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international Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
PUEY. 2y=0, va, Xr eV va, val 
; 1 2) 
TE AY teda Za TE Ed 
my gm» Ha 
DO Y Zim Su dues (3) 
i=0 j=0 k=0 
0sm,s3:.0sm, $3,0sm.s3 em, -m,m, $5 
Equations (1) are known in literature as Upward RFM as they 
allow the image coordinates to be obtained starting from the 3D 
coordinates of a ground point. In order to proceed with the 
estimation of the transformation parameters a; bj c; e d; (i = 
0219), it is necessary to trigger a least square iterative process 
afier having linearised equations (1). This procedure has been 
implemented in IDL language (Interactive Data Language) so 
as to be able to adjust for the lack of transparency of 
commercial software packages. 
Some critical elements of the algorithm are here underlined. 
First of all, in order to avoid numerical calculation problems 
(truncation or under/overflow errors), it is necessary to 
normalise both the image coordinates and those of the object in 
the interval (-1;+1) (OpenGIS-OCG, 1999). Furthermore, as the 
denominators of the polynomials assume very different values, 
in function of the distribution of the GCPs and of the altimetric 
range, it is probable that the coefficient matrix of the least 
square system results to be ill-conditioned. As a consequence, 
the normal matrix can result to be singular, in particular when 
polynomials of a high degree are used. The iterative process 
often does not converge in this case. : 
If we presume that the camera model is not available to the 
users, it is necessary to choose the ground control points in a 
conventional way, that is, through collimation of the 
homologous points on the cartography/DEM or through specific 
GPS survey campaigns. As it is not possible to obtain a regular 
distribution of the GCPs, it is therefore necessary to implement 
a numerical regularisation algorithm to make the iterative 
process converge. 
Tikhonov algorithm is one of the most commonly used 
regularisation algorithms for the resolution of ill-conditioned 
systems: it consists in the adding of an arbitrarily small constant 
A” to the diagonal of the normal matrix in order to improve the 
conditioning number. The resolution equation of the least 
square system (4) is then changed as shown in (5) 
Xo ac may um o (5) 
The choice of Tikhonov's parameter is not univocal and it is in 
fact obtained by empirically elaborating numerous solutions 
while varying the X parameter, choosing the one that minimises 
the rejects on the control points. 
The numerous tests that have been carried out have shown how, 
even though the iterative process converges, the subsequent 
orthoprojection step, in some cases, have some problems that 
are connected to distortion of the generated images which are 
probably due to the use of polynomials of too high a degree. To 
verify the correctness of this hypothesis and in order to avoid 
having to “a priori” choose the degree of the polynomials that 
has to be used (that is, the number of parameters for each 
polynomial), an adequacy analysis of the model was 
implemented. It is based on two different statistical tests in 
order to automatically determine whether a coefficient is 
necessary and which coefficients are not necessary. 
The xy? test with allows to verify whether the model was 
overparametrized. In the case in which an overparametrization 
is shown, another test on the significance of the estimated 
unknowns is performed to determine how many and which 
polynomial coefficients are not necessary. 
The standardised Z parameter is calculated according to the 
following relation: 
W=— (6) 
  
ae ^ Rn e «tl dr MT. ^ . . 
where: x, = the i" estimated coefficient and, 6, — i^ estimated 
r.m.s. In the case in which the following statistical test (7) based 
on the Student distribution, is verified the relative coefficient x; 
is placed equal to zero: 
x 
Wish (7) 
i 
  
  
Where: /,, = the value of the Student distribution for the 
relative value of redundancy (n —r), n = number of equations, r 
= number of unknowns 
The coefficient is annulled through the addition of a new 
condition equation to the initial system and by inserting an 
elevated weight in the corresponding position of the weight 
matrix.. 
2.2 Neural Network Model 
The neural network approach to the orthoprojection of 
satellite/aerial images can be considered an innovative attempt 
to solve the problem of the correction of images through non- 
parametric methods. 
The neural network consists of mathematical models whose 
operative philosophy is inspired by cerebral biological 
dynamics: the calculation process is schematised as a flow of 
distributed information whose elaboration occurs inside 
dedicated calculation units, which are known as "neurons" of 
the network. Some of these receive information from the 
external environment, others return answers to the environment 
and still others, if there are any, communicate with only the 
units inside the network: they are called input, output and 
hidden units, respectively, as shown in figure 1. 
input layer 
‘ k 
X. v output layer - 
er rf \ i E 
— 0 Kk ; + 3 dE 
au Su 7 z Ke Jd 
X x 
da” ve 
EN Nm [ 
EC , x / wA ] — ) E. 1 
/ x F 
f o eu 
bnc ion 
cba / 
hidden layer 
Figure 1 — Concept scheme of a two layer (hidden and output) 
computational MLP neural network