“Fusion frame”的意思、由来-开放百科全书

词条

Fusion frame

释义

Definition
Theorem for the relationship between fusion frames and global frames
Local frame representation
Definition of fusion frame operator
Properties of fusion frame operator
Representation of fusion frame operator
References
External links
See also

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

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 for all we have

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]

Theorem for the relationship between fusion frames and global frames

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 called 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 called a fusion frame system for with lower and upper bounds and .^[2]

Local frame representation

Let be a closed subspace, and let be an orthonormal basis of . Then for all , the orthogonal projection of onto is given by .^[3]

We can also express the orthogonal projection of onto in terms of given local frame of ,

where is a dual frame of the local frame .^[1]

Definition of fusion frame operator

Let be a fusion frame for . Let be representation space for projection. The analysis operator is defined by

Then The adjoint operator , which we call the synthesis operator, is given by

where .

The fusion frame operator is defined by

.^[2]

Properties of fusion frame operator

Given the lower and upper bounds of the fusion frame , and , the fusion frame operator can be bounded by

where is the identity operator. Therefore, the fusion frame operator is positive and invertible.^[2]

Representation of fusion frame operator

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 . With

and

where is the canonical dual frame of , the fusion frame operator is given by

The fusion frame operator is then given by an matrix.

References

1. ^¹²³⁴⁵{{cite journal|last=Casazza|first=Peter G.|author2=Kutyniok, Gitta|author3=Li, Shidong|title=Fusion frames and distributed processing|journal=Applied and Computational Harmonic Analysis|year=2008|volume=25|issue=1|pages=114–132|doi=10.1016/j.acha.2007.10.001}}
2. ^¹²{{cite book|last1=Casazza|first1=P.G.|last2=Kutyniok|first2=G.|title=Frames of subspaces|journal=Wavelets, Frames and Operator Theory|date=2004|volume=345|pages=87–113|doi=10.1090/conm/345/06242|series=Contemporary Mathematics|isbn=9780821833803}}
3. ^{{cite book|last=Christensen|first=Ole|title=An introduction to frames and Riesz bases|year=2003|publisher=Birkhäuser|location=Boston [u.a.]|isbn=978-0817642952|page=8}}

External links

Fusion Frames