Mathematical frame extension
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.
Given a Hilbert space
, let
be closed subspaces of
, where
is an index set. Let
be a set of positive scalar weights. Then
is a fusion frame of
if there exist constants
such that
![{\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a4c71ec199550b59f66bbc602441e2275899fd)
where
denotes the orthogonal projection onto the subspace
. The constants
and
are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other,
becomes a
-tight fusion frame. Furthermore, if
, we can call
Parseval fusion frame.[1]
Assume
is a frame for
. Then
is called a fusion frame system for
.[1]
Relation to global frames
[edit]
Let
be closed subspaces of
with positive weights
. Suppose
is a frame for
with frame bounds
and
. Let
and
, which satisfy that
. Then
is a fusion frame of
if and only if
is a frame of
.
Additionally, if
is a fusion frame system for
with lower and upper bounds
and
, then
is a frame of
with lower and upper bounds
and
. And if
is a frame of
with lower and upper bounds
and
, then
is a fusion frame system for
with lower and upper bounds
and
.[2]
Local frame representation
[edit]
Let
be a closed subspace, and let
be an orthonormal basis of
. Then the orthogonal projection of
onto
is given by[3]
![{\displaystyle P_{W}f=\sum \langle f,x_{n}\rangle x_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae16163a916bbbe85177c78cd79da806911ac11)
We can also express the orthogonal projection of
onto
in terms of given local frame
of
![{\displaystyle P_{W}f=\sum \langle f,f_{k}\rangle {\tilde {f}}_{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb6ebc33dd5d8f2761456e67d62f1132930b61bc)
where
is a dual frame of the local frame
.[1]
Fusion frame operator
[edit]
Let
be a fusion frame for
. Let
be representation space for projection. The analysis operator
is defined by
![{\displaystyle T_{W}\left(f\right)=\{v_{i}P_{W_{i}}\left(f\right)\}_{i\in {\mathcal {I}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d71ffec7194f8616e397e9591987cf9a4b69d6e3)
The adjoint is called the synthesis operator
, defined as
![{\displaystyle T_{W}^{\ast }\left(g\right)=\sum v_{i}f_{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead6c88ee900de740aba75caeee3dcf64bb0e9ba)
where
.
The fusion frame operator
is defined by[2]
![{\displaystyle S_{W}\left(f\right)=T_{W}^{\ast }T_{W}\left(f\right)=\sum v_{i}^{2}P_{W_{i}}\left(f\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6384fd84d5e46375201c7016068ba8729c7ed43)
Given the lower and upper bounds of the fusion frame
,
and
, the fusion frame operator
can be bounded by
![{\displaystyle AI\leq S_{W}\leq BI,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16d7a807e42937ec472633c55ad805dee5ed85d)
where
is the identity operator. Therefore, the fusion frame operator
is positive and invertible.[2]
Given a fusion frame system
for
, where
, and
, which is a dual frame for
, the fusion frame operator
can be expressed as
,
where
,
are analysis operators for
and
respectively, and
,
are synthesis operators for
and
respectively.[1]
For finite frames (i.e.,
and
), the fusion frame operator can be constructed with a matrix.[1] Let
be a fusion frame for
, and let
be a frame for the subspace
and
an index set for each
. Then the fusion frame operator
reduces to an
matrix, given by
![{\displaystyle S=\sum _{i\in {\mathcal {I}}}v_{i}^{2}F_{i}{\tilde {F}}_{i}^{T},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544af0bfaa38631b02bbe13d000f122dd7c32684)
with
![{\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}|},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa5011ddec36fe5de05e11b2f3145d2b03130db)
and
![{\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}|},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/887f34a9452291d0c058d914c24a4356a9216c6e)
where
is the canonical dual frame of
.