本文選題：MRA-框架 + 洛朗多項式��； 參考：《陜西師范大學》2015年碩士論文

【摘要】：小波框架既能克服正交小波的不足,又增加了適當?shù)娜哂嘈?不但保持了除正交之外的所有小波的性質(zhì),例如很好的時頻局部化特性和平移不變性。在實際應用中它可以把光滑性、緊支撐、對稱性(或反對稱性)等完美的結合在一起。對信號的重構較正交小波有更好的穩(wěn)定性；而且框架比正交小波更易于設計。本文首先介紹了多分辨分析和小波框架；然后給出了由MRA出發(fā)構造的框架(MRA-框架),這樣構造的框架具有極其類似于使用MRA構造的小波(MRA-小波)的分解和重構算法。這樣的算法非常簡單,僅僅是分層迭代(類似于Mallat算法)。還介紹了框架乘子,包括對應于單個生成元框架的框架乘子和對應于多個生成元框架的Fourier乘子矩陣,它們都可以從已經(jīng)構造好的小波框架出發(fā)構造出不同于已經(jīng)存在的小波框架。基于MRA-框架的這種優(yōu)勢和Fourier乘子矩陣的這種思想的基礎之上,將Fourier乘子矩陣限制在MRA-框架的范圍內(nèi),得到了高通濾波器乘子矩陣。 高通濾波器乘子矩陣可以從已經(jīng)構造好的MRA-框架出發(fā)構造出不同于已經(jīng)存在的MRA-框架。并且,給出了一個高通濾波器乘子矩陣的充分條件。然后,給出了怎樣使得具有不同對稱類型的洛朗多項式的加法和乘法運算結果仍然是對稱(或者反對稱)的洛朗多項式條件。根據(jù)洛朗多項式的這個性質(zhì),得到了構造出具有特定的對稱性的高通濾波器乘子矩陣,可以使得由它構造的MRA-框架均是對稱(或者反對稱),只要已經(jīng)構造好的MRA-框架是對稱(或者反對稱)的,這就是算法1。然后,用算法1給出了幾個例子,從已經(jīng)構造好的具有兩個(三個)高通濾波器的對稱(或者反對稱)MRA-框架出發(fā),得到不同于它們的具有兩個(三個)高通濾波器的對稱(或者反對稱)MRA-框架。最后,再用最基本的圖像處理——圖像去噪,說明了使用高通濾波器乘子矩陣構造出的MRA-框架在信號處理方面還是具有一定的使用價值。
[Abstract]:Wavelet frame can not only overcome the deficiency of orthogonal wavelet, but also increase proper redundancy. It not only preserves the properties of all wavelets except orthogonality, such as good time-frequency localization and translation invariance. In practical applications, it combines smoothness, compact support, symmetry (or antisymmetry) perfectly. The reconstruction of signal is more stable than orthogonal wavelet, and the frame is easier to design than orthogonal wavelet. In this paper, we first introduce the multi-resolution analysis and wavelet framework, and then give the frame (MRA-frame) constructed from MRA, which is very similar to the decomposition and reconstruction algorithm of MRA-constructed wavelet (MRA-wavelet). Such an algorithm is very simple and is simply a hierarchical iteration (similar to the Mallat algorithm). The frame multipliers, including the frame multipliers corresponding to a single generative meta-frame and the Fourier multiplier matrices corresponding to a plurality of generating meta-frames are also introduced. All of them can construct wavelet frames which are different from the existing ones. Based on the advantage of MRA-frame and the idea of Fourier multiplier matrix, the Fourier multiplier matrix is limited to the MRA-frame, and the high-pass filter multiplier matrix is obtained. The multiplier matrix of high pass filter can construct different MRA-frame from the MRA-frame which has already been constructed. Moreover, a sufficient condition for the multiplier matrix of high pass filter is given. Then, how to make the addition and multiplication results of Laurent polynomials of different symmetric types are still symmetric (or antisymmetric) conditions of Laurent polynomials are given. According to this property of Laurent polynomial, the multiplier matrix of high pass filter with special symmetry is constructed. We can make the MRA-frame constructed by it symmetric (or antisymmetric), so long as the constructed MRA-frame is symmetric (or anti-symmetric), this is the algorithm 1. Then, by using algorithm 1, several examples are given, starting from a symmetric (or antisymmetric) MRA-frame with two (three) high pass filters. A symmetric (or antisymmetric) MRA-frame with two (or three) high pass filters is obtained. Finally, by using the most basic image process-image denoising, it is shown that the MRA-frame constructed by using the multiplier matrix of high-pass filter is still valuable in signal processing.
【學位授予單位】：陜西師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：TN713

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相關期刊論文 前2條

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本文編號：2117498