On calculation of fractal dimension of images

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Abstract

Fractal geometry has gradually established its importance in the study of image characteristics. There are many techniques to estimate the dimensions of fractal surfaces. A famous technique to calculate fractal dimension is the grid dimension method popularly known as box-counting method. In this paper, we have found out a lower bound of the box size and provided the reason for having it. The study indicates the need for limiting the box sizes within certain bounds.

Introduction

Fractal geometry popularized by Mandelbrot (1982) has gained much support in the field of image analysis. Pentland, 1984, Pentland, 1986 provided the first theory in this respect by stating that fractal dimension correlates quite well with human perception of smoothness versus roughness of surfaces, with fractal dimension of 2 corresponding to smooth surfaces and fractal dimension of 3 corresponding to a maximum rough “salt-and-pepper” surface. Of course, he assumed the surface to be modeled by fractional Brownian function. However, it was followed by many other theories applicable to a wider class of fractals. Gangepain and Roques-Carmes (1986) described the popular reticular cell counting method and Keller et al., 1987, Keller et al., 1989, Chen et al., 1990, Chen et al., 1993 gave even more interesting theories. Many studies are made to find the upper and lower bounds of the box size and in this regard (Chen et al., 1993) have given a theoretical justification for a restriction on the smallest box size inspired by the work of Pickover and Khorasani (1986). As we studied the cell counting method, we also found out the necessity of having a lower bound of the box size.

This paper is organized as follows. Section 2 describes various methods for calculating the fractal dimension. Section 3 discusses the reason for having upper and lower bounds of the box size. Section 4 gives a formula for finding the lower bound of the box size. Section 5 shows the experimental results. Concluding remarks are made in Section 6.

Section snippets

Methods for calculating fractal dimension

From the properties of self-similarity, fractal dimension D of a set A is defined asD=log(N)/log(1/r),where N is the total number of distinct copies similar to A and A is scaled down by a ratio of 1/r.

Eq. (1) can be directly applied to geometrical fractals such as Sierpinski's Gasket or Cantor Dust and a theoretical estimation of fractal dimension is possible in such cases. However, for image surfaces, the value of N has to be computed using methods like box-counting and the fractal dimension

Bound of the box size

As we know box-counting is a very simple process. However, many researches have been performed to improve the procedures with respect to the calculation of the roughness accurately. Sometimes problems occur while estimating the roughness effectively due to improper limits and box size (Pickover and Khorasani, 1986, Chen et al., 1993, Feng et al., 1996). Of course, authors have taken various corrective measures. Many authors have assumed certain bounds according to the procedure adopted by them.

Proposed lower bound of the box size

According to Gangepain and Roques-Carmes (1986) L=⌊L×G/M⌋ and hence M/L number of vertical boxes on any grid as indicated in Fig. 1 will have a length of L, which can be a multiple of gray level units in the space. As the image intensity surface is quantized over the space, L determines how many boxes contain at least one gray level intensity surface. Studying the cell counting method, we discussed that the imposition of a lower bound on the box size is necessary in order to calculate the

Experimental results

In order to find the fractal dimensions of images generated by us (Bisoi and Mishra, 1999) which are shown in Fig. 2, we applied the various methods discussed in Section 2. The methods have been applied with and without the lower bound (as given by Eq. (9)) in order to estimate the fractal dimension; the upper bound is given by Eq. (10) being applicable in all cases. The images generated by us have used the IFS technique modified by incorporation of inverse replicas of images. The resulting

Concluding remarks

Due to growing importance of box-counting methods on calculation of fractal features from an image, we established a lower bound of the box size to ensure accurate results. Our study corroborates the following intuitive observations.

With too low a box size, the maximum number of boxes above a grid would be more than the number of available intensity levels. The resulting unaccounted boxes would lead to an underestimation of fractal dimension. On the other hand, for too high a box size, the

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