Image Processing Reference
In-Depth Information
one can write the basis change between the world and the coordinate frames, Eq.
(13.21), as follows.
W
R
T
C
=
(13.23)
The quantities
appear as row vectors in (13.22) in brackets. However, it is im-
portant to note that their elements are not scalars, but basis vectors! To mark that this
is a formalism to represent the basis vectors jointly, in Eq. (13.22) we used
C
,
W
,
which are in a different “vector” notation than the ordinary vector notation (boldface
letters). Accordingly, the multiplication in the basis change equation involves the
symbolic quantities such as
e
X
, not scalars. The brackets are used throughout this
chapter to mark that we are constructing a matrix or a vector by juxtaposing other
entities, which can be other matrices or vectors in coordinate or abstract representa-
tion.
A vector
−−−→
C
,
W
O
C
P
expressed in the camera frame can be written as
−−−→
O
C
P
=
X
C
e
X
+
Y
C
e
Y
+
Z
C
e
Z
(13.24)
are scalars, i.e., coordinates, that are the projections of
−−−→
Here,
X
C
,Y
C
,Z
C
O
C
P
on
e
X
,
e
Z
,
e
Z
, respectively. Using the definitions for
C
,
W
in (13.22) we can write the
equation of coordinates, (13.24), as
−−−→
O
C
P
=
(
−−−→
(
−−−→
O
C
P
)
C
,
O
C
P
)
C
=(
X
C
,Y
C
,Z
C
)
T
.
C
·
where
(13.25)
Note that we used the subscript (
)
C
to mark that the vector in question is no longer
an abstract vector, but it is expressed in the camera frame by means of a specific
triplet of scalars, the coordinates. Accordingly, the same vector will be represented
as a
different
triplet of scalars in the world frame. The new coordinates can be
obtained by substituting (13.22) in (13.25), yielding:
−−−→
O
C
P
=
·
(
−−−→
(
−−−→
O
C
P
)
C
=
R
T
O
C
P
)
C
C
W
·
(13.26)
Equation (13.25) represents
−−−→
O
C
P
in the camera frame. Using an analogous notation,
it can be represented in the world frame as well:
−−−→
O
C
P
=
(
−−−→
O
C
P
)
W
W
(13.27)
Representing the same vector in the same frame, the right-hand sides of Eqs. (13.26)
and (13.27) can be compared to each other, establishing the following identity be-
tween the coordinate triplets of the camera and the world frames as:
(
−−−→
(
−−−→
O
C
P
)
W
=
R
T
O
C
P
)
C
·
(13.28)
or
(
−−−→
(
−−−→
O
C
P
)
W
(13.29)
In the latter, we note that the coordinate transformation uses the matrix
R
,
which
is the
inverse
of the matrix used in the basis transformation, Eq. (13.23),
R
T
.We
summarize this result in lemma 13.4.
O
C
P
)
C
=
R
·
