gram (CALGE), 
orientation starts 
ites of all tie ang 
lock geometry is 
er in this section, 
oints and control 
that aim in any 
he block without 
ameters and for 
use RESECT to 
/ery image and a 
to compute the 
is from an image 
/T with at least 4 
e do so by an 
from a kernel of 
measured image 
pute preliminary 
is feasible. It can 
tenation of single 
, we need in the 
re), which can be 
4 GCPs). The EO 
ised to determine 
d coordinates of 
rnel. To complete 
nd coordinates of 
justment. 
Ps and tie points 
mage point list of 
ing, based on the 
)bject points: this 
the images of the 
quate so that the 
itional GCPs are 
> measured come 
ew images which 
1 until the whole 
new tie points at 
pends just on the 
distribution of tie 
ny true additional 
cedure. Take for 
ar shape: for the 
lock, we need: 
rather than the 
ock ends, at least 
h strip. 
aint (the method 
50% forward and 
t the method is 
larger number of 
ze larger than in 
re to start from a 
n every direction 
en be more than 
n object points is 
e solution. If this 
that iteration. If 
m nearby images 
he orientation of 
the image. If this is not the case, it is necessary to measure one 
or more additional tie points, until RESECT can manage to 
compute the solution. 
Since the space intersection makes use of only pairs of rays, all 
combinations of intersections for the oriented images containing 
a tie point are computed and a robust measure of location and 
dispersion of the solutions is computed, to get rid of possible 
measurement errors, if redundancy is enough. If the dispersion 
is too high, the point is discarded. Then, before running the 
block adjustment, all tie points are re-projected back to the 
images and the discrepancies analyzed. If their RMS exceed a 
threshold, the image is not included in block adjustment. 
COMPUTING APPROXIMATIONS FOR 
EXTERIOR ORIENTATIONS AND TIE POINTS 
The Algorithm for Space Resection RESECT 
The problem of space resection can be solved by setting up a 
system of collinearity equations, which require at least three 
GCPs, whose object and image coordinates are known. The 
method used to compute the solution is usually a least squares 
approach, requiring a linearization around a set of approximate 
values for the unknowns, because the collinearity equations are 
not linear. Since close-range blocks often contain many 
convergent images and objects to measure may have a 3D 
complex form, to establish a good set of approximate 
parameters is not an easy task. Many solutions have been 
proposed in literature to solve the space resection without 
requiring approximation for the EO parameters, differing in the 
camera model (with or without a-priori knowledge of the 
interior orientation) as well as in the number of object points 
required. Among the others, algorithms have been published by 
Killian (1955), Fischler and Bolles (1981), Lohse et al. 
(1989), Zeng and Wang (1992), Tan et al. (1996): a discussion 
on the merits of these proposals is beyond the scope of this 
paper; an excellent review may be found in Wrobel (2001). 
We started from the original version of the RESECT algorithm 
of Crespi and Marana published in 1995, introducing some 
changes to improve its performance. This method, like that 
proposed by Killian, is based on the use of 4 full GCPs per 
image and does not require any a priori information on the 
exterior parameters. 
The RESECT algorithm starts from the collinearity equation in 
general form: 
rj- € t 4; RE; (D 
where r; and &;. are the known object and image coordinates of a 
point while the unknowns of the problem are the rotation matrix 
R, the position of the projection centre c and the scale factor of 
each ray 4; (see Figure 1). From each vector equation (1) three 
Scalar equations can be derived, so that 3 points would be 
enough to compute the 9 unknown parameters (3 components of 
¢, 3 rotation angles and 3 scales 4;). Unfortunately, this system 
is not linear and no close solution exist for it. Hence, three 
equations (1) corresponding to 3 GCPs are combined to 
eliminate translations and rotations, while scales A; are replaced 
by a set of new parameters: 
usi fg nih. i222 (2,3) 
where Lmn is the distance between the GCPs m and n in object 
Space. By substituting we get: 
  
11-308, +l=u (a) 
2 > L 
(5 20563 + 1= u (b) (4) 
; Li, 
3 : I 
t — 20,56, +1, - (c) 
12 
where Sinn Em En- 
The system (4) is of degree 8 in three unknowns y, t, and t. 
Because its solution is very complex, the problem can be 
simplified by introducing a fourth GCP, which allows to reduce 
the system to degree 3 and to eliminate the parameter £4. The 
new resulting system is made up of two equations, the first 
written on the basis of GCPs 1-2-3, the second of 1-2-4: 
Byi A Cpj f * Dii j£. * Epi 45 7 Hy i=3,4 (5) 
where the coefficients are: 
a 
Bi = 2A,,(1 ES Ei) + (e? + ez = enc tn) Di (és = £z) 
  
  
  
L, 
Ca =2&,H 
p, : 
Dy =2 ^, (5,6 én) t DP e, és] i734 
12 
au? 
E, = 45 -£ — 
12i 2i e L, 
Hy, =(é + £2 rc d tuts ~1) 
L, = L, T. L, 
À, => 
. L, 
The equation system (5) has a solution for the parameter u 
given by the following equation of degree 3, which can be 
solved by means of the Cardano’s formula: 
aol + ai +ayp +a; =0 (6) 
where: ao = Di23E124 — E123D124 
a, = D123B124 — B123D124+ C123E124 — E123C124 
da = C123B124 — B123C124+ H123D124 — D123H124 
az = Hy23C124 — C123H124 
The number of real solutions for (6) depends on the sign of the 
discriminant and can vary from 1 to 3. For each positive 4 (it 
cannot be negative by definition), a value for t can be derived 
by equation (4a); imaginary values for t, are immediately 
discarded. The problem is then to select among possible pairs of 
parameter x and f£. In the original version of RESECT, this 
selection is performed through geometric tests, which require an 
initial approximation for x. After several tests performed by this 
method, we realized that for many geometric configurations of 
the exterior orientation, the algorithm failed to compute 
accurately the scales and then the spatial resection solution. A 
modified approach has therefore been implemented, based on 
computing a solution for the EO parameters using all possible 
values of 4 and t;. The correct solution is then selected on the 
basis of the residuals on the image coordinates, computed by 
projecting the GCPs onto the image by using the estimated 
parameters. For each pair (x, £5) all the parameters f; required 
for the determination of the scales A; are derived by equations 2, 
4b and 4c. 
Once scales have been determined, the problem is reduced to 
compute a 3D conformal transformation (7 parameters) without 
the knowledge of any initial values for the unknowns. The 
solution implemented is that proposed in (Sansd, 1973), which 
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