碎形維度分析的研究與應用

Abstract

量化非線性系統是研究系統特性非常重要的方法，而碎形維度則是常見的一種量化方法，由於傳統計算碎形維度的方法受限於參數的選擇或實現上不容易，因此本論文提出一種計算很容易而且不易受參數的變動而影響計算結果的方法，此方法和內在維度相似，稱為複雜度指標，由模型信號的分析結果得知，此方法的計算結果和碎形維度非常接近。另外本論文也提出改善計算效率的方法，此方法則能大幅降低所需的計算時間，使得複雜度指標在實際的應用上更有價值。在應用的方面，本論文首先將複雜度指標的方法應用於分析所錄製的癲癇腦電波信號，由分析的結果知道，此方法可以顯示腦電波在時間和空間上的關係以及監測特殊的腦電波狀態，最後則將複雜度指標的方法應用於分析紋理影像的碎形維度，此處是假設紋理影像為一個分數布朗運動模型，而根據複雜度指標和模型之間的關係則可以得到紋理影像的碎形維度。

Quantifying nonlinear systems is a very important method for characterizing the systems. And fractal dimension is the most use in quantification. Because the conventional methods of computing fractal dimension suffered from the selecting parameters or the difficulty of implementation, we present the method in this paper for easily computing and robust in changes of parameters. This method is similar to the intrinsic dimension and called the "complexity index". From the analyses of model systems, the computing results of complexity index are close to the fractal dimensions. Besides, we propose another method, which reduces most of time in computing, to improve the efficiency of computation. This makes the method of complexity index more valuable for real applications. In the cases of applications, we first apply the method of complexity index to analyze the recorded epileptic EEG. From the results of analyses, the method could reveal the time-space relationship and monitor the special states in EEG. Finally, we apply the method of complexity index to analyze the fractal dimensions of texture images. Here we assume the texture image as a model of fractional Brownian motion. According to the relationship between complexity index and fractional Brownian motion , we could estimate the fractal dimensions of texture images by complexity index.