EVALUATING THE FRACTAL DIMENSION OF PROFILES
EVALUATING THE FRACTAL DIMENSION OF PROFILES
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DOI：
10.1103/physreva.39.1500
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发表时间：
1989-02-01
影响因子：
2.9
通讯作者：
ZUCKER, SW
中科院分区：
文献类型：
--
作者：
DUBUC, B;QUINIOU, JF;ZUCKER, SW
物体的分形维数有很多种定义，包括盒维数、Bouligand-Minkowski维数和相交维数。虽然它们在连续域中都是等价的，但在离散化和应用于数字化数据时，它们有很大的不同。我们发现，这些定义的标准实现与已知的分形维数（维尔斯特拉斯-曼德尔布罗特，Kiesswetter，分数布朗运动）的自仿射曲线产生的结果与显着的错误。这些错误的来源的分析导致一个新的算法在一维，称为变分法，它产生准确的结果。变分法使用ε振荡的概念来测量一维函数在ε邻域中的振幅。当ε趋于零时，ε振荡积分的增长阶数（称为ε变分）与分形维数直接相关。本文提出了一维剖面的变分法，并证明了在极限情况下，它与经典的计盒法是等价的。其结果是一个算法，可靠地估计分形维数的一维配置文件，即图形的功能，一个单一的变量。该算法进行了测试与已知的分形维数的配置文件。
There are many definitions of the fractal dimension of an object, including box dimension, Bouligand-Minkowski dimension, and intersection dimension. Although they are all equivalent in the continuous domain, they differ substantially when discretized and applied to digitized data. We show that the standard implementations of these definitions on self-affine curves with known fractal dimension (Weierstrass-Mandelbrot, Kiesswetter, fractional Brownian motion) yield results with significant errors. An analysis of the source of these errors leads to a new algorithm in one dimension, called the variation method, which yields accurate results. The variation method uses the notion of ε oscillation to measure the amplitude of the one-dimensional function in an ε neighborhood. The order of growth of the integral of the ε oscillation (called the ε variation), as ε tends toward zero, is directly related to the fractal dimension. In this paper, we present the variation method for one-dimensional (1D) profiles and show that, in the limit, it is equivalent to the classical box-counting method. The result is an algorithm for reliably estimating the fractal dimension of 1D profiles; ie, graphs of functions of a single variable. The algorithm is tested on profiles with known fractal dimension.