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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