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ting of the 
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uli- Venezia 
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rmation as 
the digital 
map layers 
x a5X; + b„Y; +C2 
where: 
reference system; at. 
X.Y. are the coordinates of point i in the cartographic 
1? 1 
are the coordinates of point i in the instrumental 
reference system; 
a; ,b; ,c; (j = 1,2), u, v are the unknown coefficients of an 8- 
nr 
parameter transformation. 
To solve system (1) for the unknown coefficients of the homo- 
logous transformation and for the tie point coordinates of two 
contiguous map units, it is necessary to linearize system (1) in 
its fixed part and its stochastic part: 
l, vs, = A11X1 + A14X4 + A158 
where: 
Ay; is the coefficient matrix of the unknown transformation 
parameters and cartographic tie point coordinates; 
A,4 is the coefficient matrix of tie point coordinates for carto- 
graphic and photographic layers; 
As is the coefficient matrix relating to the stochastic carto- 
graphic control point coordinates; 
x, is the unknown vector of an 8-parameter transformation 
coefficients and cartographic tie point coordinates; 
x4 is the unknown vector of cartographic-photographic tie point 
coordinates; 
sis a stochastic vector of cartographic control point coordinates. 
To perform the updating of an already existing digital mapping 
by pictures taken on the ground with amateur cameras, Crosilla 
and Visintini (1996) have already proposed to apply the so- 
called Direct Linear Transformation (DLT): 
  
  
xoada Xt EY iit Ta rx Kir? 
J LoX; t LigY; € LZ; 1 J J 
(2) 
do L;X;+L6Y;+L7/Z;+Lge era 
LX; + LY + LZ +1 71 075 
where: 
X; , y; are the image coordinates of point j; 
Xi,Y; are the coordinates of point j in the cartographic 
reference system; 
Z; is the orthometric height of point j; 
L, , … Ly, are the DLT parameters, function of the traditional 
internal orientation parameters (x9 , yo, c) and the external 
(Xe » Ye » Ze » @, 9, k) ones; 
K, is the coefficient describing image deformation due to 
objective radial distortion, being x; =x;'-Xg, y;-yi-—yo. 
2 = 2 12 
Ij =X; tyi : 
Also in this case to solve (2) for the unknown DLT parameters 
and coordinates of points necessary to update the map, system 
(2) must be linearized in the following way: 
12 + V2 = A22X3 + A23X3 + A 74X4 + Ag58 
Where: 
195 
À 252 is the coefficient matrix of DLT parameters and unknown 
height Z of cartographic points; 
A,3 is the coefficient matrix relating to cartographic point 
coordinates to be updated; 
À»24 is the coefficient matrix of tie points for cartographic and 
photographic layers, used to orient the pictures; 
As is the coefficient matrix relating to cartographic stochastic 
control point coordinates appearing on the photos; 
X, is the unknown vector of DLT parameters and orthometric 
height Z of cartographic points; 
X3 is the unknown vector of point coordinates to be updated. 
It seeems worth to consider both systems (1) and (2) in the 
same computation process since in this way tie points for 
cartographic and photographic layers, which are necessary to 
orient the images, improve the redundancy of the global linear 
systems and make it possible to take advantage of the 
correlation existing between variables of a georeferencing 
process and those of a DLT transformation. 
The global system can be written as a "mixed linear model" 
(Dermanis, 1990): 
  
  
  
  
  
  
  
l- v- Ax Bs (3) 
where: 
i51 Vul | ae i in By qu e 
1, V2 D An Az Az A3s 
and with: 
E(v)=0  E(v-v')=Q Es)=p, 
H-u,)- 6-1, |- 0. HG-n,- v ]- 9. =0 
As a consequence it holds that: 
Qu =BQ,B"+Q 
Qu -Q 
HB = E(l)= Ax+Bp, 
Qu = QB’ 
According to the general theory of estimation and prediction 
vector x will be estimated while the stochastic vector s will be 
predicted. 
The best linear uniformly unbiased (BLUU) estimate X of x 
can be obtained by: 
x =NTATQy 0 - oBy,) (4) 
while the BLUU compatible prediction S of s is given by 
(Dermanis, 1990): 
$- op, * Q,B'Qy ! HQ - oBp,) (5) 
and v follows from: 
Y z-1— AX — BS (6) 
where: 
H=I1-AN"A"Q, ' N=ATQ, A 
The parameter o takes the values: 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996