x} «is la A A x4l« ls. 
11 ^12. ^13| [Xv | 
Y Arq App Anal [Ym Bol (18) 
Z A31 ^33. ^3s| [7M BS] - 
is commonly used for 3-0 space transforma- 
tion, where S is a scale, A= (A4; ;) is an 
orthogonal matrix and B«(B.) ie a transla- 
tion vector. B and S are evaluated from 
gravity centers and a scale ratio of two 
coordinate systems. Thus eq.(18) is re- 
duced to the form; 
Xi 74 Em > 
(121,2. .n) (19) 
where suffix i means control point No.. X. 
and Xy, are coordinate vectors associated 
with the object space coordinate system 
and the global model coordinate system 
respectively. Their origins are assumed 
already shifted to respective gravity 
centers and X, are assumed to be scaled by 
S. The matrix A is determined so as to 
minimize 
n 
E = ECA Xpy47X451 (A XX 0) 
j=1 
This problem was already solved by some 
researchers (Arun, 1987, Horn, 1888). The 
authors adopted the Arun's method: By 
expanding eg. (20) one obtains 
E x 
j 
«Ms 
T 
OGTIXGU08 Xa I Xy 
; 
= 2Xwi TATX). (21) 
E is minimized when 
n 
frace( EGO AT X) 
$21 
n 
jet 
is maximized. With appropriate orthogonal 
matrices U, V which singular-value- 
decompose X (Xy; X3 ito 
n 
Fame = oval coon 
iz1 
where A is a diagonal matrix, the solu- 
tion of the matrix A is given as 
A = yul. (23) 
4.3 Evaluation of angular elements 
After all rotation matrices (M4 ;) associ- 
ated with the object space coordinate 
system ( or global model coordinate sys- 
tem) are obtained, they are decomposed to 
angular elements. Let the matrices related 
to angular elsements K, d and Q be ex- 
pressed simply as [K],[$] and [gl]. Here 
angles are expressed by capital letters. 
If the rotation order of angles is fixed, 
the matrix (M4 ;) can be singular and 
unable to be decomposed to unique angular 
elements. In order to assure unique 
202 
decomposition, one has to change the order 
of rotations depending on the values of 
elements of the rotation matrix: i.e., 
! 
bt 
1,(M45) =[Q][6][K] 
BY Tf. May = + 1,(M452 v" [fd E019] 
c) If Mig = + 1 and Mgq = + 1, 
([K][Q] [6] 
t 
(Ms) 
Since the treatments for any cases are 
similar, here only case a) is discussed. 
From equation sin & mM. 3 one gets two 
candidates for 6 for -Ti«ésTT. Since cos & 
# 0, 
= M53/cos &, 
Q 
cos Q * M43/cos 6, 
o 
2 
ia 
= 
M44/cos ©, 
sin K = M15/cos ó. (24) 
For" each candidate for d,. 9 and K are 
determined uniquely. They are tested on 
whether to satisfy the following equations. 
-Cos Qsin K + sin Qsin écosk = M54 
cos Qcos K + sin Qsin ésinK = M55 
sin Qsin K + cos Qsin dcosK = M21(25) 
-sin Qcos K * cos Qsin &sinK = Mj, 
Sets of candidates which do not satisfy 
all the equations are discarded. 
5. EXPERIMENTS 
The procedure was applied to two experi- 
ments for validity check; A simple rela- 
tive orientation of a pair of stereo 
photographs and a camera calibration 
without control points. 
Relative orientation of a pair of 
reo photographs 
1 [on 
ig i3 
A target field of 5m x 5m x 0.5m (depth) 
was imaged by a 35mm metric camera, PENTAX 
PAMS 645, f= 44.979mm. Two photographs 
were taken vertically in stereo with a 
base length of 1.5m, overlapping each 
other 50%. Common pass-points are 12 in 
number (minimum requirement is 8). This 
configuration is not good for the proce- 
dure of automatic adjustment but very 
COMMON in industrial photogrammetry. 
In nine eigen-values obtained from 
eq.(6), three of them were 0.0598,0.146 
and 1.02, while others are greater than 
100,000. As a result of applying the 
procedure mensioned in 3., a set of rota- 
tion angles with respect to the model 
coordinate system were obtained only for 
the third minimal eigen-value. The other 
eigen-values did not produce misleading 
false solutions. Residual y-parallaxes 
obtained in the ensuing precise orienta- 
tion were 7 um in RMS. Table 2 shows the 
approximations and precise values of 
angles. 
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