Image Processing Reference
In-Depth Information
with
⎛
⎞
⎛
⎞
⎛
⎞
−
s
x
0
c
0
c
r
1
x
y
1
(
−−→
(
−−−→
⎝
⎠
,
⎝
⎠
,
⎝
⎠
CP
)
DH
=
Z
−
s
y
r
0
001
O
P
)
AH
=
Z
M
D
=
0
(13.14)
In the lemma, (
−−→
CP
)
DH
and (
−−−→
O
P
)
AH
are the homogeneous coordinates of the
point
P
, and the homogeneous coordinates of the point
P
,
respectively. By combining lemmas 13.1 and 13.2, one can thus obtain a single linear
transformation from frame
in frame
D
in frame
A
C
to frame
D
, without passing through
A
. This is made
explicit in lemma 13.3 .
Lemma 13.3.
Given a point
P
and its camera frame coordinates,
(
−−
OP
)
C
, its per-
spective transformation
P
has digital image frame coordinates that can be obtained
linearly from the former by
(
−−→
CP
)
DH
=
M
I
(
−−
OP
)
C
(13.15)
with
⎛
⎞
⎛
⎞
cZ
rZ
Z
X
Y
Z
(
−−→
(
−−
OP
)
C
=
⎝
⎠
,
⎝
⎠
CP
)
DH
=
(13.16)
and
⎛
⎞
⎛
⎞
⎛
⎞
s
x
0
c
0
0
−
s
y
r
0
001
−
f
00
0
f
0
001
−
f
x
0
c
0
⎝
⎠
⎝
⎠
=
⎝
⎠
M
I
=
M
D
M
A
=
0
f
y
r
0
001
−
(13.17)
Here
M
I
is the
intrinsic matrix
that encodes the hardware parameters of the camera
with
f
x
=
s
y
whereas
(
−−
OP
)
C
, and
(
−−→
f
s
x
, and
f
y
=
f
CP
)
DH
are coordinates of a
point in the camera frame,
, and the homogeneous coordinates corresponding to
the perspective projected point in the digital image frame,
C
D
, respectively.
The coordinates of the vector
−−
OP
are relative the basis
e
X
,
e
Y
,
e
Z
,
and the vec-
tor
−−→
CP
is represented via its 3D
homogeneous
coordinates. From the concept of
homogeneous coordinates, it then follows that
if (
−−→
CP
)
DH
=
P
x
,P
y
,P
z
T
(
−−→
P
x
,P
y
T
1
P
z
=(
c, r
)
T
(13.18)
CP
)
D
=
⇒
Conversely,
⎛
⎞
CP
)
D
=
c
r
c
r
1
(
−−→
(
−−→
CP
)
DH
=
λ
⎝
⎠
if
⇒
(13.19)
with
λ
being an arbitrary real scalar.
