makes use of Hamilton’s Quaternion representation for 
rotations. 
Experience has shown that this solution may not be very 
accurate: a least squares solution based on collinearity equations 
is therefore computed to improve it; at this second stage also 
other GCPs available in the image may be included. To this 
aim, the 7 parameters determined by RESECT are used to 
linearize equations (1). In a previous paper (Albertella and 
Scaioni, 1999), we compared the collinearity equations in 
general form (1) to that in normal form (where the scale factor 
has been eliminated): we found that the former have a larger 
convergence radius while the latter allow a more accurate 
computation of the solution. We therefore use both in that 
sequence, to ensure that the space intersection will provide good 
approximations for tie points. 
The analysis of singularities in the computation of the 
parameters or the conditions for the uniqueness of the solution 
have not been investigated analytically; a general discussion on 
the topic may be again found in Wrobel (2001). Some empirical 
tests with simulated data has been performed to gauge the 
sensitivity of the algorithm. 
In this respect, a problem coming from the original version of 
RESECT was the selection of the three points among the four to 
be used to write down equation (4): it turned out in fact that 
different combinations of them may give rise to different 
solutions. We got rid of this ambiguity by simply computing the 
solution for all possible combinations of the set of four points 
and keeping that with the smallest RMS of the residuals on the 
image coordinates, if smaller than a given threshold. From the 
simulations we performed, this always turned out to be the 
correct one. 
If more than four known object points are available, those 
defining in image space the quadrilateral with the maximum 
perimeter lenght are tried first; if no solution satisfying the 
above mentioned threshold can be found, all combinations of 
four points are explored. 
  
  
  
  
  
Figure 1 — Geometric meaning of symbols used in RESECT. 
Computation of Approximate Ground Coordinates 
The computation of a least squares bundle adjustment requires 
also the approximate object coordinates of tie points. Neglecting 
the proper definition of the stochastic model, it is 
straightforward to compute them by writing down the 
collinearity equation in linear form; nevertheless, our 
experience suggests that results are not accurate enough when 
dealing with convergent images. Then we implemented an 
approach based on a vector representation of two homologous 
rays, presented by Cooper and Robson (1996). The formulas to 
determine the object coordinates of the point are given in the 
following, according to the geometric scheme in Figure 2. 
Given a pair of oriented and calibrated images, let u and v be 
two unit vectors projecting two corresponding image points in 
object space. By properly choosing two positive scalars 7 and Ç 
the intersection of vectors 7u and Cv gives the point A in object 
space. Due to errors in image coordinates and in orientation 
parameters, we will have a residual parallax vector, given by: 
p=-uth+y (7) 
where b = [1 b,/b, b./b,] is the baseline vector. Parameters N 
and Ç can be determined so that length of p is minimum: 
.-(u: v)(v : b) * (ub) 
1-(u-v) 
_(u-v)(u-b)—(v-b) 
7] 
8 1-(u-v)? 
  
(8) 
0 
Let 7%, Ç be the solution of the minimum problem. The 
intersection point A will be located in the midpoint of p: 
a -— ru * 5 p — ^ (ru * b + wv) (9) 
while the magnitude of the parallax vector can be assumed as an 
indicator of the accuracy of the position of A: 
Po =-70u +b + Gov (10) 
This procedure yields straightforward the result and does not 
need further iterations. 
  
  
  
  
  
Figure 2 — Geometric scheme of space intersection. 
TESTS WITH SIMULATED AND REAL BLOCKS 
The check of the procedure described so far has involved 
several photogrammetric blocks, covering both terrestrial and 
aerial environments, different kind of sensors (metric analogue 
cameras and calibrated digital cameras) and geometry of camera 
stations. Some tests have been carried out also on simulated 
blocks. In this section, we refer to one simulated example and 
two real blocks, deemed representative of two different block 
geometry often met in practical applications. 
—456— 
At first, 
order t 
configui 
points, ¢ 
a series 
gravity | 
camera 
of a gric 
axis poi 
coordin: 
paramet 
test has 
whether 
to the | 
solved. 
sense) 
critical 
recover: 
for the « 
The sec 
procedu 
blocks. 
Wiki 
Blocks 
first the 
the sec 
and les: 
the foll 
The fir 
images 
sculptu: 
regular] 
at least 
block h 
a small 
have be 
by this 
comput 
the rol 
RESEC 
into the 
increas 
their ac 
every b 
constra 
recomp 
into the 
determi 
instanc 
but ime 
strengh 
The sec 
Cloiste: 
feature 
photos 
taken L 
unfortu 
acquisi 
the ima