N 
. The design and obtaining of the digital stereo pairs and GPS 
positions of cameras; 
3. The recognition and measurement of the targets images; 
4. The mathematical model of camera calibration that used is 
the self-calibrating bundle adjustment. After Fraser (1997) 
only the k1 term is used for radial lens distortion. 
f al(X-Xs) +bICY- Ys) - cl(Z- 75) 
a3(X- X9 -b3(Y- y.) -c3(7- 73 
a2(X- X3) - b2(Y- Y3) * c2(Z- Z3) 
a3(X- X9) -bJ(Y- Y) -63(2- 73) 
  
x—-x0+Ax=— 
  
y-y0+Ay=—f 
In Equation (1), (x0, y0) and f respectively are the principal 
point offsets and the principal distance of camera, (Ax, Ay) is 
the image coordinate correction due to lens distortion, (x, y) and 
(X, Y, Z) are the coordinates of image point and its 
corresponding object point respectively, (Xs, Ys, Zs) are the 
coordinates of the exposure center, and the (ai, bi, ci) i=1,3 are 
respectively the functions of three rotation angles (w,p,K): 
al- cosQ cosk 
bl- cosk sinq sino + cosw sink 
cl— -sinqQ coso cosk - sink sino 
a2— -cosqQ sink 
b2- -sink sinQ sino -- coso cosk ) 2) 
€2= sink sing cos® + sinw cosk 
a3= sing 
b3- -cosgsino 
c3= COS COS J 
Ax =klxx r+ k2xx r+ plxQx; *r) *p2x2x,y, 
Ay - kIx y,r? k2x y,r'* plx2x,y,* p2x y: « r?) 
  
Xr-X-—-xQ (3) 
vr-Vv-vg 
2 2 2 J 
[^ — XT +Yr 
These equations result in 10 unknown parameters. That means 
at least 6 targets are reqiured to provide minimum redundancy 
during calibration of one camera. After linearisation, the error 
equations of the least squares bundle adjustment are expressed 
as following matrix equations: 
V=AX-L (4) 
T 
L=(x-> y-r) 
T 
x=(x, Ys Zs € O K f x9 yo ki) (5) 
A= di di dis dis dis dis d; dis dio diio 
d do» do Qo. As As à ds Qs dio 
In Equation (5), x' and y' are the image coordinates computed 
by means of approximate values of the parameters according to 
the collinearity equations; a; (i=1,2, j=1,10) are the partial 
Since the convergence of the least squares solution depends on 
the initial values of exterior parameters, a DLT algorithm 
(Karara, 1989) is used to provide this information. The DLT 
equations with 11 unknowns from which all the camera 
parameters can be derived are expressed as Equation (7). 
  
ATAX=ATL 
Xs(aTAy lATL } (6) 
Xs - Iq X*LoYtLsZtLA4 
X LoX+L10Y+Lj12+} (7) 
2 | L5X*L6Y*L7Z* Lg 
  
Mie TY UTTIZ 
i 2 2 2 
L- V/2+L2, +12) (8) 
xo-(LixLo*L2xLio*LaxLijL? \ 
yo=(L5*L9+L6*L10+L7<L11)L2 
Fx= L213 +1312 x3 Nx 
2 
  
  
zr lli 2) 2 ti 
f-(fy*fy)/2 ) 
@ = tan 7! (EL10 /Li1) 
  
\ 
95 igno 4) 
aj 2 L(xoXL9-Lp/fx 
K 7 cos" l (a1/coso ) (10) 
Xs Laos L2m daüli2 | 54 
Vo [ES 7b scr L8 
Zslsl. L9. 1051 1 ) 
5. The relationship between cameras and GPS antennas. 
Following calibration the distances and angles among 
cameras and GPS antennas can be considered as fixed values, 
They can then be used to reconstruct the imaging geometry 
for subsequent photogrammetric intersection. 
  
  
Figure 2 the relationship between 
mapping frame and sensor frame 
whole sensor 
GPS observat 
differential c« 
cameras then ¢ 
the sensor sys 
camera positic 
observations a 
If no rotation : 
between sens( 
about y-axis) 
matrix can be 
CO! 
R =| — ci 
Let the left Gl 
and its coordi 
coordinate of 
frame as (AX 
determine the: 
position in ma 
any position 
expressions: 
In the future v 
rotation matri: 
then can be cc 
the same, the 
camera exteri 
calibration. T 
differential ro 
orientation of 
without field c 
6. Forward 
their ster 
* Comput 
image p: 
X 
Y 
Z 
X 
y 
7 
* As the 
unknown 
two conjt 
I 
derivatives with respect to the 10 unknowns for x and y. Based As illustrated in Figure 2, the relationship between mapping Z= 
on the least square adjustment, the normal equations and their frame (M-XYZ) and the sensor frame (S-xyz) can be expressed 
solution can be obtained as Equation (6). by six parameters, i.e. 3 rotations and 3 positions, because the 
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