Image Processing Reference
In-Depth Information
where
v
i,
1
≤
i
≤
3
are the vectors of 3D vertices coordinates and
(a
i
,b
i
,c
i
)
the barycentric coordinates
of
q
i
.
3.3.2.2 Reconstructing a Single Facet
Given the same
N
c
3D-to-2D correspondences lying inside one single facet as in the weak perspective
case, its vertex coordinates
v
i,
1
≤
i
≤
3
can be computed by solving the following equation where the
d
i
are treated as auxiliary variables to be recovered as well
⎡
⎣
⎤
⎦
v
1
v
2
v
3
d
1
...
d
i
...
d
N
c
M
f
=
0
,
(3.13)
with
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
u
1
v
1
1
a
1
A
b
1
A
c
1
A
−
0
...
...
...
...
...
...
...
...
...
...
...
⎡
⎣
⎤
⎦
u
i
v
i
1
a
i
A
b
i
A
c
i
A
0
...
−
0
...
M
f
=
.
...
...
...
...
...
...
...
...
⎡
⎣
⎤
⎦
u
N
c
v
N
c
1
a
N
c
A
b
N
c
A
c
N
c
A
0
...
...
...
−
For
N
c
>
4, if the columns of
M
f
had become linearly independent, the system would then have
had a unique solution. However, this is not what happens.
To prove that
M
f
is rank-deficient, we show that its last column can always be written as a
linear combination of the others as follows. From Eq.
3.12
we can write
⎡
⎣
⎤
⎦
=
a
N
c
A
λ
1
+
b
N
c
A
λ
2
+
c
N
c
A
λ
3
,
u
N
c
v
N
c
1
−
(3.14)
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