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CALIBRATION OF CLOSE-RANGE PHOTOGRAMMETRIC SYSTEMS 1483 
The inclusion of distortion and affinity parameters into Equations (3) and (4) leads to: 
(x' — xe') * dr, * dp,' t dg. 
u' = R' (y' 2 y; + dr,’ + dp,’ + dg,’ (6) 
— ze! 
(x" — x,") + dr," + dp,” + dg." 
u" > R" (y" = y") + dr,” + dp,’ dE da, (7) 
—ze" 
where dr, = (x — Xe) (ko + kar® + kar* + kar$) 
dr, = (y — yo) (ks + kir? + kort + kar®) (5 
dp, = pir? + 2(x — x?) * 2pa(x — x) (y — Ye) 
dp, 7 ps(r* + Uy — 499) + 2 (X — x9 (9 — wo (9) 
dg, = A (y = Ye) (10) 
dg, = B(y — yc) 
Note the constant term k, in Equation (8) which is necessary due to the fact that z, remains 
fixed for this iteration step. The coefficients Kk, . . . ka, py, pz, A, and B are unknowns along with 
the elements of relative orientation, while r is expressed as 
rye((x ~ m2. + cw 
To include absolute orientation into the approach, control points have to be utilized. Rather 
than using collinearity equations—which is an option in the program and avoids step by step 
iterations but leads to a simultaneous solution at the expense of requiring more object space 
control—a control restraint condition is used. Prior to its use, the scale factors A’ and A”, which 
do not affect the coplanarity equations, have to be approximated for the control points. To do 
so, the space coordinates of point P (see Figure 1) are expressed in both photo systems: 
x, TX. us 
=v. | * v» ]w 
Z , 7 , ta) 
p c (11) 
X X" fis" 
Y," = Y." + AT us 2 
X» md u," 
Then coplanarity is expressed as 
® =(X, ~ XP? +(Y, = YP +L Zu = min (12) 
or 
8o 0D _ 
prar (13) 
which leads to 
(u,'2 uy? t u2) A' — (uz'uz" * uy'u," t uu") X" 
= (B,uz' + Bu, + Bu; ) 
(ur t uuu t ulus) A — (uy? + uw," + ui?) X" (14) 
= (Bu; + By," + B,u;") 
Equations (14) are solved for A" and X", expressing the base components according to Equation 
(1). These values for À' and A" are used in the control restraint: