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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