Image Processing Reference
In-Depth Information
We can now “transport” a point represented in the world coordinates to the perspec-
tive image, where we will be able to find its coordinates in the digital image frame.
We do that first by transferring the coordinates of the point in the world frame to the
camera frame via Eq. (13.36) and then taking these to the digital image frame via
(13.15), which is realized by matrix multiplications as stated in lemma 13.7.
Lemma 13.7.
A point
P
represented in the world basis is projected to a point in the
digital image basis via a linear transformation if both coordinates are homogeneous:
(
−−→
CP
)
DH
=
M
(
−−−→
O
W
P
)
WH
=
M
I
M
E
(
−−−→
O
W
P
)
WH
(13.42)
where
M
=
M
I
M
E
is called the
camera matrix
.
5
In conclusion, for every perspective camera there exists a 3
×
4 camera matrix:
⎛
⎞
M
11
M
12
M
13
M
14
M
21
M
22
M
23
M
24
M
31
M
32
M
33
M
34
⎝
⎠
=
M
I
M
E
M
=
(13.43)
where
M
I
,
M
E
are defined by Eqs. (13.17) and (13.40), respectively. In the follow-
ing section, we outline the fundamentals of determining
M
,
M
I
,
M
E
.
13.3 Intrinsic and Extrinsic Matrices by Correspondence
In practice,
M
is not known to a sufficient degree of accuracy for a variety of rea-
sons, e.g., either or both of
M
I
,
M
E
are noisy or unavailable. Because of this,
M
will be assumed here to be unknown. We will estimate
M
,
M
I
,
M
E
from known
correspondences between a set of image and world points. Defining the pair of ho-
mogeneous position vectors (with
λ
=0) that correspond to the same point in the
world frame and in the image frame, respectively, as
=(
−−−→
=(
−−→
p
=(
X, Y, Z,
1)
T
O
W
P
)
WH
λ
p
=
λ
(
c, r,
1)
T
CP
)
DH
(13.44)
we note that these must satify Eq. (13.42):
λ
p
=0
Mp
−
(13.45)
If one uses the definition of
M
given by Eq. (13.43), this produces three equations:
XM
11
+
YM
12
+
ZM
13
+
M
14
−
cλ
=0
(13.46)
XM
21
+
YM
22
+
ZM
23
+
M
24
−
rλ
=0
(13.47)
XM
31
+
YM
32
+
ZM
33
+
M
34
−
λ
=0
(13.48)
5
The matrix
M
is also known as the
projection matrix
.
