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[
]
()
T
T
w
c
w
c
w
c
where
r
j
(
j
= 1...3) is the
j
th
column of matrix
R
, and
t
t
t
=
−
R
t
.
x
y
z
Then, from equation 25, it turns out immediately (ref. equation 23) that:
()
(
) (
)
()
T
T
T
3
4
3
,
t
=
ε
q
'
,
r
=
ε
q
'
,
x
=
q
'
1
q
'
3
,
y
=
q
'
2
q
'
3
,
u
=
−
q
'
1
×
q
'
2
q
'
2
×
q
'
3
z
z
3
z
0
0
(
)
2
3
2
3
f
y
=
q
'
−
y
q
'
r
=
−
ε
q
'
−
y
q
'
/
f
r
=
r
×
r
,
,
,
0
2
z
0
y
1
2
3
(
)
f
'
1
u
x
t
=
−
ε
q
'
2
4
−
y
t
/
f
= ε
q
−
r
−
r
,
,
x
z
2
0
3
y
z
0
z
y
(
)
t
q
'
1
4
ut
x
t
/
f
=
−
ε
−
−
,
(24)
x
z
y
0
z
x
where
ε
z
(=
±
1) is the same as in equation 24.
Constrained DLT
In some cases, the skew factor need not be considered, while in others the
nonlinear effects have been eliminated in advance by means of the techniques.
Then an additional constraint
(
) (
)
T
1
2
2
3
qq
'
×
'
q q
'
×
'
=
0
, which guarantees the
w
R
, can be added to the calculation of DLT. Taking
this into consideration, the original DLT problem should be redefined as (ref.
equations 21 and 23):
orthonormality of the matrix
()()()
2
2
2
Ap
subject to the constraints
Minimize
p
3
+
p
3
+
p
3
=
1
and
1
2
3
(
) (
T
)
qq
'
1
×
'
2
q q
'
2
×
'
3
=
0
.
(27)
a
d
=
Ap
is neither geometrically
nor statistically meaningful (Hartley & Zisserman, 2000), it is better to minimize
the following
geometric distance
:
However, because the
algebraic distance
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