# The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics

Authors: Jinsong Leng; Qixun Guo; Tingzhu Huang
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

The experimental data transmission is an important part of high energy physics experiment. In this paper, we connect fusion frames with the experimental data transmission implement of high energy physics. And we research the utilization of fusion frames for data transmission coding which can enhance the transmission efficiency, robust against erasures, and so forth. For this application, we first characterize a class of alternate fusion frames which are duals of a given fusion frame in a Hilbert space. Then, we obtain the matrix representation of the fusion frame operator of a given fusion frame system in a finite-dimensional Hilbert space. By using the matrix representation, we provide an algorithm for constructing the dual fusion frame system with its local dual frames which can be used as data transmission coder in the high energy physics experiments. Finally, we present a simulation example of data coding to show the practicability and validity of our results.


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## Body

## 1. Introduction

Because of the cross-regional feature of high energy physics experiment, there exists a huge amount of data produced in the experiment procedure which needs to be transmitted from each experimental field to the remote center for processing synthetically every day. The relevant technologies in many current data transmission systems are the data transmission protocol GridFTP, the object-related database management system PostgreSQL, the application sever JBoss, and so forth in order to ensure the real-time, reliable, and efficient data transmission [1]. But in many transmission systems, all the signals have to be retransmitted when one or more vectors of data are lost in the transmission process, which leads to wasting a lot of time and resources. Why not use fusion frames? We find that they are natural suitable tools for the experimental data transmission coding of high energy physics. In fact, this application of fusion frames can save more time and resources caused by the retransmission.Redundancy is an interesting and attractive feature of frames, because it has at least two advantages. First, it makes the construction of various classes of frames flexible; secondly, it can enhance the robustness of encoding data when erasures occur in signal transmissions. So, the theory of frames has been developed rapidly in mathematics and achieved successful applications in various areas of pure and applied sciences and engineering in the past twenty years. We only mention some applications of frames here such as signal and image processing [2], quantization [3], capacity of transmission channel [4–6], coding theory [7–12], and data transmission technology [13].With the development of signal processing systems, frames are restricted and fusion frames appear. The utility of fusion frames in handling missing data packet erasures problem is shown in [14]. The theory of fusion frames was systematically introduced in [15, 16]. Since then, many excellent results about the theory and application of fusion frames have been obtained in an amazing speed [15–22]. In fact, fusion frames are generalization of conventional frames and go beyond them, and they have been found to be good tools in large signal processing systems in which distributed or parallel processing is required. For instance, in a coding transmission process, the encoded and quantized data must be put in numbers of packets. When one or more packets are scrumped, lost, or delayed, fusion frames can enhance the robustness to the packet erasures. Furthermore, we can see the successful applications of fusion frames in sensors network [23], filter bank [24], transmission coding [14, 25, 26], and so forth. We refer to [27] and the reference therein for more details about the applications of fusion frames.We first describe how to use fusion frames for transmission coding in experiment of high energy physics to enhance the transmission efficiency and robust against erasures. According to the characteristics of the experimental data transmission system of high energy physics, the distributed, parallel, and fused processing are required in the transmission process. Hence, fusion frames can be applied to coding in the transmission scheme to improve the transmission efficiency, stability, and robust of the whole system.On the other hand, however, some related problems about fusion frames, especially in applications, are still open. Many excellent results about conventional frames have been achieved and applied successfully, but how to generalize them to fusion frames? It is a tempting subject because of the complexity of the structure of fusion frames compared with conventional frames. For the application of data transmission, we study mainly the dual fusion frames of a given fusion frame and the matrix representations of fusion frame systems in finite-dimensional Hilbert spaces for constructing dual fusion frame systems in this paper.We outline this paper as follows. In Section2, we recall the experimental data transmission course of high energy physics and propose a new transmission model in which fusion frame and its dual are used for data coding. Then, we introduce and recall some notations, conceptions, and some basic theory about frames and fusion frame systems. In Section 3, we first introduce a kind of alternate fusion duals based on the definition given in [20]. We investigate and characterize these alternate fusion duals. Then, we consider how to get the matrix representation of the fusion frame operator of a given fusion frame system in a finite-dimensional Hilbert space. So that, based on this matrix representation, a method for construction of the dual fusion frame system with its local dual frames is prescribed. A simulation example is given to show the practicality and validity of these results in experimental data transmission.


## 2. Fusion Frames for Experimental Data Transmission Coding of High Energy Physics and Preliminaries

The main function of fusion frame in data transmission procedure is data coding to implement distributed, parallel, and fused processing of the whole transmission system. A large amount of data produced by experiment sites of high energy physics is encoded by local frames and stored in some packets in the sender sides; the packets from all experiment sites are decoded/processed by dual fusion frame in the center. Based on the conventional transmission system, we establish the structure scheme of the data transmission system by using fusion frames (see Figure1) and precise the transmission procedure briefly as follows. The original data from the Data Acquisition System is transmitted to the Dropbox for Temporary Directories by the Data Buffer. When the Fetcher finds that there are new data directories in the Dropbox, the original data will be encoded by using a local frame, quantized, and stored into some packets in it. Once the Sending Directories receives these encoded and quantized packets which consist of some vectors from the Dropbox, it will send all these vectors to the processing center and wait the feedback from the receiver. The feedback is sent by the Data Checking Module of the processing center when it confirms that all data from the sender are received. Then it submits all these vectors to the Receiving Directories in which these vectors will be decoded and fusing processed by a fusion frame system and its dual. Finally, all decoded signals are submitted to the Warehouse, and the procedure is over. In the old transmission model, the Data Checking Module of the receiver will check the integrity of these received packets. When it finds that some vectors or coefficients are lost in the transmission process, it will ask the sender retransmit all signals. The re-transmission procedure is unnecessary if a fusion frame and its dual are used for data coding, and a lot of time and resources are saved. Thus, applying fusion frames for data coding in the transmission process can enhance the reliability, efficiency, and robust for erasures. of the transmission system.Figure 1
The experimental data transmission course of high energy physics.Then, let us recall and introduce some basic notations, concepts, and results about frames and fusion frames that are needed for this paper. Let us begin with the concept of frames.Letℋ be a separable (real or complex) Hilbert space. A collection of vectors F={fi}i∈I⊂ℋ is called a frame for ℋ if there exist constants 0<A≤B<∞ such that
(1)A∥f∥2≤∑i∈I‍|〈f,fi〉|2≤B∥f∥2∀f∈ℋholds for everyf∈ℋ, where I denote an index set. The optimal constants (maximal for A and minimal for B) are called lower and upper frame bounds. If A=B, then F is called a tight frame, and it is called a Parseval frame(Sometimes, a Parseval frame is also called a normalized tight frame) when A=B=1. A uniform frame is a frame when all the elements in the frame sequence have the same norm.Given a frameF={fi}i∈I, the operator ΘF:ℋ→ℓ2(I) defined by
(2)ΘF(f)=∑i∈I‍〈f,fi〉ei
is called the analysis operator of F, where {ei}i∈I is the standard orthonormal basis for ℓ2(I). The adjoint operator ΘF* of ΘF given by
(3)ΘF*(∑i∈I‍ciei)=∑i∈I‍cifi
is called the synthesis operator of F. If we let SF=ΘF*ΘF, then we have
(4)SF(f)=∑i∈I‍〈f,fi〉fi,f∈H.
Thus, SF is a positive invertible bounded linear operator on ℋ, which is called the frame operator of F.A collection of vectorsF~={f~i}i∈I in ℋ is called a dual frame for F={fi}i∈I if F~={f~i}i∈I satisfies the reconstruction formula
(5)f=∑i∈I‍〈f,fi〉f~i=∑i∈I‍〈f,f~i〉fi∀f∈ℋ.
A direct calculation yields
(6)f=∑i∈I‍〈f,fi〉SF-1(fi)=∑i∈I‍〈f,SF-1(fi)〉fi∀f∈ℋ.
This implies that {SF-1(fi)}i∈I is a dual frame of {fi}i∈I. The frame {SF-1(fi)}i∈I is called the canonical dual frame of {fi}i∈I. If a dual frame {f~i}i∈I is not the canonical dual frame, it is also called an alternate dual frame.A frameF is a tight frame if and only if SF=ΘF*·ΘF=AIℋ for some positive constant A, where Iℋ is the identity operator. A frame F is a Parseval frame if and only if SF=ΘF*·ΘF=Iℋ; that is, the canonical dual of F is itself. So, the analysis operator ΘF of a Parseval frame is an isometry operator. A linear operator P from a Hilbert space ℋ to ℋ is called an orthogonal projection if P is self-adjoint and P2=P.Given a finite frameF={fi}i=1k in an n-dimensional Hilbert space ℋ, then we necessarily have k≥n. When k=n, F is automatically a basis of ℋ.We will use the notation𝔽 when the result being stated holds for both the real number field ℝ and the complex number field ℂ. When ℋ=𝔽n, then fi(i=1,2,…,k) are column vectors, and the analysis operator ΘF for the frame F={fi}i=1k is a matrix with the row vector fi* as the ith row of the matrix for i=1,2,…,k, where the superscript “*” denotes the conjugate-transpose of a vector or a matrix. Relatively, the synthesis operator ΘF* for the frame F is the conjugate-transpose matrix of ΘF, and the frame operator SF=ΘF*·ΘF is an n×n positive invertible matrix. With respect to a fixed orthonormal basis of ℋ, any element of ℋ and any linear operator can be expressed by the coordinate vector and the matrix representation. So in most cases we will identify an n-dimensional Hilbert space with 𝔽n.Let us now recall the definitions and basic results about fusion frames which are mostly adopted from [15, 16].Definition 1.
LetI denote an index set, and let {Wi}i∈I be a family of closed subspaces of a Hilbert space ℋ with a family of weights {wi}i∈I where wi>0 for all i∈I. Then, {(Wi,wi)}i∈I is called a fusion frame for ℋ if there exist constants 0<C≤D<∞ such that
(7)C∥f∥2≤∑i∈I‍wi2∥PWi(f)∥2≤D∥f∥2∀f∈ℋ,
where PWi denotes the orthogonal projection onto the Wi. The constants C and D are called the lower and upper fusion frame bounds. The family {(Wi,wi)}i∈I is called a C-tight fusion frame if C=D, and it is called a Parseval fusion frame if C=D=1. The family {(Wi,wi)}i∈I is called an orthonormal fusion basis if ℋ=⊕i∈IWi. A Bessel fusion sequence refers to the case when {(Wi,wi)}i∈I has an upper fusion frame bound, but not necessarily a lower bound.Definition 2.
Let{(Wi,wi)}i∈I be a fusion frame for ℋ, and {fij}j∈Ji be a frame of Wi where Ji are index sets for i∈I. Then, {fij}j∈Ji, i∈I are called local frames, and {(Wi,wi,{fij}j∈Ji)}i∈I is called a fusion frame system for ℋ. The constants C and D are the associated lower and upper fusion frame bounds if they are the fusion frame bounds for {(Wi,wi)}i∈I, and A and B are the local frame bounds if there are the common frame bounds for the local frames {fij}j∈Ji for each i∈I. The dual frames {f~ij}j∈Ji, i∈I of the local frames in Wi are called local dual frames.The following result shows the relationship between a fusion frame system and its local frames, as well as their frame bounds.Theorem 3 (c.f. [16], Theorem 2.3).
For eachi∈I, let Wi be a closed subspace for ℋ, and let {fij}j∈Ji be a frame for Wi with frame bounds Ai and Bi. Suppose that 0<A=infi∈IAi≤supi∈IBi=B<∞. Then, the following conditions are equivalent.(i)
{(Wi,wi)}i∈I is a fusion frame for ℋ.(ii)
{wifij}j∈Ji,i∈I is a frame for ℋ.
In particular, if{(Wi,wi,{fij}j∈Ji)}i∈I is a fusion frame system for ℋ with fusion frame bounds C and D, then {wifij}j∈Ji,i∈I is a frame for ℋ with frame bounds AC and BD. If {wifij}j∈Ji,i∈I is a frame for ℋ with frame bounds C and D, then {(Wi,wi,{fij}j∈Ji)}i∈I is a fusion frame system for ℋ with fusion frame bounds C/B and D/A.Let𝒲={(Wi,wi)}i∈I be a fusion frame for ℋ. The analysis operator Θ𝒲 is defined by
(8)Θ𝒲:ℋ⟶(∑i∈I‍⊕Wi)ℓ2withΘ𝒲(f)={wiPWi(f)}i∈I,
where
(9)(∑i∈I‍⊕Wi)ℓ2={{fi}i∈I∣fi∈Wi,{∥fi∥}i∈I∈ℓ2(I)}
is called the representation space. The synthesis operator Θ𝒲* (the adjoint operator of Θ𝒲) can be defined by
(10)Θ𝒲*:(∑i∈I‍⊕Wi)ℓ2⟶ℋwithΘ𝒲*(f)=∑i∈I‍wifi,f={fi}i∈I∈(∑i∈I‍⊕Wi)ℓ2.
The fusion frame operator S𝒲 for 𝒲 is defined by
(11)S𝒲(f)=Θ𝒲*Θ𝒲(f)=∑i∈I‍wi2PWi(f).About dual fusion frames, the following definition was given in [15].Definition 4.
Let{(Wi,wi)}i∈I be a fusion frame for space ℋ with fusion frame operator S𝒲. Then, {(S𝒲-1Wi,wi)}i∈I is called the dual fusion frame of {(Wi,wi)}i∈I.
The dual fusion frame defined previously satisfies the following reconstruction formula(12)f=∑i∈I‍wi2S𝒲-1PWi(f)=∑i∈I‍wi2PS𝒲-1WiS𝒲-1PWi(f),f∈ℋ.
Based on (12), the following definition about alternate duals was introduced in [20].Definition 5.
Let𝒲={(Wi,wi)}i∈I be a fusion frame for space ℋ with fusion frame operator S𝒲, and, 𝒱={(Vi,vi)}i∈I be a Bessel fusion sequence. Then, 𝒱 is called an alternate dual of 𝒲 if
(13)f=∑i∈I‍wiviPViS𝒲-1PWi(f),
holds for all f∈ℋ.Then, it was proved that𝒱 is also a fusion frame [20]. We will call it an alternate fusion dual of 𝒲 in this paper.


## 3. A Class of Alternate Fusion Duals and Construction of Dual Fusion Frame Systems

We first introduce a class of alternate fusion duals𝒱={(Vi,wi)}i∈I of a given fusion frame 𝒲={(Wi,wi)}i∈I which satisfy
(14)PViS𝒲-1PWi(f)=PS𝒲-1WiS𝒲-1PWi(f),∀f∈ℋ,i∈I.
A Bessel fusion sequence which satisfies (14) naturally satisfies (13). So, we can obtain the following obvious result.Proposition 6.
Let𝒲={(Wi,wi)}i∈I be a fusion frame for space ℋ with fusion frame operator S𝒲. If a Bessel fusion sequence 𝒱={(Vi,wi)}i∈I satisfies (14), then it is an alternate fusion dual of 𝒲.All Bessel fusion sequences satisfying (14) form a special kind of alternate fusion duals of 𝒲. The following proposition will characterize these duals.Proposition 7.
Let𝒲={(Wi,wi)}i∈I be a fusion frame for space ℋ with fusion frame operator S𝒲. A Bessel fusion sequence 𝒱={(Vi,wi)}i∈I satisfies (14) if and only if it has the form
(15)Vi=S𝒲-1Wi⊕Ui,forsomesubspaceUiofℋhhhhhhhhhhhhhhhhhhhhhhhandeachi∈I.Proof.
If (14) holds for any i∈I, then we have
(16)PViS𝒲-1PWi(f)=PS𝒲-1WiS𝒲-1PWi(f)=S𝒲-1PWi(f),∀f∈ℋ,
which follows
(17)(Iℋ-PVi)S𝒲-1PWi(f)=PVi⊥S𝒲-1PWi(f)=0,∀f∈ℋ.
For any f∈S𝒲-1Wi, there exists g∈Wi such that f=S𝒲-1g. Hence,
(18)PVi⊥f=PVi⊥S𝒲-1g=PVi⊥S𝒲-1PWig=0,
which implies that f⊥Vi⊥. Hence, S𝒲-1Wi⊥Vi⊥, which follows S𝒲-1Wi⊂Vi. Let Ui=Vi-S𝒲-1Wi; then for any f=S𝒲-1g∈S𝒲-1Wi, we have
(19)PUif=(PVi-PS𝒲-1Wi)S𝒲-1PWi(g)=0,
which implies
(20)S𝒲-1Wi⊥Ui.
Hence, Vi=S𝒲-1Wi⊕Ui.
Conversely, assume that𝒱 satisfies (15). Then for any f∈ℋ and i∈I, since S𝒲-1PWi(f)∈S𝒲-1Wi, we have
(21)PViS𝒲-1PWi(f)=PS𝒲-1WiS𝒲-1PWi(f),
which implies that (14) holds.The following proposition shows that the dual fusion frame{(S𝒲-1Wi,wi)}i∈I can minimize the projection norm of any f∈ℋ in the class of alternate fusion duals introduced previously. The property is analogous to Theorem 6.8 of [28] in the case of traditional frames, and its proof is trivial.Proposition 8.
Let𝒲={(Wi,wi)}i∈I be a fusion frame for space ℋ with fusion frame operator S𝒲. Then for any alternate fusion dual 𝒱={(Vi,wi)}i∈I of 𝒲 which satisfies (14), one have
(22)∑i∈I‍wi2∥PS𝒲-1Wi(f)∥2≤∑i∈I‍wi2∥PVi(f)∥2,∀f∈ℋ.Then, we consider the construction of the dual fusion frame in a finite-dimensional Hilbert spaceℋ. In Section 2, we will see that any n-dimensional Hilbert space can be identified with 𝔽n, and the analysis, synthesis, and frame operator of any conventional frame can be expressed by their matrix representations, respectively. It is essential for this construction to obtain the matrix representation of the fusion frame operator S𝒲 and its inverse which need the local frames. Hence, we will study the construction of the dual fusion frame system {(S𝒲-1Wi,wi,{S𝒲-1fij}j∈Ji)}i∈I with its local dual frames of a given fusion frame system {(Wi,wi,{fij}j∈Ji)}i∈I.Let{(Wi,wi,{fij}j=1ni)}i=1m be a fusion frame system for space 𝔽n. Then the analysis operator of the local frame of Wi is a ni×n matrix ΘFi with fij* as its jth row, and the n×ni matrix ΘFi* is its synthesis operator. Furthermore, the ith local frame operator is an n×n matrix SFi=ΘFi*ΘFi.Remark 9.
For the purpose of coding of anyf∈𝔽n, any vector of the subspaces of 𝔽n we consider has n elements, and ΘFi always denotes the analysis operator of the system {fij}j=1ki in 𝔽n throughout this paper. Hence, it is a ki×n matrix, not a ki×(dimWi) matrix.Definition 10.
LetW be an l-dimensional subspace of 𝔽n with a local frame F={fi}i=1k. SF is the local frame operator of F. If there exists an operator A such that f=SFA(f)=ASF(f) holds for all f∈W, we call A the inverse of   SF  in  W and denote it by SF-1.For obtaining the local dual framesf~ij=SFi-1(fij) (j=1,2,…,ni,i=1,2,…,m) of a given fusion frame system {(Wi,wi,{fij}j∈Ji)}i∈I, where SFi-1 is the inverse of SFi in Wi, we must calculate SFi-1 at first. For this purpose, the following lemma holds.Lemma 11.
LetW be an l-dimensional subspace of 𝔽n with an orthonormal basis {ei}i=1l and a frame F={fi}i=1k with frame bounds A, B. Define L to be an l×n matrix with the vector ei* as its ith row for i=1,2,…,l, where ei* is the conjugate-transpose of ei. The sequence G={gi}i=1k is given by gi=L(fi) for i=1,2,…,k. Then, G is a frame of 𝔽l with the same frame bounds as F.Proof.
For anyf=(a1,a2,…,al)*∈𝔽l, we have L*(f)=∑i=1l‍aiei∈W, and ∥L*(f)∥2=∑i=1l‍|ai|2=∥f∥2, where L* is the conjugate-transpose of L. Therefore,
(23)A∥f∥2=A∥L*(f)∥2≤∑i=1k‍|〈L*(f),fi〉|2=∑i=1k‍|〈f,L(fi)〉|2=∑i=1k‍|〈f,gi〉|2≤B∥L*(f)∥2=B∥f∥2,
as required.Theorem 12.
LetW be an l-dimensional subspace of 𝔽n with an orthonormal basis {ei}i=1l and a frame F={fi}i=1k. L is defined as the previously mentiond lemma. SF is the frame operator of F. Then,
(24)SF-1=L*(LSFL*)-1L,
is the inverse of SF in W. Moreover, the orthogonal projection PW from Fn onto W is PW=SF-1SF=SFSF-1=L*L.Proof.
LetΘF and ΘF* be the analysis operator and synthesis operator of F, respectively; then LΘF*=(L(f1),L(f2),…,L(fk)) is the synthesis operator of G={gi=L(fi)}i=1k which is denoted by ΘG*. By the previous lemma lemma, G is the frame of 𝔽l; hence, the matrix SG=ΘG*ΘG=LΘF*ΘFL*=LSFL* which is the frame operator of G is invertible. Denote L*(LSFL*)-1L by SF-1.
For anyf∈W, we have
(25)f=∑i=1l‍〈f,ei〉ei=L*L(f).
Therefore, we can get
(26)SF-1SF(f)=L*(LSFL*)-1LSF(f)=L*(LSFL*)-1LSFL*L(f)=L*L(f)=f,SFSF-1(f)=SFL*(LSFL*)-1L(f)=L*LSFL*(LSFL*)-1L(f)=L*L(f)=f;
hence, SF-1 is the inverse of SF in W.
Moreover, for anyf∈𝔽n, its orthogonal projection onto W is
(27)PW(f)=∑i=1k‍〈PW(f),fi〉SF-1(fi)=∑i=1k‍〈f,fi〉SF-1(fi)=SF-1ΘF*ΘF(f)=SF-1SF(f)=∑i=1k‍〈f,SF-1(fi)〉fi=ΘF*ΘFSF-1(f)=SFSF-1(f)=∑i=1l‍〈f,ei〉ei=L*L(f),
as claimed.The proof of the following proposition is straightforward, by using Proposition 2.6 of [16] and the previous theorem, we omit it.Proposition 13.
Let{(Wi,wi,Fi={fij}j=1ni)}i=1m be a fusion frame system for 𝔽n, and let F~i={f~ij}j∈Ji, i∈I be the local dual frames given by f~ij=SFi-1(fij) for all j=1,2,…,ni, i=1,2,…,m. Then, the matrix representation of the fusion frame operator is given by
(28)S𝒲=∑i∈I‍wi2ΘF~i*ΘFi=∑i∈I‍wi2ΘFi*ΘF~i=∑i∈I‍wi2SFi-1SFi=∑i∈I‍wi2SFiSFi-1,
where ΘFi and ΘF~i are the analysis operators of Fi and F~i, respectively, and SFi is the frame operator of Fi for each i∈I.Given a fusion frame system{(Wi,wi,Fi={fij}j=1ni)}i=1m of a finite-dimensional Hilbert space 𝔽n, we summarize the previous results to provide the concrete algorithm to construct its dual fusion frame system {(S𝒲-1Wi,wi,S𝒲-1Fi={S𝒲-1fij}j=1ni)}i=1m with its local dual frames as follows.Step 1.
For eachi=1,2,…,m, search the maximally linear independent subset of Fi; we denote it by Gi={gij}j=1li.Step 2.
Use the Gram-Schmidt process onGi to compute an orthonormal basis for Wi; we denote it by Ri={eij}j=1li. Construct the matrix Li constituted by this basis as follows:
(29)Li=[⟵ei1*⟶⟵ei2*⟶⋮⟵eili*⟶].Step 3.
SincePWi=Li*Li by Theorem 12, we have S𝒲=∑i=1m‍wi2Li*Li.Step 4.
Compute the local framesS𝒲-1Fi={S𝒲-1(fij)}j=1ni of {(S𝒲-1Wi,wi)}i=1m for i=1,2,…,m.Step 5.
Use formula (24) to calculate the inverse of all local frame operators SS𝒲-1Fi-1 in S𝒲-1Wi for i=1,2,…,m.Step 6.
Calculate the local dual framesSS𝒲-1Fi-1(S𝒲-1Fi)={SS𝒲-1Fi-1(S𝒲-1(fij))}j=1ni, of {(S𝒲-1Wi,wi)}i=1m for i=1,2,…,m, as required.
The fusion frame system of the following example is given in [21].Example 14.
Assume thatn=4, m=2, and n1=n2=3. The fusion frame system {(Wi,wi,Fi={fij}j=13)}i=12 is given by w1=w2=1 and
(30)f11=(66,-66,66,-33)T,f12=(66,-66,-66,33)T,f13=(66,-66,0,0)T,f21=(66,66,0,0)T,f22=(0,0,-63,-33)T,f23=(33,33,0,0)T.
The maximally linear independent subsets of F1 and F2 are {f11,f12} and {f21,f22}, respectively. Applying the Gram-Schmidt process on them, we get the orthonormal bases of W1 and W2 as follows:
(31)e11=(15,-15,15,-25)T,e12=(310,-310,-215,415)T,e21=(12,12,0,0)T,e22=(0,0,-63,-33)T.
So that
(32)L1=[15-1515-25310-310-215415],L2=[12120000-63-33].
Then, we have S𝒲=∑i=12‍wi2Li*Li=I, where I is the identity matrix, which implies that the fusion frame system {(Wi,wi,{fij}j=13)}i=12 is a Parseval fusion frame, and the local frames of the dual fusion frame system {(S𝒲-1Wi,wi,{S𝒲-1fij}j=13)}i=12 are S𝒲-1Fi=Fi for i=1,2.
Since(33)ΘF1=[66-6666-3366-66-663366-6600],ΘF2=[66660000-63-33333300],ΘF1*=[666666-66-66-6666-660-33330],ΘF2*=[66033660330-6300-330],
we can get(34)SF1=ΘF1*ΘF1=[12-1200-1212000013-2300-2323],SF2=ΘF2*ΘF2=[434300434300002323002313].
By using (24), we obtain
(35)SF1-1=L1*(L1SF1L1*)-1L1=[12-1200-1212000013-2300-2323],SF2-1=L2*(L2SF2L2*)-1L2=[3163160031631600002323002313].
Hence, the local dual frame of F1 is
(36)f~11=SF1-1f11=f11,f~12=SF1-1f12=f12,f~13=SF1-1f13=f13,
the local dual frame of F2 is
(37)f~21=SF2-1f21=(38,38,0,0)T,f~22=SF2-1f22=f22,f~23=SF2-1f23=(38,38,0,0)T.
They are also the local dual frames of the dual fusion frame system {(S𝒲-1Wi,wi,{S𝒲-1fij}j=13)}i=12.
The quark-gluon plasma is a state of the extremely dense matter which contains the quarks and gluons in high energy physics. The gray image of quark-gluon plasma is shown in Figure2. We encode the data of the image by using the local frames of the fusion frame given by this example. Suppose that the fist element of every local vector is lost in the transmission process. Then, we decode the received data by using the dual fusion frame computed by this example. The reconstructed image is shown in Figure 3. One can observe the reconstruction effect by comparing the two figures.Figure 2
The original gray image of quark-gluon plasma.Figure 3
The reconstructed gray image of quark-gluon plasma. The data of the original image is encoded by the local frames of the fusion frame given by Example14. The first coefficient of every local vector is deleted. The remained data is decoded by the dual fusion frame obtained by Example 14.


## 4.  Conclusion

We found that fusion frames can be used for experimental data transmission coding of high energy physics and studied the application of fusion frames in this field. For this goal, we first investigated the characteristics of fusion frames. We researched a class of alternate fusion duals of a given fusion frame and obtained some results about these duals. We provided a method for the matrix representation of the fusion frame operator of a given fusion frame system in a finite-dimensional Hilbert space. Based on these results, we gave an algorithm for the construction of the dual fusion frame system with its local dual frames. A simulation example has been given to show the coding effect of a fusion frame system and its dual constructed by our methods when data erasure occurs in the transmission process.



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*Source: 837129-2013-10-09.xml*