發(fā)布時(shí)間：2018-05-13 19:48

  本文選題：框架 + 融合框架　； 參考：《電子科技大學(xué)》2017年博士論文

【摘要】：框架理論是泛函分析、非線性逼近理論、算子理論以及信號(hào)理論相結(jié)合的產(chǎn)物,它是繼小波理論之后逐步發(fā)展起來的一個(gè)新的研究方向。框架理論的發(fā)展極大地促進(jìn)了純粹數(shù)學(xué)與工程應(yīng)用的結(jié)合發(fā)展,具有十分廣闊的應(yīng)用前景。如今,框架理論已經(jīng)廣泛地應(yīng)用于圖像處理、信號(hào)處理、采樣理論、數(shù)據(jù)壓縮、系統(tǒng)建模、編碼和通信等方面。隨著現(xiàn)代信息技術(shù)的快速發(fā)展和廣泛應(yīng)用,人們更加重視信息資源的開發(fā)和利用。盡管框架理論已經(jīng)得到了較好的發(fā)展,但是它作為一個(gè)新興的研究方向仍有許多問題需要進(jìn)一步研究。本學(xué)位論文對(duì)框架的基本理論展開研究,并解決框架在信號(hào)傳輸過程中有數(shù)據(jù)丟失時(shí)的重構(gòu)問題,主要研究?jī)?nèi)容如下:1.研究基于矩陣的框架設(shè)計(jì)問題。利用矩陣的奇異值分解得到構(gòu)造特殊框架的方法,同時(shí),利用酉矩陣得到一些新的緊框架,解決了求解框架算子逆的復(fù)雜性問題。該方法操作簡(jiǎn)單,從而擴(kuò)大了框架在實(shí)際問題中的應(yīng)用。2.研究融合框架的一些等式和不等式問題。利用有界線性算子的理論和方法,建立了Hilbert空間中的融合框架的等式和不等式。此結(jié)論有助于解決融合框架在并行處理和高性能物理實(shí)驗(yàn)中的相關(guān)問題。3.由于g-框架是框架的廣義形式,我們研究g-框架的相關(guān)結(jié)論。首先通過引入g-框架算子相應(yīng)的有界線性算子研究g-框架的穩(wěn)定性。進(jìn)一步,通過引入最壞情況誤差,研究對(duì)偶g-框架在有數(shù)據(jù)丟失情況下的最優(yōu)對(duì)偶g-框架,并討論規(guī)范對(duì)偶g-框架是唯一最優(yōu)對(duì)偶g-框架的充分必要條件。最后,利用已知g-框架和有界算子給出逼近對(duì)偶g-框架關(guān)于局部框架的性質(zhì),并證明了兩個(gè)g-框架是彼此接近時(shí),它們的逼近對(duì)偶g-框架也是彼此接近的。4.研究框架理論在信號(hào)傳輸過程中有丟失時(shí)的重構(gòu)問題�；谧顑�(yōu)直接法(MOD),提出一種新的搜索最優(yōu)對(duì)偶框架的方法。在信號(hào)重構(gòu)中該方法能夠?qū)ふ业阶顑?yōu)對(duì)偶框架,解決對(duì)于特殊輸入信號(hào)不是最優(yōu)的問題。同時(shí),該方法搜索到的最優(yōu)對(duì)偶框架能夠減小重構(gòu)信號(hào)與原始信號(hào)的誤差,從而在一定程度上解決了信號(hào)傳輸過程中的重構(gòu)問題。數(shù)值實(shí)驗(yàn)也驗(yàn)證了新的方法的有效性。
[Abstract]:Frame theory is the product of the combination of functional analysis, nonlinear approximation theory, operator theory and signal theory. It is a new research direction after wavelet theory. The development of frame theory has greatly promoted the combination of pure mathematics and engineering application, and has a very broad application prospect. Nowadays, the framework theory has been widely used in image processing, signal processing, sampling theory, data compression, system modeling, coding and communication. With the rapid development and wide application of modern information technology, people pay more attention to the development and utilization of information resources. Although the frame theory has been well developed, as a new research direction, there are still many problems that need to be further studied. In this dissertation, the basic theory of the frame is studied, and the reconstruction problem of the frame with data loss in the process of signal transmission is solved. The main research contents are as follows: 1. The framework design based on matrix is studied. The method of constructing a special frame is obtained by using singular value decomposition of matrix. At the same time, some new compact frames are obtained by using unitary matrix, and the complexity problem of solving the inverse of frame operator is solved. This method is easy to operate, thus expanding the application of the framework in practical problems. 2. 2. Some equality and inequality problems of fusion frame are studied. By using the theory and method of bounded linear operator, the equality and inequality of fusion frame in Hilbert space are established. This conclusion is helpful to solve the related problems in parallel processing and high performance physics experiments. Since g-frame is a generalized form of frame, we study the relevant conclusions of g-frame. Firstly, the stability of g-frame is studied by introducing the bounded linear operator corresponding to g-frame operator. Furthermore, by introducing the worst-case error, we study the optimal dual g-frame with data loss, and discuss the sufficient and necessary conditions for the canonical dual g-frame to be the only optimal dual g-frame. Finally, by using known g- frames and bounded operators, we give the properties of approximation dual g-frames with respect to local frames, and prove that when two g- frames are close to each other, their approximation dual g-frames are also close to each other. This paper studies the reconstruction of frame theory when it is lost in the process of signal transmission. Based on the optimal direct method, a new method for searching the optimal dual frame is proposed. In signal reconstruction, this method can find the optimal dual frame and solve the problem that the special input signal is not optimal. At the same time, the optimal dual framework searched by this method can reduce the error between the reconstructed signal and the original signal, thus solving the reconstruction problem in the process of signal transmission to a certain extent. Numerical experiments also verify the effectiveness of the new method.
【學(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：TN911

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 曹懷信;Hilbert空間中的Bessel序列(英文)[J];工程數(shù)學(xué)學(xué)報(bào);2000年02期

，

本文編號(hào)：1884554