Image Processing Reference

In-Depth Information

as a linear combination of the others. Assuming point
N
c
belongs to facet
f
, we can write

u
N
c

v
N
c

A
v
f,
1
,
1

v
f,
1
,
2

A
v
f,
2
,
1

v
f,
2
,
2

A
v
f,
3
,
1

v
f,
3
,
2

,

a
N
c

d
f

b
N
c

d
f

c
N
c

d
f

−

=−

−

−

(3.9)

where
v
f,i,j
is the
j
th
coordinate of the
i
th
vertex of facet
f
. This shows that the bottom two rows

of the last column can be written as a linear function of the other columns. However, computing

this linear combination would introduce non-zero terms on the higher rows of the last column. For

points also belonging to facet
f
, these terms are directly canceled, as suggested by Eq.
3.7
. For points

belonging to facets sharing no vertices with facet
f
, these values will be zero. For point
i
belonging

to a facet
l
sharing two vertices with facet
f
, the value will be

A
v
f,
1
,
1

v
f,
1
,
2

A
v
f,
2
,
1

v
f,
2
,
2

u
i

v
i

A
v
l,
3
,
1

v
l,
3
,
2

.

a
i

d
f

b
i

d
f

d
l

d
f

c
i

d
l

−

−

=

−

+

(3.10)

Therefore, this value also is a linear combination of the other columns of
M
m
. Similar reasoning

can be done for facets sharing a single vertex with
f
. As a consequence, all terms introduced on the

last column by using the linear combination of Eq.
3.9
can be canceled, which means that this last

column is a linear combination of the others, and thus that the right half of
M
m
has at most rank

N
t
−

1. This means that for a full mesh,
M
m
has at most rank 2
N
v
+

N
t
−

1. This leaves us with

N
v
+

1 ambiguities. This seems natural due first to the scale ambiguity and second to the fact that

each vertex is free to move along its line of sight without affecting the reprojection of points inside

the facets.

3.3.2 AMBIGUITIES UNDER FULL PERSPECTIVE PROJECTION

As in the weak perspective case, we show that, given 3D-to-2D correspondences, the coordinates

of the mesh vertices must be solution to a linear system by starting with a mesh containing a single

triangle and extending our result to a complete mesh.

3.3.2.1 Projection of a 3D Surface Point

Recall from Eq.
3.3
that the perspective projection of a 3D point
q
i

expressed in camera coordinates

can be written as

⎡

⎤

u
i

v
i

1

⎣

⎦
=

d
i

Aq
i
,

(3.11)

where
A
is the internal parameters matrix, and
d
i
a scalar accounting for depth.

As before, if
q
i
lies on the facet of a triangulated mesh, it can be expressed as a weighted sum

of the facet vertices. Eq.
3.11
then becomes

⎡

⎤

u
i

v
i

1

⎣

⎦
=

d
i

A
(a
i
v
1
+
b
i
v
2
+
c
i
v
3
),

(3.12)

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