Image Processing Reference
In-Depth Information
f
Z
x
X
f
Z
y
Y
=
=
(13.5)
which yields the CT between the analog image frame and the camera frame
f
Z
X,
f
Z
Y
x
=
y
=
(13.6)
Before further discussion, it is necessary to present the
homogeneous coordi-
nates
.Any
n
dimensional vector represented in a certain frame can be written as an
n
+1-dimensional vector by adding a redundancy in terms of an extra dimension.
The coordinates in the augmented version are called the homogeneous coordinates,
including a possible common scale factor
λ
. These are obtained as follows for the
dimension
n
=2. For dimensions higher than 2 the procedure is analogous.
If
−−−→
(
−−−→
O
P
=(
x, y
)
T
O
P
)
H
=
λ
(
x, y,
1)
T
⇒
(13.7)
The real scalar
λ
is arbitrary as long as it is not 0. We will mark homogenized vec-
tors here with (
)
H
to avoid confusion, although in matrix algebra such vectors are
treated in the same way as other vectors of the same dimension. An equality sign “=”
in expressions containing homogenized vectors should be interpreted in the sense of
equivalent classes, i.e., a homogenized vector that is scaled remains the same ho-
mogenized vector. If the homogeneous coordinates of a point in the image plane are
known, then the image coordinates are recoverable as
·
(
−−−→
⇒
−−−→
1
Z
(
X, Y
)
T
O
P
)
H
=(
X, Y, Z
)
T
O
P
=
(13.8)
We note that
−−
OP
is not uniquely determined by its image, i.e.,
−−→
OP
,or
−−−→
O
P
. In fact,
not only
P
, but any point
P
(not shown in Fig. 13.2) on the infinite line represented
by the vector
−−
OP
will have the image
P
. Accordingly, the coordinates of a point
P
in the 3D world are homogeneous coordinates for the point
P
if the focal length is
assumed to have the unit length. This assumption is not a loss of generality because
when length measurements are done in the focal length, or equivalently when an
appropriate image scaling is performed, then
f
=1. However, to do this requires
an estimation of
f
, for reasons as follows. In this sense, a 3D homogeneous vector
derived from a point in the 2D image plane is thus the infinite line represented by the
vector that joins the projection center,
O
, to the image point. The line, among others,
passes through all 3D world points whose image is given by the intersection between
the line and the image plane. These results are summarized by the following lemma:
Lemma 13.1.
Let the analog picture in a projective camera be described by the basis
vectors
e
x
,
e
y
,
placed at
O
, the image of the projection center. Then a point
P
is
imaged at point
P
, with the coordinates
⎛
⎞
f
00
0
f
0
001
(
−−−→
(
−−
OP
)
C
,
with
O
P
)
AH
=
M
A
·
⎝
⎠
,
M
A
=
(13.9)
