Fractal interpolation functions [6] are included in the afore- mentioned fractal ..... [2] M. F. Barnsley, Fractals Everywhere, New York Academic,. 1988. [3] X. Zhu ...
EFFICIENT CONTOUR SHAPE DESCRIPTION BY USING FRACTAL INTERPOLATION FUNCTIONS Satoshi UEMURA, Miki HASEYAMA, Hideo KITAJIMA School of Engineering, Hokkaido University Kita-Ku Kita-13 Nishi-8, Sapporo 060-8628, Japan ABSTRACT This paper presents a novel representation method for contour shape using Fractal Interpolation Functions (FIF). In the traditional idea of the FIF, the scope of its application has been limited to the case where the signal is represented by a single-valued function. Therefore, the traditional FIF cannot be applicable to multiple-valued signals. The proposed method can model a multiple-valued signal with an extended FIF derived by introducing new parameters to the traditional one. Furthermore, the proposed method utilizes the fractal dimension known as a measure of complexity to determine the parameters in the FIF and thereby can model the signal based on its complexity. Experimental results show the validity of the proposed method.
shape description using the extended FIF is derived in Section 3, and experimental results are given in Section 4. Lastly, concluding remarks are given in Section 5. 2. FRACTAL INTERPOLATION FUNCTIONS In this section, we summarize basic ideas from the theory of FIF [6]. Let be a given discrete signal, where
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1. INTRODUCTION
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In the past several years, modeling a given discrete signal to achieve highly accurate analysis has received a great deal of attention in the field of signal processing, image processing, and computer vision. Traditionally, Lagrange’s polynomial formula, splines, and ARMA model have been used to model the discrete signal. However, recently it was reported that the best way to model the signal which has the property of self-similarity [1] or self-affinity [2] is to use a fractal model [3, 4]. It is well-known that the natural shapes such as shoreline and mountain profile have the property of statistically self-similarity [5] and are easily described by using fractal models [1, 2]. Fractal interpolation functions [6] are included in the aforementioned fractal models and are used to model a one-dimensional discrete signal. When the FIF are applied to a given discrete signal, the signal is represented by its own contraction. This indicates that the FIF are capable of producing varied signals depending on the determination of the parameters. Hence an active area of research is the inverse problem [3, 7, 8]; that is, the determination of the map parameters to produce an accurate approximation of the original. Though the FIF have the ability to model a given discrete signal well, its application has been limited to the case where the signal is represented by a single-valued function. Therefore, an extended FIF derived by introducing new parameters to the traditional one are presented, and it can model a multiple-valued signal. Furthermore, the fractal dimension known as a measure of the complexity is used to solve the inverse problem, and thereby a given signal can be modeled based on its complexity. Therefore, by using the proposed method, we can efficiently describe the contour shape. This paper is organized as follows. In Section 2, the basic idea of traditional FIF is presented in brief. An efficient contour
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Under the above constraint, if is a variable then the other four parameters , , , and can be written as follows:
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Note that the parameter chosen as a free parameter has the ability to control the vertical scaling of and it is called the contraction factor concerned with the th interpolation interval. When the FIF are used to model a given discrete signal, the inverse problem arises : how to generate the attractor approximated to the given signal, and this problem results in the problem how to determine the interpolation points and the contraction factors. We propose a solution of the problem by using the fractal dimension. Therefore, we describe the relation between the fractal dimension of the attractor and the map parameters in the following subsection. /
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Table 1. The value of the contraction factor versus the fractal dimension of the function . fractal dimension fractal dimension 0.1 1.013058 0.5 1.092842 0.6 1.109844 0.2 1.033364 0.7 1.116691 0.3 1.060195 0.4 1.076334 0.8 1.121626
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a multiple-valued function. The outline of our method is: first, with the help of new introduced parameters, two signals can be generated from ; second, the map parameters are determined by using the fractal dimension and then two attractors concerned with each new signal are generated; finally, the given signal is represented by the attractor acquired by synthesizing the two attractors. Detailed account of our proposed method is as follows.
2.1. Fractal Dimension of the Attractor W
Let denote the fractal dimension of the attractor , then the and the map parameters is presented as folrelation between lows [2]: b
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In order to understand this relation more definitely, we consider the relation between the contraction factor and the function acquired by mapping into the th interpolation interval. Since as the parameter helps to control the vertical scaling of described before, we guess that the value of affects the complexity of . That is, the smaller becomes, the smaller becomes. the complexity of We verify the validity of this consideration by the following experiments. Fig. 1(a) shows a discrete signal , and (b) shows acquired by mapping into the interval with varbeious . It can be seen from Fig. 1(b) that as the value of comes smaller, also the complexity of becomes smaller and the shape of gets closer to a straight line. Table 1 shows the value of versus the fractal dimension of computed with the yard-stick method [1]. We can find from Table 1 that the is to closer is to zero, the closer the fractal dimension of one. From these results, we can say that the contraction factor affects the complexity of . M
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3. CONTOUR SHAPE DESCRIPTION In this section, we present a novel and efficient contour shape description with an extended FIF derived by introducing new parameters to the traditional FIF. Here, let be a given discrete signal which is represented by
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3.2. Parameter Identification
First, a set of the interpolation points for the signal and must be determined. Thus we extract them with equal space from and , respectively; here, we let
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obtained because the th interpolation interval includes only a few data. However, our proposed method does not include the above problem because the fractal dimension required by the method is computed as follows. First, an interval is set on as shown in is set on the centre of the interval Fig. 3 (a); the point data points are included within the interval. Second, and we compute the fractal dimension of the data within this interval and denote the value by . Repeat above process as we move the interval toward the right endpoint of , finally a set of fractal can be obtained as shown in dimension Fig. 3 (b). It can be seen from Fig. 3 (b) that the fractal dimension corresponds with the local complexity of the signal. Thus we can use the following equation for the purpose to compute the fractal dimension of , instead of computing directly.
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We illustrate the effectiveness of our proposed method with some examples. As shown in Fig. 4, the - plane data represent the shapes of the Japanese region are used. This data were acquired
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by extracting the boundary contour with raster scanning from the map data on the web page 1 of TAKESHI TAKEDA. Fig. 5 shows the description acquired by applying our proposed method; (a)– (d) show the shapes connected adjacent interpolation points with a line segment; (e)–(h) show the shapes reconstructed by using the extended FIF. The number of the interpolation points and the mean square error of fit are given in Table 2. It can be seen from Fig. 5 that it is possible to reconstruct the shape which is similar to the original with a few data. In particular, we can find that the portions surrounded by a dashed-dotted line are approximated well to their original.
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Fig. 4. Experimental data represent the shape of Japanese regions: (a) HOKKAIDO; (b) TOHOKU; (c) SHIKOKU; (d) KYUSYU. Fig. 5. Contour shape description using the extended FIF: (a)–(d) show the shape connecting the adjacent interpolation point with line segment; (e)–(h) show the shape using fractal interpolation functions.
Table 2. Performance of the description using the FIF. Figure signal Error 18.810279 40 41/1577 22.500474 (a)HOKKAIDO 41.310753 4.211640 30 39/1122 33.373322 (b)TOHOKU 37.584962 1.523365 (c)SHIKOKU 10 119/1177 0.785212 2.308577 17.203793 40 53/2056 14.929312 (d)KYUSYU 32.133105 4
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6. REFERENCES
5. CONCLUSIONS This paper has presented a representation method for contour shape using fractal interpolation functions. We found that our method had the ability to produce the shape approximated to the original with a few data points. This method has many possible applications such as; the shape representation on the automobile navigation system and animation image production using computer graphics. 1 http://hp.vector.co.jp/authors/VA003652/wtizuK/wtizuk.html
[1] B. B. Mandelbrot, The fractal geometry of nature, W. H. Freeman, San Francisco, 1982. [2] M. F. Barnsley, Fractals Everywhere, New York Academic, 1988. [3] X. Zhu, B. Cheng, and D. M. Titterington, “Fractal model of a one-dimensional discrete signal and its implementation,” IEE Proc. vision image and signal processing, vol. 141, no. 5, pp. 318–324, Oct. 1994. [4] M. A. Marvasti and W. C. Strahle, “Fractal geometry analysis of turbulent data,” Signal Processing, vol. 41, pp. 191–201, 1995. [5] A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Analysis. Machine Intell., vol. PAMI-6, no. 6, pp. 661–674, Nov. 1984. [6] M. F. Barnsley, “Fractal functions and interpolation,” Constructive Approximation, vol. 2, pp. 303–329, 1986. [7] P. Maragos, “Fractal aspects of speech signals: dimension and interpolation,” in Proc. ICASSP, vol. 1, pp. 417–420, 1991. [8] D. S. Mazel and M. H. Hayes, “Using iterated function systems to model discrete sequences,” IEEE Trans. Signal Processing, vol. 40, no. 7, pp. 1724–1734, July 1992.
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