EE ER Eum
CARRE
e
—
vi
ln,
= L
ws
m
|
u
dM
~
where: X121 and x47j are image coordinates measuring
on left and right photo respectively for fish eve
panoramic photo pairs. Considering following formula:
Nı= B.ıY1—B,Xi = pi Nx B,Y{—B,X! = B*
Prat PIN pe
Bx=B»sin A; By =Bcos À
where: À is directional angle of base B.
Differential all variations of Eq.(3),(4), we can get
following Eq. (5)
dXe=
; ^; .
AX s XU I-dBa- X1 XT aBy-E Xie Q 1 eos A X1 sin A)dA-
7 2 N32 /N2
-NX OT Q'O' ap, wi GEO OD ae NEG dos
*yv/_V#yY/
KARO ak + N
TNXiDdfi— NaX{ Didf2— NX! E,dx,+ NaX{ E4dxj —
— NX LF dz, 4- NX ( Fodz5 t NX 4E dx 1 — NX 4E4dx$
+NX}F,d={ — N,X1F d=;
XtY,—Y*Xj
+N,X{G,d0,— N X [E EE
dYc=
/ .
AY FYC EAB.- Y (Ei-iBy Y! 5 (Ÿ; cos A+ X; sin A)JdA-
7323 1,2 7/32
—NY! XDD gg, + NY QOO p MY G dw.
Ry $3 RY
Xvi Vix: dk + NY! DEEE dk,
+ N,Y Did fi— N.Y | Dd f= NiY § Exd%o+ NoY { Exdxg — NiY 2 Fidz
4 NY LF dz - NY 1 Eydx( — N,Y { E,dx} +N,Y 1F,d=1— N2Y 1 Fidei
+N,Y{ Gdw,— NY:
dZc=
ri Hu dB, Z1 à (Y { cos A + X; sin A) dA -
1,2 Lt:
—N,Z! (REXELYIYE an az t ET an IE
dZs,- ZÍ
— Z(G0 do, 4- NZ1Gida t a LZfP*— Z1 OY 1 - Y1X 2 a4
Nz) SY pg fk, +L LoP +2 0uX 7 aY 038,
NZI Dif — pr TOP* + Zu] — YD] drt NiZ{ End]
N
Ar Cop 4 ZLOSXL- aY Dd E NZLE dni Lo P
M ve p*
*ZUOoXI-2aY)31 dx] —N,Z{E dx; + 7 LesP
TZiOX1—aY ) ]dz1 — NiZ1 Fidz] (5)
where:
7 /
G,=- 21 QGcs si isin?); G,= zi (X1cos 9— Y ; sin 92);
PS P
zm (xx) —al(z,— 20);
=bj (x= x5) —bi(2— 20);
XT
fd,(X,— 3X —4,(2,— 29; X
Y*zmbG—x)—b6O T7 29; Y
ta 8X
Zim-040—3x)—000,—29;
JT zz ficos 91— sin ei. G,-— 20) cos Kırk (x,— Xp) sin lul;
p, (SYL), p," (493512), p, "(ETATS
p* P= P*
b(X1—alY! 27 Xia (BIN!
From Eq.(5),we can calculate the estimation of
accuracy for coordinates of undetermined point and
extract the permissive value of elements of interior
and exterior orientation. The formula is also the
general equation of fish-eye panoramic photograph, that
is, applied to close range photogrammetry of arbitrary
photographic case. The following sections are discussed
respectively the mathematical formula for fish-eye
panoramic stereo photogrammetry of normal, equally
tilted, parallel-averted and convergent photography.
3. The formula of fish-eye panoramic photo
pair at normal photography
At normal photography, two photographic optical axes of
fish-eve photograph pair are horizontal, parallel each
other and perpendicular to direction of photographic-
base. The rotation matrix in this case is identity
matrix. But the normal photo in reality still remains
with elemental slender error of exterior orientation
d?, d?9,, d&;, dw,, dk;, dk,. When the case of small angle
existes, the approximation of the small value first
order is applied. Thus the rotation matrix is:
1 d? — dk
R=R=RX|—de | — dû) (6)
—dk du 1
Substituting Ro R: into Egs. (3),(4) respectively and
then get Eds. (7):
He rg]
Za=Z,+N21=Zs,+N; (^, 7
; zi
td (pre)
Xo, Xs, TN X =Xs.+N, [e
e
’
t hàn- (f. E -— Zg Jet, ]
Ye.=Ys, +N Xi=Y, +N, [-(
~/
+= (fs a]
7
Zoa9Zs;TtNZ-*ZsytN, [7 IA x )d
2 v2 =>
v» a
+ fdw,+ © ee zn )]
