ince 
at 
the 
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ith- 
ani 
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ali 
EO 
in 
llo- 
at 
ner a 
the 
of 
ated 
ma- 
ated 
(see 
to 
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ject 
:han 
get 
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ject 
the 
zom- 
ing 
is 
of 
sed, 
(13 
oto 
Xp» 
S a 
)]a- 
Yp4Zpo-Zp4Ypo = 0 (25 
is rewritten to the form; 
P4*41*9*85541X9 ay Croiod4y4X9 
*tdoY1Yo *qay4(-8)tr4C-o)x»2* ra(-c)ya 
Taceo) = 40 (3) 
where 
Pry = Nosy Maar DEM ENN 2: 
Pa 1540939 03109597 Ay 152081 ME 
da - Maoñag Mala. d3 * 55033 0329n23* 
FQ * f59n947m53n5o4. "3,9 mo430327m33n22' 
r3 = MagN33-M33"23- 643 
It is easy to see that a vector 
a= (P4 P9 P3 d4 d2 dg ^4 no r3)! 
has a relation; 
Expressing eq.(3) in the form of 
an observation equation 
Xa = v , (5) 
where X is a design matrix and v is a 
residual yector, one can solve a by mini- 
mizing y'v. An objective function for 
this purpose becomes with a Lagrangean 
multiplier u 
U = ax xa - u(ala -23. (6) 
By diferentiating eq.(5) with X, one gets 
(XTx-uI)a = 0. 7) 
Namely a is an eigen-vector and u ig: a 
variance of residuals; |vl*/2. If an 
imaging configuration is good, only one u 
that is near zero is obtained. Or other- 
wise multiple candidates of u may be 
obtained, out of which the correct one is 
determined by the following procedure. 
3.2 Determination of the rotation matrices 
a 
nd angles 
Then the rotation matrices (ma 3) and (n4) 
are evaluated from the vector a. Even 
though Fig. 2-1 is assumed to be correct, 
Figs.2-2,2-3,9-4 as well as 2-1 ere -n- 
cluded in solutions. Figs.2-1 and 2-2 are 
equivalent, whereas Figs.2-3 and 2-4 are 
false, because they are turned over into a 
negative position. 
The rotation matrices must be defined as; 
(mg j= 
cos B 1 0 -sing, cosky sink, 0 
0 1 0 =s ink cosky 0 
sin DE 0 cOSg 0 0 1 
201 
= cos f4cos k4 cos B,sin k4 -sin 64 
-sin k4 cos k4 
sin gqcos k4 sin g4sin k1 COS f 
(8-1) 
(n4 27 
1 0 0 cos Bo 0 -—sin Ba 
0 Cos Wo sin Wo 0 1 0 
0 Sin wa cos Mo} |sin Bo 0 cos £5 
cos ko sin ko 0 
coin La cos ko 0 
= cos Bo cos ko 
-COS w^ Sin ko + Sin wa sin d cos k^ 
sin Wo sin ko + COS Wo sin Bo cos ko 
cos D» sin ka 
GOS W», COS ko * sin w4 sin g4 sin k 
2 9 2 2 ? 2 2 
-Sin wa cos Ko + cos Ww, sin Bo sin ko 
-sin $£» (8-2) 
sin Wo COS Bo 
COs Wy COS Bs 
It should be noted that the rotation order 
in the definition is unique. For other 
orders it can be shown that there are some 
angles at which the rotation matrix be- 
comes singular and fails to be decomposed 
to angular elements. 
3.3 Evaluation of £, 
Since Mog =0, from eqs.(4) 
Masha 7"^4: 
33020 7765! 
m33n53 ?-n"3. (9) 
And then 
£3 
m35? (n247*n557n53?) » r,* «notera? 
Since the photographs are assumed dia- 
positive, m44 ^» 0. From the orthogonality 
of (n3). 
m33 = rq? +r, 2+ Pat. (10) 
From eq.(10) two candidates of d 4 are 
obtained. Which is correct is suspended 
here. Then from eqs.(9) 
N91=-r1/M33 
N297-72/M33- 
n23*77n3/m33. (11) 
Multiplying the first, second and third of 
eqs.(4) with n54,, noo and n3 respectively 
and summing them up, one obtains 
May * -(p4nD54*P2n525*P3D23). (12-1)