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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

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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

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

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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

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Let be a closed subspace, and let be an orthonormal basis of . Then 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]

Fusion frame operator

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Definition

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

The adjoint is called the synthesis operator , defined as

where .

The fusion frame operator is defined by[2]

Properties

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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

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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

with

and

where is the canonical dual frame of .

See also

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References

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  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. arXiv:math/0605374. doi:10.1016/j.acha.2007.10.001. S2CID 329040.
  2. ^ a b c Casazza, P.G.; Kutyniok, G. (2004). "Frames of subspaces". Wavelets, Frames and Operator Theory. Contemporary Mathematics. Vol. 345. pp. 87–113. doi:10.1090/conm/345/06242. ISBN 9780821833803. S2CID 16807867.
  3. ^ Christensen, Ole (2003). An introduction to frames and Riesz bases. Boston [u.a.]: Birkhäuser. p. 8. ISBN 978-0817642952.
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