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The non-geometric and irregular objects are considered as complex patterns. The geometric complexity is measured as space lling capacity by a factor known as a fractal dimension. Dierent techniques are proposed to nd this complexity measure according to the properties of the pattern. This paper is aimed to introduce a method for counting the dimension of the lled Julia fractal set generated by the Escape Time Algorithm using the method of spreading the points inside the proposed window. The resulted dimension is called Escape Time dimension. A new method to compute a correlation dimension of the Filled Julia fractal set is also proposed based on the Grassberger-Procaccia algorithm by computing the correlation function. A log-log graph of the correlation function versus the distances between every pair of points in the lled Julia fractal set is an approximation of the correlation dimension. Finally, a comparison between these two fractal dimensions of the led Julia fractal set which is generated by the Escape Time Algorithm is presented to show the efficiency of the proposed method.

Fractal dimension is a mathematical concept used to measure the geometrical complexity of fractal set. It is defined for fractal geometric images, and considered as global features for them. There are many methods to estimate the fractal dimension of an object. The box counting dimension is an easier and a widely used one, while the escape time dimension is another method used to estimate the dimension of fractals generated using escape time algorithm. These methods are used to calculate the dimension for monochrome images (2Dimages). The necessity to generalize these concepts to be applicable for real world application (e.g. gray scale image, or colored images) has motivating us to introducing the concepts of fuzzy sets. Fuzzy fractal dimension is proposed as the fractal feature for n-dimensional image. In this paper, a new approach to determine fractal dimension is proposed and a new local fuzzy fractal dimension based on this approach is proposed also. It will help to extend this feature to be used in many real world applications that cannot be served based on traditional fractal dimension. By this new approach, the FD is estimated with a reduced number of computational processes. This will helps to improve the complexity of the escape time algorithm that is considered as an NP-Hard problem, and with high precision results.

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