1.1) of
by a
ise of
3.1)
on the
| case.
! used,
nd the
). YN 7C,
3.2)
AN ap-
irallel
nts in
'ormula
3.3)
1. The
rom P’
d from
je ex-
3.4)
om the
photo-
at the
|, the
ng K’,
ng K'.
by the
2.1.1)
[.2)
ecause
i1s*i2"-i2*i3" ^3" ja "712! ja" - ia" ka" - do" Kg"
Cz | js?i2"7-J2?da" ^ 3s? ja "-ja" ja" ja'ke"-je" Ks"
ks'i2"-k2'ia" ks'ja"-ke'ja" Ks'ka"-k2'ka"
(2.13.3)
and contains only the second and third components
of the 1, j, k.
2.2 Computation of the parameters
First of all it is to be assumed that the coor-
dinates xo’, yo’ and xo", yo" of the epipoles are
already calculated from
xg’ = 0 and ZX" =. 0.
They are the images of the base given by
Ao’xo’ = R’Tb and Ao"Xo" z-R'Tb,
or
xo’ cos?! cos Xo" cose"cosK"
^o'|yo' ||-cose'sinK' | and Ao" |yo" |-- |-cose" sin" |,
-C sine -C sine"
from which independently from Q" follow
yo -C
and tan¢ =
tank = -
Xo 4 Xo? *yo?
(2.3.1)
for both images. By means of these parameters R’=
z[i',j',k"] is known.
The still missing parameter Q' of R" may be cal-
culated now from any component of (2.1.3). The best
way is to use the third column
(iz'cosQ" * is'sinQ")cose" - cs2z13a
(j2'cosQ" + js'sinQ")cose" - cs2zes
- ks'sinQ" cos?" z ca2233a
and to eliminate cos" by
C32233
cose” =
- ks'sin2"
Therefrom the two symmetric possibilities
233 i2’
tan@'= ———————— —
Z13 ks! - za3 is?
233 j2'
= re eme (2.3.2)
Z23 Ks’ - z3s js”
orientation has been linearized by more obser-
vations than necessary. Moreover, C is calculated
irrespective of the conditions of rectangularity
and normalization of the unit vectors 1, j. k, so
that an iterative post-processing must take place
in order to get an algebraically and stochastically
consistent set of parameters.
2.3 Adjustment
The rotation matrices of section 2.2 undoubtedly
will be very close approximations (R) to the most
probable solutions R. Hence small additional ro-
tations dR will give the final position of the
images according to
0 -dK d4
R = dR(R) = (E+dA)(R), dA = ax 0 -do
-dé "dO 0
By means of a vector v'z(vx,vy,0) of the residuals
of coordinate measurement and by neglecting quanti-
ties of second order, (1.2.3) turns to
(x’+v’)T{(E+dA’ ) (R’)}TB{(E+dA") (R") } (X"+v")=
zX'T(C)x" *v'T(C)x" *x'T(C)v" * (p' )T dA'TB(p")4
+(p’)TBdA"(p")=o,
wherein (C)=(R’)TB(R") and (p)=(R)x. Because of
0. -dé -dk
dA'TB - 0 0 0 »
0 0 0
0 0 0
BdA" - dé)" -doO" 0
dK” 0 -00"
and using the substitutions dp=x’T(C)x"(=parallax),
v'T(CO)x'zv'T(h'), x''(C)v'z(h")'v" (according to
equ. (1.2.6)), one linearized coplanarity equation
(without round brackets at h and pi) reads
op + h'.v' * h'.v" = p1’p2"de®’ + p1’ps"dK’ -
- (pa'"pe'*ps"ps')dQ" - p1"p2’de’ - p1"ps’dK"
and represents formally the general case of least
squares adjustment, i.e. conditions with unknowns.
But as the residuals of one equation do not appear
in any other equation (Tab. 2.3), the procedure can
be simplified by introduction of the fictitious
residuals (Wolf 1968, p.105, Rinner 1972, p.402)
Ww = h1’vx’+h2’vy’+h1 "vx "+h2 "vy"
and the related weights
1 h1’2 hz’? hi"? h2"?
arise for the determination of ©", which result me m + + + ;
from the fact that the transcendental problem of g gx"? Oy’ ax" dy"
Residuals Unknowns Par.
i|vx'vy'vx" vy" vx'vy? vx" vy" ...... vx'vy'vx" vy" ]de'dC de" de"dK"| Sp
1/59 8 B H B B B B B Sp1
B H B BR BH
2 8B HB NH HN
8
ôp2
B B B H|H EH BR HB B | Ops
Tab. 2.3: Scheme of the linearized equations of coplanarity
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