2 respectively, 
multiplication 
affine coordi- 
ear, i.e., affine 
ates. The col- 
e model of the 
ase component 
tion matrix. 
ntal matrix Ë 
TR (20) 
ric matrix. As 
multiplication 
account in the 
nez 1. Thus, 
(21) 
(22) 
it matriz ana- 
|. E of Eq.(7), 
and important 
composed as a 
tew-symmeiric 
nponents, rep- 
nd perspective 
and R is an 
orientation of 
rst one. 
omponents 
ise component 
(23) 
is. By setting 
then could ob- 
ts By and Bg. 
Now we may move on to computing matrix R. Eq.(21) 
is now written columnwise as 
Br; =e; (=1,2.3) (24) 
where r; and e; are the column component vectors 
of R and E respectively. Since rank(B)=2, for each 
column of R we can only determine two parameters, 
namely three indepedent parameters (degrees of rank 
deficiency) remain unknown altogether. By choosing 
F11, T12, 713 as independent parameters, in such a way 
a solution will always be assured, we have 
Fo, = (ByF11 + €31)/Bx 731 = (Bz711 — é21)/Bx 
F99 = (By F12 + €32)/Bx 732 = (Bz¥12 — &23)/ Bx (25) 
7a3 = (By 713 + €33)/Bx 733 = (Bz¥13 — €23)/Bx 
Moreover, we can also get the length ratio, i.e., 42/31, 
of the two conjugate projective rays. Equalizing the 
two equations in Eq.(16) and multiplying B in both 
sides yields 
Xs = - 
Ex, = Bx, (26) 
À 
Its least squares solution is 
p.25. (Be) (Ba) (27) 
^; (EX2)"(Ex2) 
In summary, with pure image correspondences or the 
fundamental matrix, we could recover the two ratios 
of the three affine base components, as well as siz 
relationships among the nine components of the affine 
rotation matrix. Moreover, for each image correspon- 
dence, we could determine its length ratio of the two 
conjugate projective rays. 
Since there are three degrees of rank deficiency, the 
affine model can not be fully reconstructed without 
known object points. 
5. OBJECT RECONSTRUCTION 
In this section we first present the transformation 
between object space and the affine model. Then a 
linear algorithm is designed to perform the exterior 
orientation of the partially recovered affine model. 
It is trivial to show that the transformation between 
object space and the affine model takes the form 
(affine transformation) 
U 
a ag az a4 V 
p= Au= b; by ba ba W (28) 
C1 C9 C3 C4 1 
where u” = (U V W 1) is the homogeneous coordi- 
nates of a point in object space, À is the transfomation 
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matrix specified by twelve independent parameters. 
Inserting Eq.(16) into Eq.(28) yields 
Au = A1 Au = A,RXxs T b (29) 
which are elementary to perform the exterior orienta- 
tion. Eq.(29) has 12 (from matrix A)--3 (from matrix 
R)=15 independent orientation parameters. 
To design a linear solution, we eliminate A; in Eq.(29) 
and get a DLT-type equation 
aU + a3V -- a3W + aa 
€c1U -F coV + c3W + Ca 
b4U + baV + b3W + ba 
eoU + coV +c3W + c4 
  
à (30) 
yi 
  
The twelve parameters could be linearly determined 
up to a scale factor with six given object points 
appearing on the first image, namely the ratio 
a; = a;/ca,b; = bi/ca,C; = ci/c4 (c4 — 1) are ob- 
tained. 
Immediately after that, the remainder four parameters 
are determined linearly with the second set of Eq.(29) 
by minimum four given object points, i.e., 
A'u= kX Rx, + P (31) 
C4 
/ 4 1. a 
b.,c; similar to 
. ,;—. 
where matrix A is composed of aj,b;, c; 
matrix A, 
X, = qU+cV+cW+1 (32) 
and k is computed from Eq.(27). 
The object reconstruction is then finally performed by 
inversing Eq.(28) 
U a az as =t X — 04 
V — bi ba b3 Y -— ba (33) 
W € Ca €3 Z — C4 
Comparing to the DLT algorithm which relates every 
image independently to the object space, our algo- 
rithm fully employs the information within a stereo. 
Therefore, only 15 instead of 2 x 11 parameters are 
dependent on known object points. To ensure a lin- 
ear solution, one image of a stereo may have mini- 
mum six known object points, while minimum four of 
them appear on the other one. It is apparent that in 
this minimum configuration, the DLT algorithm fails, 
while a linear solution is available in our algorithm. 
Furthermore, this algorithm could be feasibly applied 
to successive images for object reconstruction if only 
each of them has minimum four conjugate known 
points.