Design and implementation of an estimator of fractal dimension using fuzzy techniques
Introduction
The study of fractal geometry leads us to a better comprehension of complex systems in the nature which show fractal characteristics. These characteristics, i.e. the phenomena of auto-similarity or auto-affinity, exist everywhere in natural textures. Since the publication of Mandelbrot's provocative book [1] on fractal geometry, this concept has been widely used to characterize the behavior of chaotic systems [2], to define models of natural objects [1]. It has also been applied to the general area of image analysis as a means for compressing images [3], as a vehicle for segmenting images [4], etc.
Fractal dimension is the most popular parameter for explaining and describing natural textures. The accuracy of estimation of fractal dimension influences the final results in many applications of fractal geometry. A great number of estimators, such as the box counting method [5], [6] the variation method [7], the Bouligand–Minkowski method [7], the power spectrum method [6], have been proposed. According to the experimental results reported in the literature, the accuracy of these estimators is significantly affected by true value of fractal dimension D, resolution and quantization effect [8]. Apart from these elements, trend of profile or surface is also strongly related to the accuracy of estimators [9], [10].
The previous elements affect the accuracy of estimation in a systematic way. According to the results of Huang et al. [8] and those found in our experiments, the estimation error roughly increases with D and the ranges of the estimates change more consistently toward the correct direction with the increase of resolution. However, it is very difficult to precisely determine how such an element affects the accuracy of estimation because there exist a strong correlation between all involved elements.
The accuracy of estimation can also be affected in a stochastic way. Some researchers remarked that the accuracy of the existing estimators is related to the way in which experimental points are distributed on the log–log plane, i.e. the choice of the sequence {εk} representing the sizes of structuring elements at different scales in a decreasing order. They tried to find the optimistic estimators, which permit to decide with what sequence {εk} the estimate should be computed so that the estimation error is minimized [7], [8]. According to our observation, some sequences {εk} can lead to higher estimation accuracy than other ones but the optimal sequence {εk} is an uncertain element which cannot be approximated by a determinist function. The estimated fractal dimension based on only one measure is not very significant.
Therefore, the difference between the estimate of fractal dimension De and its real value D is affected by two kinds of errors: (1) stochastic errors, which are mainly caused by the choice of the sequence {εk} and (2) systematic errors, which vary with D, resolution, trends of profiles or surfaces and several other elements. Both errors cannot be modeled by any explicit function because of the complexity and the uncertainty in the relationship between the fractal dimension and the measurable variables related to it.
In this paper, we propose a new method for improving the accuracy of estimation of fractal dimension by approximating a continuous function. In this function, the real fractal dimension D is considered as output variable and the corresponding measurable variables such as De are considered as input variables. As discussed before, the relationship between D and its input variables is not a determinist function. Both stochastic and systematic errors exist in each estimate De. Then, we decompose the procedure of function approximation into two following steps:
(1) The averaging step: it permits to obtain a weighted averaged estimate De1 from a number of measures on fractal dimension. These measured data correspond to different sequences {εk}. Stochastic errors can be largely decreased by this step and a determinist relationship between D and De1 can be obtained. For each measured value, we calculate the corresponding weight based on its distance to the averaged value of all the measured data.
(2) The approximation step: it is based on a fuzzy logic controller (FLC), which permits to estimate the value of the difference between D and De1, denoted by ΔD. A FLC is composed of a fuzzy rule base, a fuzzification interface, a fuzzy inference mechanism and a deffuzzification interface. According to the literature [11], [12], it has been proved that a FLC is capable of approximating any real continuous function on a compact set to arbitrary accuracy.A great deal of fuzzy controllers, such as Sugeno [13] controller, Mandami [14] controller have been successfully applied to a great number of practical approximation problems.
In a FLC dealing with a multi-input/single-output function approximation problem, the fuzzy rule base is composed of the following rules:
where xi′s () and y are the input and the output variables of the FLC.
The fuzzification interface calculates the membership function of each input variable and the defuzzification interface gives an output value according to the membership function of y and the inference mechanism.
In the FLC used in the approximation step, the fuzzification permits to partition the universe of discourse of ΔD into sub intervals and that of the input variables into subspaces. The fuzzy rules can be extracted from a set of fractal objects as learning samples, where real fractal dimensions are known. These extracted fuzzy rules permit to explain the complex relationship between the accuracy of estimation of fractal dimension and the involved systematic elements such as real value of D and resolution. By inferring from the fuzzy rules, we can calculate the approximate value of ΔD, which leads to the final estimate of fractal dimension.
The proposed method is based on the box counting estimator for taking measures on fractal dimensions, but it differs from the existing estimators in that it is a model-free estimator, i.e. fractal dimensions can be estimated without requiring a specific mathematical description of how the output functionally depends on the inputs. This estimator is obtained only from learning data. With this method, estimation error of fractal dimension can be largely decreased because stochastic errors related to different sequences of {εk} can be compensated by each other at the averaging step and systematic errors related to a set of correlated measurable variables can be estimated using the FLC.
A value of fractal dimension can be included between 1 and 2 (profiles) or between 2 and 3 (surfaces). The higher the value of fractal dimension, the more the fractal structure “fills” the underlying space and “rougher” the profile or the surface appears. In this paper, we study only fractal dimensions of one dimensional profiles, i.e., graphs of continuous mono-valued functions of a single variable.
This paper is organized as follows. Section 2 reviews two well-known fractal functions and two existing estimators of fractal dimension. Section 3 describes the details of the averaging step. Section 4 designs the output/input variables of the FLC. Section 5 presents a method for extracting fuzzy rules directly from learning profiles. Given a new one-dimensional profile, the value of D can be estimated according to the extracted fuzzy rules. The membership functions of the FLC are optimized in Section 6. Next, the simulation results and the corresponding analysis are shown in Section 7. The application of the proposed estimator to crimp analysis of wool fibers is also presented in this section. Finally, in Section 8, we give a conclusion and discuss the possibility of extending the proposed estimator to other profiles and surfaces.
Section snippets
Fractal functions
In order to build a learning base of profiles for the proposed estimator, we implemented two self-affine curves with known fractal dimensions (Takagi function and Weierstrass–Mandelbrot function).
A Takagi function, presented by Dubuc [15], is constructed from a superposition of sawtooth functions with frequency and amplitude scaled geometrically:where Δ(t) is the distance of t to the nearest integer. The function Ψ(·) is called the generating kernel
Averaging step
In fact, our estimator is a correction procedure based on an existing estimator: the box counting estimator. This procedure permits to correct errors caused by the previous elements affecting the accuracy of estimation. According to Section 1, existing estimators yield significant stochastic errors and systematic errors between estimated and real fractal dimensions. In the proposed estimator, the first step is to calculate the weighted average of a number of estimated fractal dimensions De's so
Design of the FLC
At the approximation step, D is considered as a determinist continuous function of the elements affecting its precision. This implicit function is approximated by a FLC whose output and input are given as follows.
Denote ΔD the difference between D and its estimate, i.e. ΔD=D−De1. We take ΔD as output variable of the FLC because it is more sensitive to different values of De1. If De1 is close enough to D, ΔD should be designed to be in a small interval close to 0, which cannot amplify the error
Fuzzy rules extraction
In the design of a FLC, an important problem is the extraction of fuzzy rules. Fuzzy rules can be extracted from expert knowledge or numerical data. As no expert knowledge is available in the approximation of the relationship between D and the involved input variables, we use a procedure developed by Abe and Lan [18] for extracting fuzzy rules from learning numerical input/output data. This procedure has shown its effectiveness in applications [18], [19].
The basic idea of this method is
Optimization of the FLC
Genetic algorithms have shown increasing interest in engineering applications since the publication of the paper of Holland [20] in 1975. Genetic algorithms are designed to find a global maximum of a function of many variables by performing a particular kind of genetic-inclined search of the space of these variables. This kind of search is especially interesting when solving complex optimization tasks, including the structure and parameters optimization of fuzzy models [21]. The main principle
Simulation results
The simulation results are given in this section. Two fractal functions presented in Section 2 (the WM function and the Takagi function) with different values of D are applied to test the effectiveness of the proposed estimator. The learning base is built from a series of WM profiles with b=2.1, domain Df=[0.6, 0.61] and various values of D.
The weights {wi} and {vi} of the averaging step and the membership functions of the approximation step are optimized using genetic algorithms with following
Conclusion
We propose in this paper a new method for estimating fractal dimensions. It permits to reduce significantly two types of estimation errors: stochastic errors and systematic errors. Stochastic errors are solved at the averaging step and systematic errors are solved at the approximation step using a FLC in which several elements affecting the estimation precision of fractal dimension are taken as input variables. In this FLC, both the extraction of fuzzy rules and the optimization of parameters
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