Image Processing Reference
In-Depth Information
Next we discuss
triangulation
, which is the process of reconstructing the posi-
tion vector of a 3D point in the reference (here left) coordinate frame. The name
triangulation refers to the triangle
O
L
PO
R
because the sought
−−→
O
L
P
is obtained by
summing up the involved vectors as
−−→
PO
L
+
−−−→
O
L
O
R
+
−−→
O
R
P
=
0
(13.66)
where coordinates of all vectors are represented in the same frame, the left camera
coordinates. Triangulation is evidently not possible when
−−→
OO
=0. Joint de-
termination of the position vector corresponding to a world point and the camera
parameters is a difficult problem in the presence of measurement noise. Unsurpris-
ingly, there have been numerous research studies [98] on the topic. We describe in
the following a basic technique that solves the triangulation problem after that the
camera parameters have been determined. We have chosen this technique more for
its simplicity and reasonable efficiency than for its superiority.
TLS Triangulation in a Projective Space
Assume that we know the positions of two points
P
L
,P
R
in the left and right
camera images of the same world point
P
(Fig. 13.7). Then, according to lemma
13.7, we have
⎧
⎨
O
W
P
)
WH
=(
−−−−→
M
L
(
−−−→
C
L
P
L
)
DHL
(13.67)
O
W
P
)
WH
=(
−−−−→
M
R
(
−−−→
⎩
C
R
P
R
)
DHR
where
M
L
,
M
R
are the camera parameters obtained from calibrating the left and
the right cameras individually against a known world frame placed at the point
O
W
.
Remembering that the cross-products of parallel lines vanish, we rewrite these as
⎧
⎨
(
−−−−→
M
L
(
−−−→
C
L
P
L
)
DHL
×
O
W
P
)
WH
=
T
L
M
L
p
=
0
(13.68)
(
−−−−→
M
R
(
−−−→
⎩
C
R
P
R
)
DHR
×
O
W
P
)
WH
=
T
R
M
R
p
=
0
Here
p
=(
−−−→
O
W
P
)
WH
is the unknown, and we expressed the cross-product operation
as a matrix multiplication according to:
⎛
⎞
⎛
⎞
c
L
r
L
1
1
r
L
10
0
−
(
−−−−→
C
L
P
L
)
DHL
×·
⎝
⎠
×· ⇒
T
L
=
⎝
⎠
·
c
L
=
−
(13.69)
r
L
c
L
−
0
⎛
⎞
⎛
⎞
c
R
r
R
1
0
1
r
R
10
−
(
−−−−→
C
R
P
R
)
DHR
×·
⎝
⎠
×· ⇒
⎝
⎠
·
T
R
=
−
c
R
=
(13.70)
−
r
R
c
R
0
where the elements of the matrices are derived from known image coordinates of
correspondence points in respective cameras. Consequently, we obtain:
