image coordinates along 
the linear feature 1 in 
image j, 
polynomial coefficients in 
image j, and 
m, order of polynomial in 
image j. 
All GPS observations along the linear feature 
on the ground must belong to this function, when 
projected into image space. In other words, the image 
coordinates of theses GPS observations must satisfy (3). 
This constraint will be treated as an additional condition 
equation with parameters in the bundle adjustment. The 
transformation of GPS points of the linear feature into 
the image can be achieved with the collinearity 
equations (4)(5). 
7 X -X * y -Y tr Z -Z 
u s 2m Ye 2 3» Ye oJ 
  
d. j J. J dd J 
he @ 
l y» (X -X Y+r (Y -Y )4r (2 —-Z ) 
J B- 6. o 23 G. o. 3 G o 
J J J J J J J J J 
r (X -X )«r (Y 
12. G 0 2 G. j . . 3 
j j j J J T j 
y =-f (5) 
(x. -X Yer (Y —Y )er (2 -2 ) 
3. 363. o. 3 G o. 3 6G ; 
j j j j 3 j j j j 
-Y 2 -2 
oe Ze ) 
  
GPS observations along 
the linear feature within 
image j, 
corresponding image 
coordinates, 
ground coordinates of the 
perspective center of 
image j, 
Rp e ,734, elements of the rotation 
j 
matrix of that particular 
image which are functions 
of the rotation angles 
o ,$9 ,x ,and 
LJ J 
f focal length. 
In equation (3) the right hand sidcs of 
equations (4) and (5) are substituted for X, and y,, 
J 
which yields (6). 
EA ER A i, 9 9 uud aeu a 
3.0 0 eye y e m. 
Thus, we introduce the orientation parameters 
of the photograph into this geometric constraint. 
Equation (6) can be treated as a condition equation with 
parameters in the bundle adjustment. In this equation, 
the perspective center coordinates and rotation angles 
are unknowns, while the GPS observations along the 
linear feature and the polynomial coefficients are 
observed quantities. This model also accounts for the 
accuracy of the observed GPS coordinates along the 
linear feature and the computed polynomial coefficients 
as obtained from (3). 
There are two practical problems in 
implementing this algorithm. Namely, choosing the 
GPS observations along the linear feature that belong to 
a particular image and selecting the order of the 
polynomial. 
The GPS observations along the linear feature 
within image j can be obtained as follows: the corners of 
the image are projected into object space using the 
initial values of the exterior orientation parameters of 
image j as obtained from the GPS observation in the 
aircraft and the approximations of the rotation angles. 
The GPS observations along the linear feature that fall 
within the projected area plus a threshold are selected. 
The order of the polynomial can be chosen fully 
automatically by statistical testing of the computed 
parameters. We begin with a high order polynomial and 
check the significance of each coefficient by testing of 
hypothesis, (7), which tests whether the coefficient 
under consideration is significantly different from zero. 
Ha —N(0,07) © i22,...m (7) 
with 
o? is the variance of 
coefficient à; as obtained 
from equation (3), and 
a; highest order coefficient 
of the polynomial. 
This null hypothesis (H,) can be tested through Chi- 
squared tables, as follows: 
206 
if 
otherwise, 
where: 
o is 
df is 
If 
that the : 
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is reduced 
coefficient 
hypothesis 
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