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Jinsong Leng, Qixun Guo, Tingzhu Huang, "The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics", Advances in High Energy Physics, vol. 2013, Article ID 837129, 9 pages, 2013. https://doi.org/10.1155/2013/837129
The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics
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.
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 . 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 , quantization , capacity of transmission channel [4–6], coding theory [7–12], and data transmission technology .
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 . 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 , filter bank , transmission coding [14, 25, 26], and so forth. We refer to  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 . 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.
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 be a separable (real or complex) Hilbert space. A collection of vectors is called a frame for if there exist constants such that
holds for every , where denote an index set. The optimal constants (maximal for and minimal for ) are called lower and upper frame bounds. If , then 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 uniform frame is a frame when all the elements in the frame sequence have the same norm.
Given a frame , the operator defined by is called the analysis operator of , where is the standard orthonormal basis for . The adjoint operator of given by is called the synthesis operator of . If we let , then we have Thus, is a positive invertible bounded linear operator on , which is called the frame operator of .
A collection of vectors in is called a dual frame for if satisfies the reconstruction formula A direct calculation yields This implies that is a dual frame of . The frame is called the canonical dual frame of . If a dual frame is not the canonical dual frame, it is also called an alternate dual frame.
A frame is a tight frame if and only if for some positive constant , where is the identity operator. A frame is a Parseval frame if and only if ; that is, the canonical dual of is itself. So, the analysis operator of a Parseval frame is an isometry operator. A linear operator from a Hilbert space to is called an orthogonal projection if is self-adjoint and .
Given a finite frame in an -dimensional Hilbert space , then we necessarily have . When , is automatically a basis of .
We will use the notation when the result being stated holds for both the real number field and the complex number field . When , then are column vectors, and the analysis operator for the frame is a matrix with the row vector as the th row of the matrix for , where the superscript “*” denotes the conjugate-transpose of a vector or a matrix. Relatively, the synthesis operator for the frame is the conjugate-transpose matrix of , and the frame operator is an positive invertible matrix. With respect to a fixed orthonormal basis of , any element of and any linear operator can be expressed by the coordinate vector and the matrix representation. So in most cases we will identify an -dimensional Hilbert space with .
Definition 1. Let denote an index set, and let be a family of closed subspaces of a Hilbert space with a family of weights where for all . Then, is called a fusion frame for if there exist constants such that where denotes the orthogonal projection onto the . The constants and are called the lower and upper fusion frame bounds. The family is called a C-tight fusion frame if , and it is called a Parseval fusion frame if . The family is called an orthonormal fusion basis if . A Bessel fusion sequence refers to the case when has an upper fusion frame bound, but not necessarily a lower bound.
Definition 2. Let be a fusion frame for , and be a frame of where are index sets for . Then, , are called local frames, and is called a fusion frame system for . The constants and are the associated lower and upper fusion frame bounds if they are the fusion frame bounds for , and and are the local frame bounds if there are the common frame bounds for the local frames for each . The dual frames , of the local frames in 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. , Theorem 2.3). For each , let be a closed subspace for , and let be a frame for with frame bounds and . Suppose that . Then, the following conditions are equivalent.(i) is a fusion frame for .(ii) is a frame for .
In particular, if is a fusion frame system for with fusion frame bounds and , then is a frame for with frame bounds and . If is a frame for with frame bounds and , then is a fusion frame system for with fusion frame bounds and .
Let be a fusion frame for . The analysis operator is defined by where is called the representation space. The synthesis operator (the adjoint operator of ) can be defined by The fusion frame operator for is defined by
About dual fusion frames, the following definition was given in .
Definition 4. Let be a fusion frame for space with fusion frame operator . Then, is called the dual fusion frame of .
The dual fusion frame defined previously satisfies the following reconstruction formula Based on (12), the following definition about alternate duals was introduced in .
Definition 5. Let be a fusion frame for space with fusion frame operator , and, be a Bessel fusion sequence. Then, is called an alternate dual of if holds for all .
Then, it was proved that is also a fusion frame . We will call it an alternate fusion dual of 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 of a given fusion frame which satisfy A Bessel fusion sequence which satisfies (14) naturally satisfies (13). So, we can obtain the following obvious result.
Proposition 6. Let be a fusion frame for space with fusion frame operator . If a Bessel fusion sequence satisfies (14), then it is an alternate fusion dual of .
All Bessel fusion sequences satisfying (14) form a special kind of alternate fusion duals of . The following proposition will characterize these duals.
Proposition 7. Let be a fusion frame for space with fusion frame operator . A Bessel fusion sequence satisfies (14) if and only if it has the form
Proof. If (14) holds for any , then we have
For any , there exists such that . Hence,
which implies that . Hence, , which follows . Let ; then for any , we have
Conversely, assume that satisfies (15). Then for any and , since , we have which implies that (14) holds.
The following proposition shows that the dual fusion frame can minimize the projection norm of any in the class of alternate fusion duals introduced previously. The property is analogous to Theorem 6.8 of  in the case of traditional frames, and its proof is trivial.
Proposition 8. Let be a fusion frame for space with fusion frame operator . Then for any alternate fusion dual of which satisfies (14), one have
Then, we consider the construction of the dual fusion frame in a finite-dimensional Hilbert space . In Section 2, we will see that any -dimensional Hilbert space can be identified with , 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 and its inverse which need the local frames. Hence, we will study the construction of the dual fusion frame system with its local dual frames of a given fusion frame system .
Let be a fusion frame system for space . Then the analysis operator of the local frame of is a matrix with as its th row, and the matrix is its synthesis operator. Furthermore, the th local frame operator is an matrix .
Remark 9. For the purpose of coding of any , any vector of the subspaces of we consider has elements, and always denotes the analysis operator of the system in throughout this paper. Hence, it is a matrix, not a matrix.
Definition 10. Let be an -dimensional subspace of with a local frame . is the local frame operator of . If there exists an operator such that holds for all , we call the inverse of ????in?? and denote it by .
For obtaining the local dual frames () of a given fusion frame system , where is the inverse of in , we must calculate at first. For this purpose, the following lemma holds.
Lemma 11. Let be an -dimensional subspace of with an orthonormal basis and a frame with frame bounds , . Define to be an matrix with the vector as its th row for , where is the conjugate-transpose of . The sequence is given by for . Then, is a frame of with the same frame bounds as .
Proof. For any , we have , and , where is the conjugate-transpose of . Therefore, as required.
Theorem 12. Let be an -dimensional subspace of with an orthonormal basis and a frame . is defined as the previously mentiond lemma. is the frame operator of . Then, is the inverse of in . Moreover, the orthogonal projection from onto is .
Proof. Let and be the analysis operator and synthesis operator of , respectively; then is the synthesis operator of which is denoted by . By the previous lemma lemma, is the frame of ; hence, the matrix which is the frame operator of is invertible. Denote by .
For any , we have Therefore, we can get hence, is the inverse of in .
Moreover, for any , its orthogonal projection onto is as claimed.
The proof of the following proposition is straightforward, by using Proposition 2.6 of  and the previous theorem, we omit it.
Proposition 13. Let be a fusion frame system for , and let , be the local dual frames given by for all , . Then, the matrix representation of the fusion frame operator is given by where and are the analysis operators of and , respectively, and is the frame operator of for each .
Given a fusion frame system of a finite-dimensional Hilbert space , we summarize the previous results to provide the concrete algorithm to construct its dual fusion frame system with its local dual frames as follows.
Step 1. For each , search the maximally linear independent subset of ; we denote it by .
Step 2. Use the Gram-Schmidt process on to compute an orthonormal basis for ; we denote it by . Construct the matrix constituted by this basis as follows:
Step 3. Since by Theorem 12, we have .
Step 4. Compute the local frames of for .
Step 5. Use formula (24) to calculate the inverse of all local frame operators in for .
Step 6. Calculate the local dual frames , of for , as required.
The fusion frame system of the following example is given in .
Example 14. Assume that , ,?and . The fusion frame system is given by and
The maximally linear independent subsets of and are and , respectively. Applying the Gram-Schmidt process on them, we get the orthonormal bases of and as follows:
Then, we have , where is the identity matrix, which implies that the fusion frame system is a Parseval fusion frame, and the local frames of the dual fusion frame system are for .
we can get By using (24), we obtain Hence, the local dual frame of is the local dual frame of is They are also the local dual frames of the dual fusion frame system .
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.
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.
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).
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