Game Development Reference
In-Depth Information
T
im
im
im
sponding 2-D image coordinates
x
=
x
y
(
i
= 1 ...
N
) by the following
i
i
i
equation:
Ap 0
⋅=
,
(21)
21
N
×
where
21
0
is a column vector with all
2
N
elements being
0
and
N
×
w
w
w
−
w
im
−
w
im
−
w
im
−
im
xyz
10 0 00
0000 1
10 0 00
0000
xx
yx
z x
x
1
1
1
1
1
1
1
1
1
1
w
w
w
w
im
w
im
w
im
im
xyz
−
xy
−
yy
−
z y
−
y
1
1
1
1
1
1
1
1
1
1
w
w
w
w
im
w
im
w
im
im
xyz
−
xx
−
yx
−
z x
−
x
2
2
2
2
2
2
2
2
2
2
A
=
w
w
w
w
im
w
im
w
im
im
xyz
1
−
xy
−
yy
−
z y
−
y
2
2
2
2
2
2
2
2
2
2
,
w
w
w
w
im
wim
wim
im
xyz
10 0 00
−
xx
−
y
x
−
z
x
−
x
N
N
N
N
N
N
N
N
N
N
w
w
w
w
im
w
im
w
im
im
0000
xyz
1
−
xy
−
yy
−
z
y
−
y
N
N
N
N
N
N
N
N
N
N
T
pppppppppppp
.
1
1
1
1
2
2
2
2
3
3
3
3
p
=
1
2
3
4
1
2
3
4
1
2
3
4
Since the overall scaling of the 12 unknowns is irrelevant, a certain constraint
should be imposed. This constraint is, in fact, used to get rid of the scale
randomicity of the camera projection (multiple 3-D objects with different scales
may correspond to the same 2-D projections). A simple form is to let one
unknown be equal to one, for example,
3
4
p
=
. In this case a simpler linear
equation can be derived. The remaining 11 unknowns can thus be calculated
from this new equation by employing various methods, e.g., least squares.
However, because of the possibility of singularity of this assumption, that is
3
4
1
p
=
0
, other forms of constraints should be imposed instead. One possibility is
()()()
2
2
2
+ + =
. In this case, the problem to be solved is
reformulated as (ref. equation 21) the minimization of
Ap
subject to the
constraint that
the constraint
p
3
p
3
p
3
1
1
2
3
i
j
i
j
Cp
=
1
c
=
0
c
.
C
is defined such that
(
,
i j
=
1 2
), where
represents the element at the
i
th
row and
j
th
column, except
===
. A
closed-form solution to this problem can be obtained by the method described in
Faugeras (1993) and Hartley & Zisserman (2000).
9
10
11
c
cc
1
9
10
11
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