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

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.

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 Section 2, 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 Figure 1) 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.

Let ℋ be a separable (real or complex) Hilbert space. A collection of vectors 𝐹={𝑓_𝑖}_𝑖∈𝐼⊂ℋ is called a frame for ℋ if there exist constants 0<𝐴≤𝐵<∞ such that 𝐴𝑓²≤_𝑖∈𝐼𝑓,𝑓_𝑖²≤𝐵𝑓²∀𝑓∈ℋ (1)

holds for every 𝑓∈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 frame 𝐹={𝑓_𝑖}_𝑖∈𝐼, the operator Θ_𝐹∶ℋ→ℓ²(𝐼) defined by Θ_𝐹𝑓=_𝑖∈𝐼𝑓,𝑓_𝑖𝑒_𝑖 (2) is called the analysis operator of 𝐹, where {𝑒_𝑖}_𝑖∈𝐼 is the standard orthonormal basis for ℓ²(𝐼). The adjoint operator Θ^∗_𝐹 of Θ_𝐹 given by Θ^∗_{𝐹𝑖∈𝐼}𝑐_𝑖𝑒_𝑖=_𝑖∈𝐼𝑐_𝑖𝑓_𝑖 (3) is called the synthesis operator of 𝐹. If we let 𝑆_𝐹=Θ^∗_𝐹Θ_𝐹, then we have 𝑆_𝐹𝑓=_𝑖∈𝐼𝑓,𝑓_𝑖𝑓_𝑖,𝑓∈𝐻. (4) Thus, 𝑆_𝐹 is a positive invertible bounded linear operator on ℋ, which is called the frame operator of 𝐹.

A collection of vectors 𝐹={𝑓_𝑖}_𝑖∈𝐼 in ℋ is called a dual frame for 𝐹={𝑓_𝑖}_𝑖∈𝐼 if 𝐹={𝑓_𝑖}_𝑖∈𝐼 satisfies the reconstruction formula 𝑓=_𝑖∈𝐼𝑓,𝑓_𝑖𝑓_𝑖=_𝑖∈𝐼𝑓,𝑓_𝑖𝑓_𝑖∀𝑓∈ℋ. (5) A direct calculation yields 𝑓=_𝑖∈𝐼𝑓,𝑓_𝑖𝑆_𝐹⁻¹𝑓_𝑖=_𝑖∈𝐼𝑓,𝑆_𝐹⁻¹𝑓_𝑖𝑓_𝑖∀𝑓∈ℋ. (6) This implies that {𝑆_𝐹⁻¹(𝑓_𝑖)}_𝑖∈𝐼 is a dual frame of {𝑓_𝑖}_𝑖∈𝐼. The frame {𝑆_𝐹⁻¹(𝑓_𝑖)}_𝑖∈𝐼 is called the canonical dual frame of {𝑓_𝑖}_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 frame 𝐹 is a tight frame if and only if 𝑆_𝐹=Θ^∗_𝐹⋅Θ_𝐹=𝐴𝐼_ℋ for some positive constant 𝐴, where 𝐼_ℋ is the identity operator. A frame 𝐹 is a Parseval frame if and only if 𝑆_𝐹=Θ^∗_𝐹⋅Θ_𝐹=𝐼_ℋ; that is, the canonical dual of 𝐹 is itself. So, the analysis operator Θ_𝐹 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 frame 𝐹={𝑓_𝑖}^𝑘_𝑖=1 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 notation 𝔽 when the result being stated holds for both the real number field ℝ and the complex number field ℂ. When ℋ=𝔽^𝑛, then 𝑓_𝑖(𝑖=1,2,…,𝑘) are column vectors, and the analysis operator Θ_𝐹 for the frame 𝐹={𝑓_𝑖}^𝑘_𝑖=1 is a matrix with the row vector 𝑓^∗_𝑖 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 Θ^∗_𝐹 for the frame 𝐹 is the conjugate-transpose matrix of Θ_𝐹, and the frame operator 𝑆_𝐹=Θ^∗_𝐹⋅Θ_𝐹 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 𝔽^𝑛.

Let us now recall the definitions and basic results about fusion frames which are mostly adopted from [15, 16].

Definition 1. Let 𝐼 denote an index set, and let {𝑊_𝑖}_𝑖∈𝐼 be a family of closed subspaces of a Hilbert space ℋ with a family of weights {𝑤_𝑖}_𝑖∈𝐼 where 𝑤_𝑖>0 for all 𝑖∈𝐼. Then, {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 is called a fusion frame for ℋ if there exist constants 0<𝐶≤𝐷<∞ such that 𝐶𝑓²≤_𝑖∈𝐼𝑤²_𝑖𝑃_𝑊𝑖𝑓²≤𝐷𝑓²∀𝑓∈ℋ, (7) where 𝑃_𝑊𝑖 denotes the orthogonal projection onto the 𝑊_𝑖. The constants 𝐶 and 𝐷 are called the lower and upper fusion frame bounds. The family {(𝑊_𝑖,𝑤_𝑖)}_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 {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 is called an orthonormal fusion basis if ℋ=⊕_𝑖∈𝐼𝑊_𝑖. A Bessel fusion sequence refers to the case when {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 has an upper fusion frame bound, but not necessarily a lower bound.

Definition 2. Let {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for ℋ, and {𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖} be a frame of 𝑊_𝑖 where 𝐽_𝑖 are index sets for 𝑖∈𝐼. Then, {𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖}, 𝑖∈𝐼 are called local frames, and {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}_{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 {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼, and 𝐴 and 𝐵 are the local frame bounds if there are the common frame bounds for the local frames {𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖} for each 𝑖∈𝐼. The dual frames {𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖}, 𝑖∈𝐼 of the local frames in 𝑊_𝑖 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 each 𝑖∈𝐼, let 𝑊_𝑖 be a closed subspace for ℋ, and let {𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖} be a frame for 𝑊_𝑖 with frame bounds 𝐴_𝑖 and 𝐵_𝑖. Suppose that 0<𝐴=inf_𝑖∈𝐼𝐴_𝑖≤sup_𝑖∈𝐼𝐵_𝑖=𝐵<∞. Then, the following conditions are equivalent.(i){(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 is a fusion frame for ℋ.(ii){𝑤_𝑖𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖,𝑖∈𝐼} is a frame for ℋ.
In particular, if {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖})}_𝑖∈𝐼 is a fusion frame system for ℋ with fusion frame bounds 𝐶 and 𝐷, then {𝑤_𝑖𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖,𝑖∈𝐼} is a frame for ℋ with frame bounds 𝐴𝐶 and 𝐵𝐷. If {𝑤_𝑖𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖,𝑖∈𝐼} is a frame for ℋ with frame bounds 𝐶 and 𝐷, then {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖})}_𝑖∈𝐼 is a fusion frame system for ℋ with fusion frame bounds 𝐶/𝐵 and 𝐷/𝐴.

Let 𝒲={(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for ℋ. The analysis operator Θ_𝒲 is defined by Θ_𝒲∶ℋ⟶_𝑖∈𝐼⊕𝑊_𝑖ℓ2withΘ_𝒲𝑓=𝑤_𝑖𝑃_𝑊𝑖𝑓_𝑖∈𝐼, (8) where _𝑖∈𝐼⊕𝑊_𝑖ℓ2=𝑓_{𝑖𝑖∈𝐼}∣𝑓_𝑖∈𝑊_𝑖,𝑓_{𝑖𝑖∈𝐼}∈ℓ²(𝐼) (9) is called the representation space. The synthesis operator Θ^∗_𝒲 (the adjoint operator of Θ_𝒲) can be defined by Θ^∗_𝒲∶_𝑖∈𝐼⊕𝑊_𝑖ℓ2⟶ℋwithΘ^∗_𝒲𝑓=_𝑖∈𝐼𝑤_𝑖𝑓_𝑖,𝑓=𝑓_{𝑖𝑖∈𝐼}∈_𝑖∈𝐼⊕𝑊_𝑖ℓ2. (10) The fusion frame operator 𝑆_𝒲 for 𝒲 is defined by 𝑆_𝒲𝑓=Θ^∗_𝒲Θ_𝒲𝑓=_𝑖∈𝐼𝑤²_𝑖𝑃_𝑊𝑖𝑓. (11)

About dual fusion frames, the following definition was given in [15].

Definition 4. Let {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for space ℋ with fusion frame operator 𝑆_𝒲. Then, {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 is called the dual fusion frame of {(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼.
The dual fusion frame defined previously satisfies the following reconstruction formula 𝑓=_𝑖∈𝐼𝑤²_𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=_𝑖∈𝐼𝑤²_𝑖𝑃_𝑆𝒲⁻¹_𝑊𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓,𝑓∈ℋ. (12) Based on (12), the following definition about alternate duals was introduced in [20].

Definition 5. Let 𝒲={(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for space ℋ with fusion frame operator 𝑆_𝒲, and, 𝒱={(𝑉_𝑖,𝑣_𝑖)}_𝑖∈𝐼 be a Bessel fusion sequence. Then, 𝒱 is called an alternate dual of 𝒲 if 𝑓=_𝑖∈𝐼𝑤_𝑖𝑣_𝑖𝑃_𝑉𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓, (13) holds for all 𝑓∈ℋ.

Then, it was proved that 𝒱 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 duals 𝒱={(𝑉_𝑖,𝑤_𝑖)}_𝑖∈𝐼 of a given fusion frame 𝒲={(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 which satisfy 𝑃_𝑉𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=𝑃_𝑆𝒲⁻¹_𝑊𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓,∀𝑓∈ℋ,𝑖∈𝐼. (14) A Bessel fusion sequence which satisfies (14) naturally satisfies (13). So, we can obtain the following obvious result.

Proposition 6. Let 𝒲={(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for space ℋ with fusion frame operator 𝑆_𝒲. If a Bessel fusion sequence 𝒱={(𝑉_𝑖,𝑤_𝑖)}_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. Let 𝒲={(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for space ℋ with fusion frame operator 𝑆_𝒲. A Bessel fusion sequence 𝒱={(𝑉_𝑖,𝑤_𝑖)}_𝑖∈𝐼 satisfies (14) if and only if it has the form 𝑉_𝑖=𝑆_𝒲⁻¹𝑊_𝑖⊕𝑈_𝑖,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒𝑈_𝑖𝑜𝑓ℋ𝑎𝑛𝑑𝑒𝑎𝑐ℎ𝑖∈𝐼. (15)

Proof. If (14) holds for any 𝑖∈𝐼, then we have 𝑃_𝑉𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=𝑃_𝑆𝒲⁻¹_𝑊𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓,∀𝑓∈ℋ, (16) which follows 𝐼_ℋ−𝑃_𝑉𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=𝑃_𝑉^⟂_𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=0,∀𝑓∈ℋ. (17) For any 𝑓∈𝑆_𝒲⁻¹𝑊_𝑖, there exists 𝑔∈𝑊_𝑖 such that 𝑓=𝑆_𝒲⁻¹𝑔. Hence, 𝑃_𝑉^⟂_𝑖𝑓=𝑃_𝑉^⟂_𝑖𝑆_𝒲⁻¹𝑔=𝑃_𝑉^⟂_𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑔=0, (18) which implies that 𝑓⟂𝑉^⟂_𝑖. Hence, 𝑆_𝒲⁻¹𝑊_𝑖⟂𝑉^⟂_𝑖, which follows 𝑆_𝒲⁻¹𝑊_𝑖⊂𝑉_𝑖. Let 𝑈_𝑖=𝑉_𝑖−𝑆_𝒲⁻¹𝑊_𝑖; then for any 𝑓=𝑆_𝒲⁻¹𝑔∈𝑆_𝒲⁻¹𝑊_𝑖, we have 𝑃_𝑈𝑖𝑓=𝑃_𝑉𝑖−𝑃_𝑆𝒲⁻¹_𝑊𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑔=0, (19) which implies 𝑆_𝒲⁻¹𝑊_𝑖⟂𝑈_𝑖. (20) Hence, 𝑉_𝑖=𝑆_𝒲⁻¹𝑊_𝑖⊕𝑈_𝑖.
Conversely, assume that 𝒱 satisfies (15). Then for any 𝑓∈ℋ and 𝑖∈𝐼, since 𝑆_𝒲⁻¹𝑃_𝑊𝑖(𝑓)∈𝑆_𝒲⁻¹𝑊_𝑖, we have 𝑃_𝑉𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓=𝑃_𝑆𝒲⁻¹_𝑊𝑖𝑆_𝒲⁻¹𝑃_𝑊𝑖𝑓, (21) which implies that (14) holds.

The following proposition shows that the dual fusion frame {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖)}_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. Let 𝒲={(𝑊_𝑖,𝑤_𝑖)}_𝑖∈𝐼 be a fusion frame for space ℋ with fusion frame operator 𝑆_𝒲. Then for any alternate fusion dual 𝒱={(𝑉_𝑖,𝑤_𝑖)}_𝑖∈𝐼 of 𝒲 which satisfies (14), one have _𝑖∈𝐼𝑤²_𝑖𝑃_𝑆𝒲⁻¹_𝑊𝑖𝑓²≤_𝑖∈𝐼𝑤²_𝑖𝑃_𝑉𝑖𝑓²,∀𝑓∈ℋ. (22)

Then, we consider the construction of the dual fusion frame in a finite-dimensional Hilbert space ℋ. In Section 2, we will see that any 𝑛-dimensional Hilbert space can be identified with 𝔽^𝑛, 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 𝑆_𝒲 and its inverse which need the local frames. Hence, we will study the construction of the dual fusion frame system {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖,{𝑆_𝒲⁻¹𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖})}_𝑖∈𝐼 with its local dual frames of a given fusion frame system {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖})}_𝑖∈𝐼.

Let {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}^𝑛𝑖_𝑗=1)}^𝑚_𝑖=1 be a fusion frame system for space 𝔽^𝑛. Then the analysis operator of the local frame of 𝑊_𝑖 is a 𝑛_𝑖×𝑛 matrix Θ_𝐹𝑖 with 𝑓^∗_𝑖𝑗 as its 𝑗th row, and the 𝑛×𝑛_𝑖 matrix Θ^∗_𝐹𝑖 is its synthesis operator. Furthermore, the 𝑖th local frame operator is an 𝑛×𝑛 matrix 𝑆_𝐹𝑖=Θ^∗_𝐹𝑖Θ_𝐹𝑖.

Remark 9. For the purpose of coding of any 𝑓∈𝔽^𝑛, any vector of the subspaces of 𝔽^𝑛 we consider has 𝑛 elements, and Θ_𝐹𝑖 always denotes the analysis operator of the system {𝑓_𝑖𝑗}^𝑘𝑖_𝑗=1 in 𝔽^𝑛 throughout this paper. Hence, it is a 𝑘_𝑖×𝑛 matrix, not a 𝑘_𝑖×(dim𝑊_𝑖) matrix.

Definition 10. Let 𝑊 be an 𝑙-dimensional subspace of 𝔽^𝑛 with a local frame 𝐹={𝑓_𝑖}^𝑘_𝑖=1. 𝑆_𝐹 is the local frame operator of 𝐹. If there exists an operator 𝐴 such that 𝑓=𝑆_𝐹𝐴(𝑓)=𝐴𝑆_𝐹(𝑓) holds for all 𝑓∈𝑊, we call 𝐴 the inverse of ??𝑆_𝐹??in??𝑊 and denote it by 𝑆_𝐹⁻¹.

For obtaining the local dual frames 𝑓_𝑖𝑗=𝑆_𝐹⁻¹_𝑖(𝑓_𝑖𝑗) (𝑗=1,2,…,𝑛_𝑖,𝑖=1,2,…,𝑚) of a given fusion frame system {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖})}_𝑖∈𝐼, where 𝑆_𝐹⁻¹_𝑖 is the inverse of 𝑆_𝐹𝑖 in 𝑊_𝑖, we must calculate 𝑆_𝐹⁻¹_𝑖 at first. For this purpose, the following lemma holds.

Lemma 11. Let 𝑊 be an 𝑙-dimensional subspace of 𝔽^𝑛 with an orthonormal basis {𝑒_𝑖}^𝑙_𝑖=1 and a frame 𝐹={𝑓_𝑖}^𝑘_𝑖=1 with frame bounds 𝐴, 𝐵. Define 𝐿 to be an 𝑙×𝑛 matrix with the vector 𝑒^∗_𝑖 as its 𝑖th row for 𝑖=1,2,…,𝑙, where 𝑒^∗_𝑖 is the conjugate-transpose of 𝑒_𝑖. The sequence 𝐺={𝑔_𝑖}^𝑘_𝑖=1 is given by 𝑔_𝑖=𝐿(𝑓_𝑖) for 𝑖=1,2,…,𝑘. Then, 𝐺 is a frame of 𝔽^𝑙 with the same frame bounds as 𝐹.

Proof. For any 𝑓=(𝑎₁,𝑎₂,…,𝑎_𝑙)^∗∈𝔽^𝑙, we have 𝐿^∗(𝑓)=∑^𝑙_𝑖=1𝑎_𝑖𝑒_𝑖∈𝑊, and ‖𝐿^∗(𝑓)‖²=∑^𝑙_𝑖=1|𝑎_𝑖|²=‖𝑓‖², where 𝐿^∗ is the conjugate-transpose of 𝐿. Therefore, 𝐴𝑓²=𝐴𝐿^∗𝑓²≤^𝑘_𝑖=1𝐿^∗𝑓,𝑓_𝑖²=^𝑘_𝑖=1𝑓,𝐿𝑓_𝑖²=^𝑘_𝑖=1𝑓,𝑔_𝑖²≤𝐵𝐿^∗𝑓²=𝐵𝑓², (23) as required.

Theorem 12. Let 𝑊 be an 𝑙-dimensional subspace of 𝔽^𝑛 with an orthonormal basis {𝑒_𝑖}^𝑙_𝑖=1 and a frame 𝐹={𝑓_𝑖}^𝑘_𝑖=1. 𝐿 is defined as the previously mentiond lemma. 𝑆_𝐹 is the frame operator of 𝐹. Then, 𝑆_𝐹⁻¹=𝐿^∗𝐿𝑆_𝐹𝐿^∗−1𝐿, (24) is the inverse of 𝑆_𝐹 in 𝑊. Moreover, the orthogonal projection 𝑃_𝑊 from 𝐹^𝑛 onto 𝑊 is 𝑃_𝑊=𝑆_𝐹⁻¹𝑆_𝐹=𝑆_𝐹𝑆_𝐹⁻¹=𝐿^∗𝐿.

Proof. Let Θ_𝐹 and Θ^∗_𝐹 be the analysis operator and synthesis operator of 𝐹, respectively; then 𝐿Θ^∗_𝐹=(𝐿(𝑓₁),𝐿(𝑓₂),…,𝐿(𝑓_𝑘)) is the synthesis operator of 𝐺={𝑔_𝑖=𝐿(𝑓_𝑖)}^𝑘_𝑖=1 which is denoted by Θ^∗_𝐺. By the previous lemma lemma, 𝐺 is the frame of 𝔽^𝑙; hence, the matrix 𝑆_𝐺=Θ^∗_𝐺Θ_𝐺=𝐿Θ^∗_𝐹Θ_𝐹𝐿^∗=𝐿𝑆_𝐹𝐿^∗ which is the frame operator of 𝐺 is invertible. Denote 𝐿^∗(𝐿𝑆_𝐹𝐿^∗)⁻¹𝐿 by 𝑆_𝐹⁻¹.
For any 𝑓∈𝑊, we have 𝑓=^𝑙_𝑖=1𝑓,𝑒_𝑖𝑒_𝑖=𝐿^∗𝐿𝑓. (25) Therefore, we can get 𝑆_𝐹⁻¹𝑆_𝐹𝑓=𝐿^∗𝐿𝑆_𝐹𝐿^∗−1𝐿𝑆_𝐹𝑓=𝐿^∗𝐿𝑆_𝐹𝐿^∗−1𝐿𝑆_𝐹𝐿^∗𝐿𝑓=𝐿^∗𝐿𝑓𝑆=𝑓,_𝐹𝑆_𝐹⁻¹𝑓=𝑆_𝐹𝐿^∗𝐿𝑆_𝐹𝐿^∗−1𝐿𝑓=𝐿^∗𝐿𝑆_𝐹𝐿^∗𝐿𝑆_𝐹𝐿^∗−1𝐿𝑓=𝐿^∗𝐿𝑓=𝑓; (26) hence, 𝑆_𝐹⁻¹ is the inverse of 𝑆_𝐹 in 𝑊.
Moreover, for any 𝑓∈𝔽^𝑛, its orthogonal projection onto 𝑊 is 𝑃_𝑊𝑓=^𝑘_𝑖=1𝑃_𝑊𝑓,𝑓_𝑖𝑆_𝐹⁻¹𝑓_𝑖=^𝑘_𝑖=1𝑓,𝑓_𝑖𝑆_𝐹⁻¹𝑓_𝑖=𝑆_𝐹⁻¹Θ^∗_𝐹Θ_𝐹𝑓=𝑆_𝐹⁻¹𝑆_𝐹𝑓=^𝑘_𝑖=1𝑓,𝑆_𝐹⁻¹𝑓_𝑖𝑓_𝑖=Θ^∗_𝐹Θ_𝐹𝑆_𝐹⁻¹𝑓=𝑆_𝐹𝑆_𝐹⁻¹𝑓=^𝑙_𝑖=1𝑓,𝑒_𝑖𝑒_𝑖=𝐿^∗𝐿𝑓, (27) 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 {(𝑊_𝑖,𝑤_𝑖,𝐹_𝑖={𝑓_𝑖𝑗}^𝑛𝑖_𝑗=1)}^𝑚_𝑖=1 be a fusion frame system for 𝔽^𝑛, and let 𝐹_𝑖={𝑓_𝑖𝑗}_{𝑗∈𝐽𝑖}, 𝑖∈𝐼 be the local dual frames given by 𝑓_𝑖𝑗=𝑆_𝐹⁻¹_𝑖(𝑓_𝑖𝑗) for all 𝑗=1,2,…,𝑛_𝑖, 𝑖=1,2,…,𝑚. Then, the matrix representation of the fusion frame operator is given by 𝑆_𝒲=_𝑖∈𝐼𝑤²_𝑖Θ^∗𝐹_𝑖Θ_𝐹𝑖=_𝑖∈𝐼𝑤²_𝑖Θ^∗_𝐹𝑖Θ𝐹_𝑖=_𝑖∈𝐼𝑤²_𝑖𝑆_𝐹⁻¹_𝑖𝑆_𝐹𝑖=_𝑖∈𝐼𝑤²_𝑖𝑆_𝐹𝑖𝑆_𝐹⁻¹_𝑖, (28) where Θ_𝐹𝑖 and Θ𝐹_𝑖 are the analysis operators of 𝐹_𝑖 and 𝐹_𝑖, respectively, and 𝑆_𝐹𝑖 is the frame operator of 𝐹_𝑖 for each 𝑖∈𝐼.

Given a fusion frame system {(𝑊_𝑖,𝑤_𝑖,𝐹_𝑖={𝑓_𝑖𝑗}^𝑛𝑖_𝑗=1)}^𝑚_𝑖=1 of a finite-dimensional Hilbert space 𝔽^𝑛, we summarize the previous results to provide the concrete algorithm to construct its dual fusion frame system {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖,𝑆_𝒲⁻¹𝐹_𝑖={𝑆_𝒲⁻¹𝑓_𝑖𝑗}^𝑛𝑖_𝑗=1)}^𝑚_𝑖=1 with its local dual frames as follows.

Step 1. For each 𝑖=1,2,…,𝑚, search the maximally linear independent subset of 𝐹_𝑖; we denote it by 𝐺_𝑖={𝑔_𝑖𝑗}^𝑙𝑖_𝑗=1.

Step 2. Use the Gram-Schmidt process on 𝐺_𝑖 to compute an orthonormal basis for 𝑊_𝑖; we denote it by 𝑅_𝑖={𝑒_𝑖𝑗}^𝑙𝑖_𝑗=1. Construct the matrix 𝐿_𝑖 constituted by this basis as follows: 𝐿_𝑖=⟵𝑒^∗_𝑖1⟶⟵𝑒^∗_𝑖2⟶⋮⟵𝑒^∗_𝑖𝑙𝑖⟶. (29)

Step 3. Since 𝑃_𝑊𝑖=𝐿^∗_𝑖𝐿_𝑖 by Theorem 12, we have 𝑆_𝒲=∑^𝑚_𝑖=1𝑤²_𝑖𝐿^∗_𝑖𝐿_𝑖.

Step 4. Compute the local frames 𝑆_𝒲⁻¹𝐹_𝑖={𝑆_𝒲⁻¹(𝑓_𝑖𝑗)}^𝑛𝑖_𝑗=1 of {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖)}^𝑚_𝑖=1 for 𝑖=1,2,…,𝑚.

Step 5. Use formula (24) to calculate the inverse of all local frame operators 𝑆_𝑆⁻¹_{𝒲𝐹𝑖−1} in 𝑆_𝒲⁻¹𝑊_𝑖 for 𝑖=1,2,…,𝑚.

Step 6. Calculate the local dual frames 𝑆_𝑆⁻¹_{𝒲𝐹𝑖−1}(𝑆_𝒲⁻¹𝐹_𝑖)={𝑆_𝑆⁻¹_{𝒲𝐹𝑖−1}(𝑆_𝒲⁻¹(𝑓_𝑖𝑗))}^𝑛𝑖_𝑗=1, of {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖)}^𝑚_𝑖=1 for 𝑖=1,2,…,𝑚, as required.
The fusion frame system of the following example is given in [21].

Example 14. Assume that 𝑛=4, 𝑚=2,?and 𝑛₁=𝑛₂=3. The fusion frame system {(𝑊_𝑖,𝑤_𝑖,𝐹_𝑖={𝑓_𝑖𝑗}³_𝑗=1)}²_𝑖=1 is given by 𝑤₁=𝑤₂=1 and 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. (30) 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: 𝑒₁₁=15,−15,15,−25^𝑇,𝑒₁₂=310,−310,−2,15415^𝑇,𝑒₂₁=1√2,1√2,0,0^𝑇,𝑒₂₂=√0,0,−63√,−33^𝑇. (31) So that 𝐿₁=15−1515−253−103−102154,𝐿15₂=1√21√2√0000−63−√33. (32) Then, we have 𝑆_𝒲=∑²_𝑖=1𝑤²_𝑖𝐿^∗_𝑖𝐿_𝑖=𝐼, where 𝐼 is the identity matrix, which implies that the fusion frame system {(𝑊_𝑖,𝑤_𝑖,{𝑓_𝑖𝑗}³_𝑗=1)}²_𝑖=1 is a Parseval fusion frame, and the local frames of the dual fusion frame system {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖,{𝑆_𝒲⁻¹𝑓_𝑖𝑗}³_𝑗=1)}²_𝑖=1 are 𝑆_𝒲⁻¹𝐹_𝑖=𝐹_𝑖 for 𝑖=1,2.
Since Θ_𝐹1=√66−√66√66−√33√66−√66−√66√33√66−√66,Θ00_𝐹2=6666√0000−63−√33√33√33,Θ00_∗𝐹1=√66√66√66−√66−√66−√66√66−√660−√33√330,Θ^∗_𝐹2=660√33660√33√0−630√0−330, (33)
we can get 𝑆_𝐹1=Θ^∗_𝐹1Θ_𝐹1=12−12−100212100003−√23√00−2323,𝑆_𝐹2=Θ^∗_𝐹2Θ_𝐹2=4343400343200003√23√002313. (34) By using (24), we obtain 𝑆_𝐹⁻¹₁=𝐿^∗₁𝐿₁𝑆_𝐹1𝐿^∗₁⁻¹𝐿₁=12−12−100212100003−√23√00−2323,𝑆_𝐹⁻¹₂=𝐿^∗₂𝐿₂𝑆_𝐹2𝐿^∗₂⁻¹𝐿₂=33163160031621600003√23√002313. (35) Hence, the local dual frame of 𝐹₁ is 𝑓₁₁=𝑆_𝐹⁻¹₁𝑓₁₁=𝑓₁₁,𝑓₁₂=𝑆_𝐹⁻¹₁𝑓₁₂=𝑓₁₂,𝑓₁₃=𝑆_𝐹⁻¹₁𝑓₁₃=𝑓₁₃, (36) the local dual frame of 𝐹₂ is 𝑓₂₁=𝑆_𝐹⁻¹₂𝑓₂₁=38,38,0,0^𝑇,𝑓₂₂=𝑆_𝐹⁻¹₂𝑓₂₂=𝑓₂₂,𝑓₂₃=𝑆_𝐹⁻¹₂𝑓₂₃=√38,√38,0,0^𝑇. (37) They are also the local dual frames of the dual fusion frame system {(𝑆_𝒲⁻¹𝑊_𝑖,𝑤_𝑖,{𝑆_𝒲⁻¹𝑓_𝑖𝑗}³_𝑗=1)}²_𝑖=1.
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 Figure 2. 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 Example 14. 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.

Acknowledgments

The authors are thankful to the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (11271001, 61170311), 973 Program (2013CB329404), and Sichuan Province Science and Technology Research Project (12ZC1802).