ince
at
the
3ys-
ith-
ani
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ali
EO
in
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at
ner a
the
of
ated
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ated
(see
to
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don
ject
the
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ing
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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)