Enhanced BC Algorithm Incorporating a Novel Sampling Step and a Fractional Box Count

Abstract

The Box-Counting (BC) method is one of the most commonly used algorithms for fractal dimension calculation of binary images in the fields of Engineering, Science, Medical Science, Geology and so on due to its simplicity and reliability.

One of the issues related to fractal dimension is data sampling that involves a process where a certain size of box is taken from a given image and it has a direct effect on the precision of the fractal dimension estimation. The Geometric Step (GS) method, arithmetic step method, and divisor step method are the representative methods. The GS method is mainly used because of its efficiency. However, the GS method has some drawbacks in nature. If the image size is large, it provides insufficient data for regression analysis. It can be applied to the image of pixel size for 100 [%] pixel utilization. Application of the GS method to an image of may waste pixels in the calculation and degrade the estimation accuracy.

In this thesis, a novel sampling method is proposed in order to resolve the shortcomings of the GS method on the basis of the intuitive observation that an estimate may have a higher degree of precision if more pixels are utilized in each step and a sufficiently large number of fitting data are guaranteed. The proposed sampling method is an improved version of the conventional GS method, called the modified GS (MGS) method. The MGS method selects some additional step sizes with higher pixel utilization rate among the middle values between the integer powers of 2 to constitute the overall step set with the GS method.

Not all sampling methods including the MGS method can guarantee 100 [%] pixel utilization when the BC method is applied to images of an arbitrary size. This study suggests a novel fractional counting method to resolve the problem of pixel waste. The proposed counting method counts pixels of fractal within a discarded box (not of size) and adds its fractional count normalized by both the average pixel number of all boxes with size and step size to integer count.

The performance of the enhanced BC method incorporating the MGS method and fractional counting method is verified on a set of deterministic fractal images whose theoretical dimensions are well known and compared it with those of the existing BC methods. The experimental results show that the proposed method outperforms the conventional BC method and triangle BC method.