point P and its measured image point Dc(Xc, Yc) are
described as
Xo — a4X * 25 Y +2a3ZL + 24 (1)
Vc — 85X + aY + a7Z + ag
in which a (1=1,---, 8) are independent coefficients.
Geometrically, the eight orientation parameters of the
affine image are considered to be three rotation
parameters (W, §, K) of the image, two translation
elements (Xo, Loc) which indicate two of the
three-dimensional coordinates of the origin of the
measured image coordinate system (Xc, Yc) with
respect to the object space coordinate system (X, Y, Z),
the image scale s, and two rotation parameters (e, [)
describing the relationship between projected rays and
the normal to the image plane. The eight orientation
. parameters of a single affine image can thus be provided
uniquely if four control points are available.
3 rj ;
me 8ht affine image
S
PmlXM, Ym, ZM)
modcl ems
es
Oe 2e . POGT, 25
Cc Cl
object space
0
a
a
Figure-2 : relative and absolute orientation of a
stereopair of affine images
Next, we will consider the orientation problem of a
stereopair of two-dimensional affine images (See
Figure-2.). The basic equations are written down as
a, 1 X zi a12Y +a13Z + 844
a15X + ajgY + aj7Z + agg
Xc1
2
AE (2)
for the left 1mage, and in the form
221X + 299 Ÿ 4 223Z. - 224
255X T 256Y + 25 77. * 258
Xc2
3
Yc2 )
for the right one, respectively. The condition that
Equations 2 and 3 are valid for all object points
photographed in common on the left and right images
can be formulated as
612
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
411. 412 213 2144-Xc1
415 2416 317 AB Yet
221 222 23503 Y4-Xc2
Qs dhe 227 23258 Yyc2
S b
which is equivalent to the copíanarity condition of
corresponding rays. Under the condition of Equation 4
we can form a three-dimensional space (XM, Yu, Zu)
which can be transformed into the object space (X,Y,Z)
by a three-dimensional affine transformation having 12
independent coefficients, i.e.,
XM = B,X + BY + BaZ + Ba
YM BsX + BgY + B7Z + Be (5)
ZM = BaX + BioY+BıiiZ+Bı2z
Also, the space (XM, YM, ZM) is equivalent to
the model space. From the results obtained above we
can find the following characteristics of the orientation
problem of overlapped affine images:
1) The coplanarity condition of corresponding rays
can mathematically provide four orientation
parameters among the twelve ones of the stereopair
of affine images, and
2) The one-to-one correspondence relating the model
and object spaces can be uniquely determined, if
four control points are given in the object space.
MODEL CONNECTION THEORY WITH ADJA-
CENT STEREO MODELS
We will assumed that an object was imaged on four
different planes based on parallel projection. The first
stereo model is constructed with the first and second
images, and the second stereo model with third and
fourth images. We will investigate what relationship is
valid between the first and second stereo models (See
Figure-3.). The general three-dimensional affine
Les
1 m
Y Jes cet
0., y y ORE em E
^ € cdr Je T MM
c Pez(Xe2,Y 2) / a C HE =
e dinens} jonal
Se trans Joraa tion
D > Y Pua X2, Yu, 12)
Ty Qni Yun. M1 277277
7
soon] second sterco vodel
first stereo model
hree aires onal
NS 4 7 i alfine trans Torna tion
2 T ue
^
A. Y object space
C X
0
ec dinensi onal
au Mi trans lornation
Figure-3 : model connection problem of multiple
affine images
Or 1
N = Dd
the
