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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