Image Processing Reference
In-Depth Information
degeneracies. We start with a mesh containing a single triangle and extend our result to a complete
one.
3.3.1.1 Projection of a 3D Surface Point
Recall from Eq.
3.2
that under a weak perspective model, the projection to a 2D image plane of a
3D point
q
i
whose coordinates are expressed in the camera referential can be written as
d
u
i
v
i
A
I
2
×
2
0
P
q
i
,
P
=
=
(3.4)
where
d
is a depth factor associated to the weak perspective camera and
A
is a 2
×
2 matrix
representing the camera internal parameters.
If
q
i
lies on the facet of a triangulated mesh, it can be expressed as a weighted sum of the facet
vertices. Eq.
3.4
becomes
d
u
i
v
i
P
(a
i
v
1
+
=
b
i
v
2
+
c
i
v
3
),
(3.5)
where
v
i,
1
≤
i
≤
3
are the vectors of 3D vertex coordinates and
(a
i
,b
i
,c
i
)
the barycentric coordinates
of
q
i
.
3.3.1.2 Reconstructing a Single Facet
Let us assume that we are given a list of
N
c
3D-to-2D correspondences for points lying inside one
single facet. The coordinates of its three vertices
v
i,
1
≤
i
≤
3
can be computed by solving the linear
system
⎡
⎣
u
1
v
1
⎤
⎦
a
1
P
b
1
P
c
1
P
−
⎡
⎣
⎤
⎦
=
...
...
...
...
v
1
v
2
v
3
d
u
i
v
i
a
i
P
b
i
P
c
i
P
−
0
,
(3.6)
...
...
...
...
u
N
c
v
N
c
a
N
c
P
b
N
c
P
c
N
c
P
−
where
d
is treated as an auxiliary variable to recover as well. Since we only have one facet, we also
only have one projection matrix. Thus, only a single
d
corresponding to the average depth of the
facet is necessary and all [
u
i
,v
i
]
T
can be put in the same column.
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