In mathematics, a fusion frame of a vector space is a natural extension of a frame . It is an additive construct of several, potentially "overlapping" frames. The motivation for this concept comes from the event that a signal can not be acquired by a single sensor alone (a constraint found by limitations of hardware or data throughput), rather the partial components of the signal must be collected via a network of sensors, and the partial signal representations are then fused into the complete signal.
By construction, fusion frames easily lend themselves to parallel or distributed processing[1] of sensor networks consisting of arbitrary overlapping sensor fields.
Definition
edit
Given a Hilbert space
H
{\displaystyle {\mathcal {H}}}
, let
{
W
i
}
i
∈
I
{\displaystyle \{W_{i}\}_{i\in {\mathcal {I}}}}
be closed subspaces of
H
{\displaystyle {\mathcal {H}}}
, where
I
{\displaystyle {\mathcal {I}}}
is an index set. Let
{
v
i
}
i
∈
I
{\displaystyle \{v_{i}\}_{i\in {\mathcal {I}}}}
be a set of positive scalar weights. Then
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
is a fusion frame of
H
{\displaystyle {\mathcal {H}}}
if there exist constants
0
<
A
≤
B
<
∞
{\displaystyle 0<A\leq B<\infty }
such that
A
‖
f
‖
2
≤
∑
i
∈
I
v
i
2
‖
P
W
i
f
‖
2
≤
B
‖
f
‖
2
,
∀
f
∈
H
,
{\displaystyle A\|f\|^{2}\leq \sum _{i\in {\mathcal {I}}}v_{i}^{2}{\big \|}P_{W_{i}}f{\big \|}^{2}\leq B\|f\|^{2},\quad \forall f\in {\mathcal {H}},}
where
P
W
i
{\displaystyle P_{W_{i}}}
denotes the orthogonal projection onto the subspace
W
i
{\displaystyle W_{i}}
. The constants
A
{\displaystyle A}
and
B
{\displaystyle B}
are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other,
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
becomes a
A
{\displaystyle A}
-tight fusion frame. Furthermore, if
A
=
B
=
1
{\displaystyle A=B=1}
, we can call
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
Parseval fusion frame.[1]
Assume
{
f
i
j
}
i
∈
I
,
j
∈
J
i
{\displaystyle \{f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}}
is a frame for
W
i
{\displaystyle W_{i}}
. Then
{
(
W
i
,
v
i
,
{
f
i
j
}
j
∈
J
i
)
}
i
∈
I
{\displaystyle \{\left(W_{i},v_{i},\{f_{ij}\}_{j\in J_{i}}\right)\}_{i\in {\mathcal {I}}}}
is called a fusion frame system for
H
{\displaystyle {\mathcal {H}}}
.[1]
Relation to global frames
edit
Let
{
W
i
}
i
∈
H
{\displaystyle \{W_{i}\}_{i\in {\mathcal {H}}}}
be closed subspaces of
H
{\displaystyle {\mathcal {H}}}
with positive weights
{
v
i
}
i
∈
I
{\displaystyle \{v_{i}\}_{i\in {\mathcal {I}}}}
. Suppose
{
f
i
j
}
i
∈
I
,
j
∈
J
i
{\displaystyle \{f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}}
is a frame for
W
i
{\displaystyle W_{i}}
with frame bounds
C
i
{\displaystyle C_{i}}
and
D
i
{\displaystyle D_{i}}
. Let
C
=
inf
i
∈
I
C
i
{\textstyle C=\inf _{i\in {\mathcal {I}}}C_{i}}
and
D
=
inf
i
∈
I
D
i
{\textstyle D=\inf _{i\in {\mathcal {I}}}D_{i}}
, which satisfy that
0
<
C
≤
D
<
∞
{\displaystyle 0<C\leq D<\infty }
. Then
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
is a fusion frame of
H
{\displaystyle {\mathcal {H}}}
if and only if
{
v
i
f
i
j
}
i
∈
I
,
j
∈
J
i
{\displaystyle \{v_{i}f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}}
is a frame of
H
{\displaystyle {\mathcal {H}}}
.
Additionally, if
{
(
W
i
,
v
i
,
{
f
i
j
}
j
∈
J
i
)
}
i
∈
I
{\displaystyle \{\left(W_{i},v_{i},\{f_{ij}\}_{j\in J_{i}}\right)\}_{i\in {\mathcal {I}}}}
is a fusion frame system for
H
{\displaystyle {\mathcal {H}}}
with lower and upper bounds
A
{\displaystyle A}
and
B
{\displaystyle B}
, then
{
v
i
f
i
j
}
i
∈
I
,
j
∈
J
i
{\displaystyle \{v_{i}f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}}
is a frame of
H
{\displaystyle {\mathcal {H}}}
with lower and upper bounds
A
C
{\displaystyle AC}
and
B
D
{\displaystyle BD}
. And if
{
v
i
f
i
j
}
i
∈
I
,
j
∈
J
i
{\displaystyle \{v_{i}f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}}
is a frame of
H
{\displaystyle {\mathcal {H}}}
with lower and upper bounds
E
{\displaystyle E}
and
F
{\displaystyle F}
, then
{
(
W
i
,
v
i
,
{
f
i
j
}
j
∈
J
i
)
}
i
∈
I
{\displaystyle \{\left(W_{i},v_{i},\{f_{ij}\}_{j\in J_{i}}\right)\}_{i\in {\mathcal {I}}}}
is a fusion frame system for
H
{\displaystyle {\mathcal {H}}}
with lower and upper bounds
E
/
D
{\displaystyle E/D}
and
F
/
C
{\displaystyle F/C}
.[2]
Local frame representation
edit
Let
W
⊂
H
{\displaystyle W\subset {\mathcal {H}}}
be a closed subspace, and let
{
x
n
}
{\displaystyle \{x_{n}\}}
be an orthonormal basis of
W
{\displaystyle W}
. Then the orthogonal projection of
f
∈
H
{\displaystyle f\in {\mathcal {H}}}
onto
W
{\displaystyle W}
is given by[3]
P
W
f
=
∑
⟨
f
,
x
n
⟩
x
n
.
{\displaystyle P_{W}f=\sum \langle f,x_{n}\rangle x_{n}.}
We can also express the orthogonal projection of
f
{\displaystyle f}
onto
W
{\displaystyle W}
in terms of given local frame
{
f
k
}
{\displaystyle \{f_{k}\}}
of
W
{\displaystyle W}
P
W
f
=
∑
⟨
f
,
f
k
⟩
f
~
k
,
{\displaystyle P_{W}f=\sum \langle f,f_{k}\rangle {\tilde {f}}_{k},}
where
{
f
~
k
}
{\displaystyle \{{\tilde {f}}_{k}\}}
is a dual frame of the local frame
{
f
k
}
{\displaystyle \{f_{k}\}}
.[1]
Fusion frame operator
edit
Definition
edit
Let
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
be a fusion frame for
H
{\displaystyle {\mathcal {H}}}
. Let
{
∑
⨁
W
i
}
l
2
{\displaystyle \{\sum \bigoplus W_{i}\}_{l_{2}}}
be representation space for projection. The analysis operator
T
W
:
H
→
{
∑
⨁
W
i
}
l
2
{\displaystyle T_{W}:{\mathcal {H}}\rightarrow \{\sum \bigoplus W_{i}\}_{l_{2}}}
is defined by
T
W
(
f
)
=
{
v
i
P
W
i
(
f
)
}
i
∈
I
.
{\displaystyle T_{W}\left(f\right)=\{v_{i}P_{W_{i}}\left(f\right)\}_{i\in {\mathcal {I}}}.}
The adjoint is called the synthesis operator
T
W
∗
:
{
∑
⨁
W
i
}
l
2
→
H
{\displaystyle T_{W}^{\ast }:\{\sum \bigoplus W_{i}\}_{l_{2}}\rightarrow {\mathcal {H}}}
, defined as
T
W
∗
(
g
)
=
∑
v
i
f
i
,
{\displaystyle T_{W}^{\ast }\left(g\right)=\sum v_{i}f_{i},}
where
g
=
{
f
i
}
i
∈
I
∈
{
∑
⨁
W
i
}
l
2
{\displaystyle g=\{f_{i}\}_{i\in {\mathcal {I}}}\in \{\sum \bigoplus W_{i}\}_{l_{2}}}
.
The fusion frame operator
S
W
:
H
→
H
{\displaystyle S_{W}:{\mathcal {H}}\rightarrow {\mathcal {H}}}
is defined by[2]
S
W
(
f
)
=
T
W
∗
T
W
(
f
)
=
∑
v
i
2
P
W
i
(
f
)
.
{\displaystyle S_{W}\left(f\right)=T_{W}^{\ast }T_{W}\left(f\right)=\sum v_{i}^{2}P_{W_{i}}\left(f\right).}
Properties
edit
Given the lower and upper bounds of the fusion frame
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
,
A
{\displaystyle A}
and
B
{\displaystyle B}
, the fusion frame operator
S
W
{\displaystyle S_{W}}
can be bounded by
A
I
≤
S
W
≤
B
I
,
{\displaystyle AI\leq S_{W}\leq BI,}
where
I
{\displaystyle I}
is the identity operator. Therefore, the fusion frame operator
S
W
{\displaystyle S_{W}}
is positive and invertible.[2]
Representation
edit
Given a fusion frame system
{
(
W
i
,
v
i
,
F
i
)
}
i
∈
I
{\displaystyle \{\left(W_{i},v_{i},{\mathcal {F}}_{i}\right)\}_{i\in {\mathcal {I}}}}
for
H
{\displaystyle {\mathcal {H}}}
, where
F
i
=
{
f
i
j
}
j
∈
J
i
{\displaystyle {\mathcal {F}}_{i}=\{f_{ij}\}_{j\in J_{i}}}
, and
F
~
i
=
{
f
~
i
j
}
j
∈
J
i
{\displaystyle {\tilde {\mathcal {F}}}_{i}=\{{\tilde {f}}_{ij}\}_{j\in J_{i}}}
, which is a dual frame for
F
i
{\displaystyle {\mathcal {F}}_{i}}
, the fusion frame operator
S
W
{\displaystyle S_{W}}
can be expressed as
S
W
=
∑
v
i
2
T
F
~
i
∗
T
F
i
=
∑
v
i
2
T
F
i
∗
T
F
~
i
{\displaystyle S_{W}=\sum v_{i}^{2}T_{{\tilde {\mathcal {F}}}_{i}}^{\ast }T_{{\mathcal {F}}_{i}}=\sum v_{i}^{2}T_{{\mathcal {F}}_{i}}^{\ast }T_{{\tilde {\mathcal {F}}}_{i}}}
,
where
T
F
i
{\displaystyle T_{{\mathcal {F}}_{i}}}
,
T
F
~
i
{\displaystyle T_{{\tilde {\mathcal {F}}}_{i}}}
are analysis operators for
F
i
{\displaystyle {\mathcal {F}}_{i}}
and
F
~
i
{\displaystyle {\tilde {\mathcal {F}}}_{i}}
respectively, and
T
F
i
∗
{\displaystyle T_{{\mathcal {F}}_{i}}^{\ast }}
,
T
F
~
i
∗
{\displaystyle T_{{\tilde {\mathcal {F}}}_{i}}^{\ast }}
are synthesis operators for
F
i
{\displaystyle {\mathcal {F}}_{i}}
and
F
~
i
{\displaystyle {\tilde {\mathcal {F}}}_{i}}
respectively.[1]
For finite frames (i.e.,
dim
H
=:
N
<
∞
{\displaystyle \dim {\mathcal {H}}=:N<\infty }
and
|
I
|
<
∞
{\displaystyle |{\mathcal {I}}|<\infty }
), the fusion frame operator can be constructed with a matrix.[1] Let
{
W
i
,
v
i
}
i
∈
I
{\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}
be a fusion frame for
H
N
{\displaystyle {\mathcal {H}}_{N}}
, and let
{
f
i
j
}
j
∈
J
i
{\displaystyle \{f_{ij}\}_{j\in {\mathcal {J}}_{i}}}
be a frame for the subspace
W
i
{\displaystyle W_{i}}
and
J
i
{\displaystyle J_{i}}
an index set for each
i
∈
I
{\displaystyle i\in {\mathcal {I}}}
. Then the fusion frame operator
S
:
H
→
H
{\displaystyle S:{\mathcal {H}}\to {\mathcal {H}}}
reduces to an
N
×
N
{\displaystyle N\times N}
matrix, given by
S
=
∑
i
∈
I
v
i
2
F
i
F
~
i
T
,
{\displaystyle S=\sum _{i\in {\mathcal {I}}}v_{i}^{2}F_{i}{\tilde {F}}_{i}^{T},}
with
F
i
=
[
⋮
⋮
⋮
f
i
1
f
i
2
⋯
f
i
|
J
i
|
⋮
⋮
⋮
]
N
×
|
J
i
|
,
{\displaystyle F_{i}={\begin{bmatrix}\vdots &\vdots &&\vdots \\f_{i1}&f_{i2}&\cdots &f_{i|J_{i}|}\\\vdots &\vdots &&\vdots \\\end{bmatrix}}_{N\times |J_{i}|},}
and
F
~
i
=
[
⋮
⋮
⋮
f
~
i
1
f
~
i
2
⋯
f
~
i
|
J
i
|
⋮
⋮
⋮
]
N
×
|
J
i
|
,
{\displaystyle {\tilde {F}}_{i}={\begin{bmatrix}\vdots &\vdots &&\vdots \\{\tilde {f}}_{i1}&{\tilde {f}}_{i2}&\cdots &{\tilde {f}}_{i|J_{i}|}\\\vdots &\vdots &&\vdots \\\end{bmatrix}}_{N\times |J_{i}|},}
where
f
~
i
j
{\displaystyle {\tilde {f}}_{ij}}
is the canonical dual frame of
f
i
j
{\displaystyle f_{ij}}
.
See also
edit
References
edit
External links
edit