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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