=
of
and
rior
the
that
rary
ssed
-eye
ally
oto
S of
each
hic-
tily
ains
on
ngle
irst
and
(7)
|. The formula for fish-eye panoramic photograph when
equallv tilted photography.
Photography with equal tilt is that both two optical
axes of camera for fish-eve lens are relative to
horizontal direction to tilting jointly a same angle,
right now: A=4=kh1=Kk=0, 0,=0.=0. But in parctical
operation, the value of small angle is existence
throughout 4#:.d%:.d% and dkz , therefore the rotation
matrix of directional cosine R», R: has the following
form:
1 dfcosw) — (dK + sin wd?) 7
R= Ri =R= | — (d¢ + sin wdk) cos @ — sin W
cos wdk sin @ cos W
1 de — dk
Xl —d? cos — sin (9
; (8)
| dk sin @ cos W
According to Taylor's formula to expand Egs.(8),the
first order terms are taken and substitute into
Fas. (3),(4), then get the formula of equally tilted
photography (9):
; vf. xl
Xam Xt ii = out [Uri )*
wil,
e fàni- (mtn) ]
v fi-r-:
-YstNYi- em ET
Va =Y at Mt =Y st Mi [= (0 rir =n ens
ol
«^
1 +
4 ficos (0,— (5. IIT me — 2) sin o.
Ji 1 I
i = 21 ;
Zoi= Zot N24 = Zeit M3 (1 rms = thot
"A
T f,sin 014 ( mr = Jeo M
; : x
Yo,—Y Ni j-Y 3 A y fi-:1/ 72 x Jin,
24
T ficos 94— (6 rH) sin
T 2
x!
Zu= Zu NZi= Zu | (1 rrr = Joka +
2 2 2
=! :
T f,sin 0; (rmm j
(9)
5. The formula of fish-eye panoramic photo pair in
parallel-averted photography
Parallel-averted photography is that both two
photographic optical axes of fish-eye photo pair are
horizontal, perpendicular to the direction of
photographic-baseline and avert a certain angle %
this time, 9 =/,=/=0,0,=02=k=k=0,but in reality.
0,=d0;, w,=d®,, k,=dk, k,=dk,» so that the rotation
matrix form of directional cosine follows: :
cos® sin$ — (cosPdk+ sin Pdw)
R=R=R=| —sin® ‘cos? sin Fdk— cos Pdw
dk do 1
cos? sin? —dk
| — sin? cos? do
dk do 1
(10)
The first order term is still taken when derivated
Eqs.(10).Substituting Eqs.(10) in Eas.(3), (1), we can
obtain the formula of parallel-averted photography, the
eri d meathod is the same as before and it is omitted
ere.
6. The formula of fish-eye panoramic photograph under
convergent photography
The convergent photography is that both two
photographic optical axes of fish-
eve photograph pair are horizontal and rotated over a
certain convergent angle Y .
According to reason,the angle should be: 0,=k=9=02
=k=0, 9,—,but in practice it always retains error.
(177 dU; ky dk» 9,7 d95 (47 d05; k,=dk,, 91—Y. $0 the
formula of relative to Eqs.(10),(6) follows:
cos) sinY —dkı 1 de, -—dk
R,Z| —sinY cosy —dw; R,X| — dP, 1 —d6),
dk, dw; 1 dk, dw, 1
Substituting Rı R; in Eqs. (3) , (4) respectiv
can obtain Eqs. (11) pectivolv, then we
NE LAM =Xs, +N; [(*. = eos +
: ( zi
sfr Ts]
A = xi
Yc,zYs,- NiY1 =Ys+N; [fn umm)
PA
Zi Y
4- ficos Y -(5 rr ==, Jao]
Za=Z,+NZ!=Z;, +N, (s. ThE" Jaki +
~/
T fido (ro Tr -2)]
2 vw’
Xa=Xsat MXi =X s+ Ne [( dmt —X, ) + fod,
2 2. 2
22
At) |
x!
Za=ZntNZi=Zs EN, (s. JOD )akı+
2 2 7€
Tfdoit (1, um)
; (11)
After differential Egs.(7)-(11) and deriving the local
derivative of all coefficients, we can obtain the
differential formula applied to the photographs of
normal, parallel-averted, equal tilt and convergent.
According to differential formula, we can calculate the
valuation of coordinated accuracy to be located point
of all photographic case and derive permiting value for
elemental error of inner and exterior orientation, by
the way, the meathod derived in this paper is also
applied to aerophotogrammetry and space photogrammetry.
m
7. Experiment
The indoor and outdoor experiments are performed in
order to examine that the theoretic formula of stereo
photogrammetry carring on a variety of photographic
cases using fish-eye lens is correct or not and has the
possibility for using it.The stereo cameras developed
by Shanghai 832 works subordinate to ministry of public
