Image Processing Reference
In-Depth Information
where
(
−−−→
xOP
)
C
=(
X, Y, Z
)
T
and
(
−−−→
(
x, y,
1)
T
are the coordinates
of the point
P
in the camera frame, and the homogeneous coordinates of the point
P
O
P
)
AH
=
Z
·
in the analog image frame, respectively.
Not surprisingly, the homogeneous coordinate concept is the result of geometry
studies in mathematics, to which A.F. Mobius (1790-1868) contributed greatly. It
has turned out to be an important tool of computer vision as well as computer graph-
ics in modern times. Among others, it is used to make affine transformations linear,
e.g., a 3D rotation and translation can be implemented as a single 4
4 matrix mul-
tiplication, thanks to the relationship between the projective spaces and Euclidean
spaces. Besides that, the representation allows a structured treatment of geometric
concepts such as points, infinite lines, planes, parallelism, and bundles [70, 91].
Above, the quantities
x,
and
y
were in length units, e.g., meters or focal length,
whereas one refers to a point in a digital image by its column count in the left-right
direction,
c
, and by its row count in the top-down direction,
r
. Having this, and the
geometric relationship
×
O
P
=
−−→
O
C
+
−−→
−−−→
CP
(13.10)
in mind, the basis vectors
e
c
,
e
r
placed at point
C
define the digital image frame,
represented by the column and row basis vectors,
e
c
,
e
r
placed at the point
C
, which
is, in general, not the same as
O
. The quantities
c, r
are then coordinates in this
basis. The coordinates of a point in the digital image frame can be transformed to the
analog image frame by the following equations:
x
=
−
(
c
−
c
0
)
s
x
(13.11)
y
=
−
(
r
−
r
0
)
s
y
The pixel counts (
c, r
)
T
are with reference to the central point,
C
. The two minus
signs in Eq. (13.11) are motivated as follows. For the
x
-direction one should note
that the delivered digital image is the one observed by an observer behind the “image
screen” at point
O
so that an increase in
c
count results in a decrease in
x
. For the
y
-direction one needs a minus sign because the row count
r
grows in the opposite
direction of
y
. By using Eq. (13.11) one can then obtain
⎧
⎨
c
=
x
cZ
=(
−
s
x
+
c
0
)
Z
x
−
s
x
+
c
0
y
s
y
⇔
rZ
=(
−
+
r
0
)
Z
(13.12)
y
s
y
r
=
−
+
r
0
⎩
Z
=
Z
In matrix form, this result is restated as follows:
Lemma 13.2.
Let the frame representing the digital picture in a projective camera
be
with the basis vectors
e
c
,
e
r
,
placed at a point
C
in the image. Then the coor-
dinates of a point
P
represented in the analog picture frame
D
A
are transformed to
frame
D
as follows:
(
−−→
CP
)
DH
=
M
D
(
−−−→
O
P
)
AH
(13.13)
