Fusion frame

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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 , 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[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 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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

  1. ^ a b c d e f Casazza, Peter G.; Kutyniok, Gitta; Li, Shidong (2008). "Fusion frames and distributed processing". Applied and Computational Harmonic Analysis. 25 (1): 114–132. doi:10.1016/j.acha.2007.10.001. 
  2. ^ a b c Casazza, P.G.; Kutyniok, G. (2004). "Frames of subspaces". Wavelets, Frames and Operator Theory. 345: 87–113. 
  3. ^ Christensen, Ole (2003). An introduction to frames and Riesz bases. Boston [u.a.]: Birkhäuser. p. 8. ISBN 0817642951. 

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