image lines are treated as observations. 
Use of image line parameters as observations 
Here, as formulated in Petsa & Patias (1994), parameters a, b of 
the ‘intercept form’ of image lines 
— + — + 1 = 0 (4) 
a b 
are assumed to be known after individual line fitting. Vector n 
is then a function of interior orientation and line parameters: 
n l =[-cb -ca ab + bx 0 -ay 0 ] (5) 
Setting 
R8 = [U V W] (6) 
and introducing Eqs. (5) and (6) into Eq. (3), finally results in 
cbU + caV -(ab + bx 0 -ay 0 ) W = 0 (7) 
Every object line of known direction gives rise to one orthogo 
nality condition (7). Image lines are first fitted and subsequently 
carried into the adjustments as observations, weighted by means 
of their variance-covariance matrix. This is approach B. 
Thus, contrary to method A, here a two-step process is adopted. 
First, points are simply constrained on lines (lens distortion can 
be estimated along with the line parameters); next, line parame 
ters are introduced into the perspective equations (7) to recover 
the values of the involved parameters c, x 0 , y 0 and co, (p, k. This 
approach has been reformulated here in a more rigorous one- 
step procedure, as follows. 
Use of image point measurements as observations 
In this case, an image line is expressed by each of its individual 
points and the line slope, assumed as t = Ax/Ay (under certain 
circumstances the equivalent formulation with t = Ay/Ax may be 
required). Thus, Eq. (5) takes the following form: 
n|=[ c -ct (xi-x 0 )-(yi-y 0 )t] (8) 
Accordingly, using again the quantities of Eq. (6), Eq. (3) final 
ly becomes: 
cU-ctV + [(xj -x 0 )-(yi -y 0 )t]W = 0 (9) 
All measured image points Xj, y; on all lines contribute one such 
condition to the adjustment. Hence, line fitting, camera calibra 
tion and image orientation are performed in one single step. The 
standard error of unit weight a Q provides the precision estimate 
for the adjustment. This is approach C. 
The coefficients of radial lens distortion can also be incorpora 
ted into the solution, as follows: 
cU-ctV + [(xj - x 0 ) - (yi - y 0 )t](l - kp 2 -k 2 r 4 ) W = 0 (10) 
Using Eq. (10) for every observed image point, it is possible to 
estimate simultaneously the 8 involved parameters (c, Xo, y 0 , kj, 
k 2 ; co, cp, k) along with the slopes t; of all image lines. Of course, 
such an adjustment might be somewhat sensitive regarding esti 
mation of distortion, as it depends on length and distribution (as 
well as noise) of measured image line segments. Yet, in the per 
formed tests estimation of distortion was in accordance with the 
results from self-calibrating bundle adjustment (see 3.3 below). 
Concluding the presentation of mathematical models, it is noted 
that approaches A and C are indeed straightforward, as the raw 
observations (namely, points on image lines) are adjusted in one 
single step under the constraint of central projection (in case A, 
the unknowns are then directly extracted from an equal number 
of equations). In this sense, approach B differs since the raw ob 
servations are initially subject only to a linearity constraint, and 
line parameters thus obtained become in fact the ‘fictitious’ ob 
servations in the final adjustment step (hence, weighting is here 
indispensable). 
3. PRACTICAL INVESTIGATIONS 
In the first tests, objects with no ground control were used. The 
purpose was to check the performance of the algorithms and the 
agreement of their results. In the next series of investigations, 
the results for a known 3D object were evaluated in relation to a 
multi-image bundle adjustment. 
3.1 Results from the different single-image approaches 
The eight images used here, seen in Fig. 1, had been taken with 
small format analogue cameras, using different lenses (35 mm; 
50 mm; the extremes of a 24-45 mm zoom lens). Since the en-