小波轉換應用於碎形圖、影像編碼及類神經網路判別混沌系統的研究

Abstract

小波轉換提供了一種分析的工具，它可在彈性的時域與頻域平面上將信號 分解為數個成分。尤其結合正交有限支持小波與多解析度分析的離散小波 轉換，更被視為有效率並且易於實現的方法。在這篇論文中，我們先研究 連續式小波轉換，並呈現其在碎形圖上的應用。根據 Daubechies 的研究 ，我們找出另一種頻域分解因式法來建構正交精簡支持小波。接著我們將 離散小波轉換應用在兩個實際的情況。在影像編碼上，我們利用推廣的二 維離散小波轉換，將影像轉成數個小波係數，並將其量化及編碼以達到壓 縮的效果。在第二個應用中，我們提出了一個以小波為基底之類神經網路 來判別混沌系統，此架構克服了在傳統類神經網路訓練過程中費時的缺點 。 The wavelet transform (WT) provides an analyzing tool to decompose a signal into several components in a flexible time- frequency plane. The discrete wavelet transform (DWT), particularly combined orthonormal finite supported wavelets with multiresolution analysis, has been regarded as an efficient and easy-implemented WT. In this thesis, we first investigate the continuous WT and present its application of fractals. Then, based on the research of Daubechies, we explore another spectral factorization to establish some compactly supported wavelets for the DWT. Next, we apply the DWT to two physical cases. Using the extended version of the DWT$-$two- dimensional DWT, we convert an image into several wavelet coefficients. Then these subimages are quantized and encoded in order to perform compression. In the second application, we propose a wavelet-based neural network (WBNN) to identify chaotic systems. The WBNN structure overcomes a disadvantage of time-consuming, existing in the training process of conventional neural networks.