Image Processing Reference
In-Depth 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 re-write 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 outlier-free 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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