ace 
point 
ents. 
rs of 
ation 
ation 
hree- 
mea- 
spect 
), the 
@.B) 
rays 
t ori- 
e can 
oints 
1 of a 
(See 
down 
  
Xc1 = à11X + 312Y t a13Z t a14 
| (2) 
yc1 = a15X + a16Y t a17Z + 318 
for the left image, and in the form 
Xc2 = a21X + a22Y + a232 t a24 
(3) 
x 2 (Xr Yo Zei) 
Z / 
model space + Á Z Ded en 
e 
Y Cor 10.0 — 
"ans SR 
Ce 
X object space 
Figure-2 : relative and absolute orientation of a 
stereopair of affine images 
Yc2 = a25X + a26Y + a27Z + a28 
for the right one, respectively. The condition that 
Equations 1 and 2 are valid for all object points 
photographed in common on the left and right im- 
ages can be formulated as 
ail aı2 913 314 -Xe1l 
815 aie 28317 aig -Yeil 
a1 a2 a3 a4 -Xc2 
a25 3826 327 2828-yc2 
- 0 (4) 
which is equivalent to the coplanarity condition of 
corresponding rays. Under the condition of Equa- 
tion 4 we can form a three-dimensional space 
(Xm, YM, YM) which can be transformed into the 
object space (X,Y,Z) by a three-dimensional affine 
transformation having 12 independent coefficients, 
Les 
XM B1X + B2Y + B3Z + B4 
YM BsX + BY + B7Z + Bg (5) 
ZM = B9X + B10Ÿ + B11Z + B12 
Also, the space (XM,YM,ZM) is equivalent to the 
model space. From the results obtained above we 
can find the following characteristics of the orien- 
tation problem of overlapped affine images: 
1) The coplanarity condition of corresponding 
rays can mathematically provide four orienta- 
tion parameters among the sixteen ones of the 
stereopair of affine images, and 
2) The one-to-one correspondence relating the 
model and object spaces can be uniquely deter- 
mined, if four control points are given in the 
object space. 
GENERAL FREE NETWORK THEORY FOR 
AFFINE IMAGES 
In the simultaneous determination of both orienta- 
tion parameters of overlapped affine images and 
coordinates of object points, the basic equations are 
given in the form of Equation 1(Okamoto and 
Akamatu(1991)). These basic equations can be lin- 
earized as 
Xc = Xe + XAA1 + YAA2 + ZAA3 + 1°AA4 
+ AJAX + A2AY + A3AZ 
(6) 
Ye = Yc + XAAs + YAAs + ZAA7 + 1°AAg 
+ AsAX + AgAY + AJAZ 
Setting up Equation 6 for all affine images under 
consideration simultaneously, we have a system of 
linear equations in a matrix form as 
AA =¢ (7) 
in which 
A : a coefficient matrix of the system of linear 
equations 
Ax : a vector of corrections to unknowns 
€ : a vector of constants. 
Without object space controls the matrix A is sin- 
gular and its rank deficiency is twelve. In the gen- 
eral free network theory for affine images we have 
therefore twelve linearly independent vectors satis- 
fying the following relationship: 
AG = 0 (8) 
where G is a matrix constructed from the 12 lin- 
early independent vectors, i.e., 
G = (gi, 82: ,12) (9) 
g1 2(A1,0,0,0,A5,0,0,0,-X, 0,0) 
gl =(0,A,0,0,0,A5,0,0,-Y,0,0) 
gl =(0,0,A1,0,0,0,A5,0,-Z,0,0) 
ge! =(0,0,0,A1,0,0,0,As,1,0,0) 
gl =(A2,0,0,0,A60,0,0,0,-X,0) 
ef =(0,A2,0,0,0,A6,0,0,0,-Y,0) (10) 
ge} =(0,0,A,0,0,0,A5,0,0,-Z,0) 
ef =(0,0,0,A,0,0,0,A6,0,1,0) 
gi (45, 0,0, 0,45,0,0,0,0;0 X) 
T 
210=(0,A3,0,0,0,A;,0,0,0,0,-Y) 
el, =(0,0,A3,0,0,0,A7,0,0,0,-Z) 
gl.=(0,0,0,A3,0,0,0,A7,0,0,1) 
These 12 vectors can easily be found by linearizing 
the three-dimensional affine transformation (Equa- 
tion 5) and expressed in the form of Equation 10. 
Seven of the twelve vectors are related to the 
three-dimensional similarity transformation and 
the last five vectors pertain to a model deforma- 
tion.