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.
Lik
mul:
sec
the
pre
cal
lef:
mo
Th:
for
The
eac
fro
bet
a)
b)
ito
fun
Froi
The
eac
ate!