Interpolatory Multiresolution Analysis And Its Applications

• Main Authors: Shyu ,Wang-Shin, 徐旺興
• Other Authors: Sun, Tien-Yu
• Format: Others
• Language: en_US
• Published: 1996
• Online Access: http://ndltd.ncl.edu.tw/handle/00985060828548653374

碩士 === 中原大學 === 應用數學研究所 === 84 === Daubechies 在 $L^2({\bf R})$. 上建構緊緻涵蓋的正交小波基底的架 構. 在早期的發展, 這些 Daubechies 小波已經被電子工程和影影像處理 密切的注意. 它提供一個次帶過濾方案的類別來正確的重建信號. Beylkin, Coifman 和 Rokhlin 使用這些緊緻涵蓋的小波經由截斷矩陣的 多重解析度分析表示中很小的元素來壓縮矩陣成為稀疏的矩陣. 基於這些 理論, 矩陣對向量的快速乘法的演算法已被發展而且而且被應用於將解數 值偏微分方程和積分方程的多層級演算法化為公式. Beylkin 討論過多變 數的微分和積分算子在正交的小波基底下表示的問題. 另外一個小波基底 理論的重要進展是由 Cohen, Daubechies, 和 Feauneau 所做的 biorthogonal 小波基底.我們將在第二節開始描述 Harten 所提的離散的 多重解析度分析的概念. 在此一架構下, 第三節將詳細的討論 Harten 的 內插多重解析度分析. 而第四節, 我們將建構矩陣多重解析度分析的表 示: 標準型和非標準型都將被討論. 然而, 矩陣的資料壓縮可以經由截段 一些多重解析度分析中較小的尺度係數而穫得. 基於矩陣的資料壓縮, 矩 陣對向量和矩陣對矩陣的快速乘法應運而生. 在第四節的最後將把這些乘 法的演算法做一翻比較. 最後在第五節中, 將提出解線性偏微分方程的多 層級方案. 主要的目的在發展多層級的擬轉換方和導出快速的 Poisson solver. Daubechies established the framework of orthonormal wavelet bases of compact support on $L^2({\bf R})$. In its early development, these Daubechies wavelets have received closed attention in electrical engineering and image processing, providing a class of subband filtering schemes with exact reconstruction in which the synthesis and analysis filters coincide. Then, Beylkin, Coifman and Rokhlin used these compactly supported wavelets to compress a large class of matrices to sparse matrices by truncating small entries in the multiresolution representations of matrices. Based on these theories, fast algorithm for matrix-vector multiplication is developed and is applied to formulate multilevel algorithms for solving partial differential equations and integral equations numerically. for such applications. Beylkin also discussed the problem of representing various differential and integral operators in orthonormal wavelet bases in his paper Another important advance in the theory of wavelet bases is the work of Cohen, Daubechies, and Feauneau on biorthogonal wavelet bases; In this work, it is shown that, under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters lead to two Riesz bases of compactly supported wavelets. We begin by describing Harten's notion of discrete multiresolution analysis in section 2,Under this framework, a detailed discussion of Harten's interpolatory multiresolution analysis is given in section 3. In section 4, we establish the multiresolution representation of matrices. Both the standard and nonstandard form are treated. Then data compression of matrices can be achieved by truncating scale coefficients in the multiresolution representation which are smaller than a selected threshold. Based on the data compression of matrices, fast algorithms for matrix-vector multiplication are developed. A comparison of these fast matrix- vector multiplication algorithms is given at the end of section