Image Processing Reference
InDepth Information
Recall from Eq.
5.2
that the projection under the perspective camera model of a 3D point
q
i
represented by its homogeneous coordinates
q
i
,
1
]
T
q
i
=[
˜
can be written as
⎡
⎤
u
i
v
i
1
⎣
⎦
=
d
i
P
q
i
,
P
˜
=
A
[
R

t
]
,
(5.9)
where
A
is the matrix of internal camera parameters,
R
is the camera rotation matrix, and
t
is the
camera translation vector. Although
A
is often assumed to be unknown, calibrating the camera better
constrains the problem
Hartley and Vidal
[
2008
]. Recall that unlike in the weak perspective case,
the scalar accounting for depth
d
i
is different for every point
i
.
As before, we can group in matrix form the equations corresponding to
N
c
such points found
in a single frame, which yields
⎡
⎤
d
1
u
1
···
d
N
c
u
N
c
P Q
,
⎣
⎦
=
d
1
v
1
···
d
N
c
v
N
c
(5.10)
d
1
···
d
N
c
Q
is the 4
where
×
N
c
matrix of homogeneous point coordinates. By assuming again that the shape
can be described as a linear combination of basis shapes
S
k
, we can rewrite the previous system as
⎡
⎤
d
1
u
1
···
d
N
c
u
N
c
N
s
⎣
⎦
=
d
1
v
1
···
d
N
c
v
N
c
AR
c
k
S
k
+
AT
,
(5.11)
d
1
···
d
N
c
k
=
1
where we have explicitly decomposed the projection matrix into internal parameters, rotation and
translation, and where each column of
T
contains the translation vector
t
.
From the outlierfree correspondences between points in
N
f
frames, we can build the system
of equations representing all projections of all points as
⎡
⎣
⎤
⎦
d
1
u
1
d
N
c
u
1
N
c
···
d
1
v
1
d
N
c
v
N
c
···
⎡
⎤
⎡
⎤
S
1
.
S
N
s
1
d
1
d
N
c
···
c
1
AR
1
c
N
s
AR
1
At
1
···
⎣
⎦
.
.
.
⎣
⎦
.
.
.
.
=
,
(5.12)
d
N
f
1
u
N
f
1
d
N
f
N
c
u
N
N
c
c
N
f
1
c
N
f
···
AR
N
f
N
s
AR
N
f
At
N
f
···
d
N
f
1
v
N
f
1
d
N
f
N
c
v
N
f
···
N
c
C
d
N
f
d
N
f
B
···
1
N
c
W
where
W
is the 3
N
f
×
N
c
matrix of scaled measurements,
C
is a 3
N
f
×
(
3
N
s
+
1
)
matrix, and
B
is the
(
3
N
s
+
1
)
×
N
c
matrix containing the shape basis.
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