Information Technology Reference
In-Depth Information
17.
Papadimitriou, T., Diamantaras, K.I., et al.: Video Scene Segmentation Using Spatial
Contours and 3D Robust Motion Estimation. IEEE Trans. On Circuits and Systems For
Video Technology 14(4) (2004)
18.
Video sequences,
http://www.cipr.rpi.edu/resource/sequences/
Appendix
Segmenting Hyperplanes of Dimension K-1 in
R
{
}
N
j
(
j
)
K
Given a set of points
X
=
x
∈
R
in a homogeneous coordinate system, and
=
1
{
}
n
i
K
linear hyperplanes
S
⊂
R
of dimension
k
=
dim(
S
)
=
K
−
1
, we need to
i
i
i
=
(
i
)
(
i
)
K
identify
S
. Usually the subspace is given as,
b
⋅
x
=
0
b
∈
R
. Then, this
hyperplane can be represented as,
{
}
.
Furthermore, an arbitrary point
x
lies on one of the hyperplanes if and only if,
K
:
T
(
i
)
0
S
=
x
∈
R
x
b
=
i
n
∪∪ ∪
Ti
()
(
xS
∈
)
...
(
xS xb
∈
)
≡
=
0
1
n
i
n
=
1
∏
Ti
()
≡
px xb
()
=
=
0
,
n
i
=
1
T
=
yc
=
0
n
(
)
T
n
0
2
0
n
−
1
1
2
0
0
1
0
n
K
m
m
where,
y
=
x
x
...
x
,
x
x
...
x
,...,
x
...
x
x
∈
R
and
c
∈
R
is a coeffi-
1
K
1
K
K
−
1
n
(
n
+
K
−
1
)!
(
i
)
cient vector consisting of a set of monomials of
{
b
}
and
m
(
n
,
K
)
=
.
(
K
−
1
)!
n
!
For a given point set
X
, we have a linear system on
c
as follows,
⎛
(
T
⎞
y
(
x
)
⎜
⎟
N
L
c
=
⎜
...
⎟
c
=
0
∈
R
,
(A1)
n
n
n
⎜
⎜
⎟
⎟
(
N
)
T
y
(
x
)
⎝
⎠
N
×
m
where,
. When the number of hyperplanes
n
is known,
c
can be ob-
tained from the null space of
L
. In practice,
n
is always determined in terms of
L
. For a unique solution of the coefficient vector
L
n
R
∈
c
, it is expected that
rank L
()
=
m n K
(,
)1
−
, which is a function of variable
n
. In the presence of noise,
n
r
∑
, where
let
rank L
(
)
=
r
when
n
=
i
and
λ
λ
<
ε
L
is the data matrix with
r
+
1
j
i
j
=
1
rank
=
r
,
λ
is the
j
th singular value of
L
and
¶
is a given threshold.
j
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