Image Processing Reference

In-Depth Information

3, it has at most rank 2. Moreover, we can show that the last column

of the global matrix also is a linear combination of the two first columns of
P
by writing

u
i

v
i

Since
P
is of size 2

×

1

d
(a
i
v
1
+
b
i
v
2
+
c
i
v
3
)

P

=

A

1

d
(a
i
v
1
+
b
i
v
2
+
c
i
v
3
)

=

0

d
A
v
1
,
1

d
A
v
2
,
1

d
A
v
3
,
1

,

a
i

b
i

c
i

=

+

+

(3.7)

v
1
,
2

v
2
,
2

v
3
,
2

where
v
i,j
is the
j
th
coordinate of vertex
v
i
. The coefficients of Eq.
3.7
are independent of the

correspondence considered and are therefore valid for any row
i
of the matrix. This means that the

entire last column can be expressed as a linear combination of the other columns of the matrix. Thus,

when
N
c
≥
3, the rank of the matrix of Eq.
3.6
is always 6.

3.3.1.3 Reconstructing theWhole Mesh

As discussed above, when there are several triangles, using the weak perspective model amounts to

introducing a projection matrix per facet. However, since in reality we only have one camera, its

internal parameters, rotation matrix, and center are bound to be the same for each triangle. This

only lets us with a variable depth factor
d
f
for each facet
f
among the
N
t
facets of the mesh. We

can then write the system

⎡

⎤

v
1

...

v
N
v

d
1

...

d
N
t

⎣

⎦

M
m

=

0
,

(3.8)

with

u
1

v
1

⎡

⎣

⎤

⎦

a
1
P

b
1
P

c
1
P

0

...

...

−

0

...

...

...

...

...

...

...

...

...

...

...

...

...

...

u
j

v
j

b
j
P

c
j
P

d
j
P

0

0

...

0

−

0

...

...

M
m
=

.

...

...

...

...

...

...

...

...

...

...

...

u
l

v
l

a
l
P

c
l
P

e
l
P

0

0

...

0

...

−

0

...

...

...

...

...

...

...

...

...

...

...

...

The left half of
M
m
, which is of size 2
N
c
×

3
N
v
,
N
c
being the total number of correspondences,

has at most rank 2
N
v
because
P

has rank 2. We can then show that its right half, which is of size

2
N
c
×
N
t
, has at most rank
N
t
−

1. To this end, we need to show that its last column can be expressed

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