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cannot assume that the representatives in equation (4.13) all have a certain form. We
must allow for the scalar multiple in the choice of representatives at least to some
degree. Fortunately, however, the equations can be rewritten in a more convenient
form that eliminates any explicit reference to such scalar multiples. It turns out that
we can always concentrate the scalar multiple in the “
b
” vector of equation (4.13).
Therefore, rather than choosing the usual representative of the form
b
= (b
1
,b
2
,1) for
[
b
], we can allow for scalar multiples by expressing the representative in the form
(
)
b
=◊
cbcb c
,
◊
,
.
(4.15)
1
2
Let
a
= (a
1
,a
2
,a
3
,a
4
) and M = (m
ij
). Let
m
j
= (m
1j
,m
2j
,m
3j
,m
4j
), j = 1,2,3, be the column
vectors of M. Equation (4.13) now becomes
(
)
=◊
(
)
amamam
∑
,
∑
,
∑
cbcb c
,
◊
,
.
1
2
3
1
2
It follows that c =
a
•
m
3
. Substituting for c and moving everything to the left, equa-
tion (4.13) can be replaced by the equations
(
)
=
am
∑
-
b
m
0
(4.16a)
(4.16b)
1
1
3
(
)
=
am
∑
-
b
m
0
.
2
2
3
It is this form of equation (4.13) that will be used in computations below. They have
a scalar multiple for
b
built into them.
After these preliminaries, we proceed to a solution for the first of our two recon-
struction problems. The object reconstruction problem is basically a question of
whether equation (4.13) determines
a
if M and
b
are known. Obviously, a single point
b
is not enough because that would only determine a ray from the camera and provide
no depth information. If we assume that we know the projection of a point with
respect to two cameras, then we shall get two equations
a
M
¢=
b
¢
(4.17a)
(4.17b)
a
¢¢ = ¢M.
b
At this point we run into the scalar multiple problem for homogeneous coordinates
discussed above. In the present case we may assume that M¢=(m
ij
¢) and M≤=(m
ij
≤)
are two fixed predetermined representatives for our projections and that we are
looking for a normalized tuple
a
= (a
1
,a
2
,a
3
,1) as long as we allow a scalar multiple
ambiguity in
b
¢=(c¢b
1
¢,c¢b
2
¢,c¢) and
b
≤=(c≤b
1
≤,c≤b
2
≤,c≤). Expressing equations (4.17)
in the form (4.16) leads, after some rewriting, to the matrix equation
(
)
aa a A
123
=
d
,
(4.18)
where
Ê
¢
¢
¢
¢
¢
¢
¢¢
¢¢
¢¢
¢¢
¢¢
¢¢
ˆ
mbmmbmm bm m bm
mbmmbmm bm m bm
mbm mbm m
-
-
-
-
11
1
31
21
2
31
11
1
31
21
1
31
Á
Á
Á
˜
˜
˜
¢
¢
¢
¢
¢
¢
¢¢
¢¢
¢¢
¢¢
¢¢
¢¢
A
=
-
-
-
-
12
1
32
22
2
32
12
1
32
22
1
32
¢
¢
¢
¢
¢
¢
13
¢¢ ¢
¢¢
¢¢
¢¢
¢¢
¢¢
-
-
-
bm
m
-
bm
Ë
¯
13
1
33
23
2
33
1 3
3
1 3
