Performance Evolution of a Fractal Dimension Estimated by an Escape Time Algorithm

A fractal dimension is a non-integer number that used as an index of the complexity and characteristics of an image. Using fractal dimensions, we can distinguish the properties of images and compare their characteristics, which is of great importance in many applications. Nowadays, there are two algorithms that are initialized to construction a fractal attractor set. The first is the iterated function system (IFS), and the second is the escape time algorithm (ETA). In this paper, we present a modified method for calculating two kinds of fractal dimensions, i.e., the box dimension and the correlation dimension of a fractal attractor set, created by ETA, such as for a filled Julia set, and those created by iterated IFS, such as for a Sierpinski gasket. Since IFS can only use a certain iteration to create fractal attractors, this limitation has made the ETA the most general and efficient algorithm for creating fractal attractors when the iteration functions are complex. It is one of the earliest colouring algorithms, and, in many applications, it is considered to be the only available option. Therefore, this motivates us to introduced a new algorithm to convert the fractal attractor created by the IFS into a fractal attractor set created by the ETA. This conversion will be accomplished by finding the shift dynamical system of the totally disconnected or non-overlapping IFS. In addition, we modify the correlation and box dimensions to calculate the dimensions of fractal attractor sets created by the ETA. Moreover, we compare the proposed algorithms for fractal dimensions with the previously known algorithms from the literature in terms of the points and computational time needed.