Image Processing Reference
In-Depth Information
W
R
T
, with
R
Lemma 13.4.
If the two bases
C
,
W
are related to each other as
C
=
being an invertible matrix and
P
being a point in the space
E
3
, then
(
−−−→
(
−−−→
O
W
P
)
C
=
R
O
W
P
)
W
·
(13.30)
where
(
−−−→
O
W
P
)
C
and
(
−−−→
O
W
P
)
W
are the coordinates of
−−−→
O
W
P
in the bases
C
and
W
,
respectively.
To reach this conclusion, we used only fundamental principles of linear algebra.
It is worth pointing out that even if the CT matrix
R
had not been an orthogonal ma-
trix, it would still relate to the basis change matrix inversely. Also, because a vector
is equivalent to all its translated versions, the above result is not only applicable to a
point, but to any vector. Also, the two involved frames,
, and the vector space
which they describe do not need to be 3D. We summarize this generalization as a
lemma,
C
,
W
R
−
1
with
R
being an invertible matrix, and
−−
AB
being a vector in the vector space, then
(
−−
AB
)
C
=
R
Lemma 13.5.
If two bases
C
,
W
are related to each other as
C
=
W
(
−−
AB
)
W
·
(13.31)
where
(
−−
AB
)
C
and
(
−−
AB
)
W
are the coordinates of
−−
AB
in the bases
C
and
W
respec-
tively.
Returning to the 3D vector space containing the world and camera frames, we
sum the vectors between the three points
O
W
,O
C
,P
(Fig. 13.3) to obtain the equa-
tion
−−−−→
O
W
O
C
+
−−−→
O
C
P
+
−−−→
PO
W
=0
(13.32)
Evidently, the relationship holds in any frame of the vector space, provided that all
three vectors are represented in that same frame. Using the world frame, we obtain
(
−−−→
O
W
P
)
W
=(
−−−−→
O
W
O
C
)
W
+(
−−−→
O
C
P
)
W
(13.33)
In practice, however, the vector
−−−→
O
C
P
is available in the camera frame. The substitu-
tion of Eq. (13.28) in this equation yields
(
−−−→
O
W
P
)
W
=(
−−−−→
(
−−−→
O
W
O
C
)
W
+
R
T
O
C
P
)
C
·
(13.34)
or
(
−−−→
(
−−−→
(
−−−−→
O
C
P
)
C
=
R
·
O
W
P
)
W
−
R
·
O
W
O
C
)
W
(13.35)
The latter is an affine relationship between the vectors (
−−−→
O
C
P
)
C
,
(
−−−→
O
W
P
)
W
, i.e.,
the position vectors of the point
P
expressed in the camera and world frames.
