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tilinear relationships can be used for calibrating mul-
tiple dimensional cameras and for transfering corre-
sponding points to different dimensional cameras.
2 PROJECTION FROM P
k
TO P
n
Let us consider a projective camera P which projects
a point X in the kD projective space to a point x in the
nD projective space.
x ∼ PX (1)
where, ∼ denotes equality up to a scale. The camera
matrix, P, of this camera is (n+ 1) × (k+ 1) and has
((k+ 1) × (n+ 1) − 1) DOF.
3 MULTIPLE VIEW GEOMETRY
OF MIXED DIMENSIONAL
CAMERAS
We next consider the properties of the multiple view
geometry of mixed dimensional cameras, which rep-
resent geometric relationships of multiple cameras
with various dimensions. Let us consider kD pro-
jective space, P
k
, and a set of various dimensional
cameras in the space. Consider k types of cameras
C
i
(i = 1, ··· , k) which induce projections from P
k
to
P
i
(i = 1, ··· , k) respectively. For example, C
1
type
cameras project a point in P
k
to a point in P
1
, and C
2
type cameras project a point in P
k
to a point in P
2
.
Suppose there are n
i
cameras of type C
i
(i = 1, ··· , k)
in the kD space. Then, we have totally N =
∑
k
i=1
n
i
cameras in the kD space. In this paper, a set of these
cameras is represented by a k dimensional vector, n,
as follows:
n = [n
1
, n
2
, ··· , n
k
]
⊤
(2)
Now, we consider DOF of N view geometry of
mixed dimensional cameras, and specify the number
of points required for computing the N view geometry
of mixed dimensional cameras. Since camera matri-
ces from P
k
to P
i
are (k + 1) × (i+ 1), the DOF of N
cameras is N((k + 1)(i + 1) − 1). The kD homogra-
phy is represented by (k+ 1) × (k +1) matirx, and so
it has (k+ 1)
2
− 1 DOF. Since these N cameras are in
a single kD projective space, the total DOF of these N
cameras is as follows:
L =
k
∑
i=1
n
i
((k+ 1)(i+ 1)− 1) − (k + 1)
2
+ 1 (3)
= (k+1)(n
⊤
i− k) + kN −k (4)
where, i = [1, 2, ··· , k]
⊤
. Thus, the N view geometry
of mixed dimensional cameras has L DOF. The very
Table 1: The minimum number of corresponding points re-
quired for computing the multiple view geometry of mixed
dimensional cameras in the 3D space. Note, the multiple
view geometries of n
⊤
= [3, 0, 0], [2, 0, 0] and [1, 1, 0] do not
exist, since the image information is not enough for defining
the multiple view geometry in these cases.
4 Views 3 Views 2 Views
n
⊤
# n
⊤
# n
⊤
#
[4,0,0] 13 [2,1,0] 10 [0,2,0] 7
[3,1,0] 9 [1,2,0] 7 [1,0,1] 7
[2,2,0] 7 [0,3,0] 6 [0,1,1] 6
[1,3,0] 7 [2,0,1] 7 [0,0,2] 5
[0,4,0] 6 [1,1,1] 6
[3,0,1] 7 [0,2,1] 6
[2,1,1] 7 [1,0,2] 6
[1,2,1] 6 [0,1,2] 6
[0,3,1] 6 [0,0,3] 5
[2,0,2] 6
[1,1,2] 6
[0,2,2] 6
[1,0,3] 6
[0,1,3] 6
[0,0,4] 5
special case of the multiple view geometry of mixed
dimensional cameras is the traditional multiple view
geometry of 2D cameras which induce projections
from P
3
to P
2
. In this case, k = 3 and n = [0, 2, 0]
⊤
.
Suppose M points in the kD space are projected
to these N cameras. Then we have image informa-
tion with Mn
⊤
i DOF from these cameras. Thus, the
following inequality must hold for fixing all the ge-
ometry of N cameras and M points in the kD space.
Mn
⊤
i ≥ L+ kM (5)
By substituting (4) into (5), we find that the following
condition must hold for computing the multiple view
geometry of mixed dimensional cameras.
M ≥ k + 1+
kN − k
n
⊤
i− k
(6)
(6) shows the minimum number of corresponding
points required for computing the multiple view ge-
ometry of mixed dimensional cameras.
The complete table of the minimum number of
corresponding points for mixed dimensional cameras
in the 3D space is as shown in table 1.
In the following part of this paper, we show the
detail of the multiple view geometry of some example
combinations of different dimensional cameras.
VISAPP 2008 - International Conference on Computer Vision Theory and Applications
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