Estimation of Fractal Dimension in Different Color Model

Introduction

Mandelbrot in 1982 has invented fractal theory that provides both mathematical and descriptive model for several apparently complex structures present in nature (B. B. Mandelbrot, 1982). Irregular shapes for example; clouds, mountains and coastlines are not simply defined by traditional geometry (Euclidean geometry). They often possess a significant invariance with the changes of magnification. This nature of self-similarity is a crucial eminence of fractal in nature. Usually it is used to estimate fractal dimension (FD). It provides several mathematical models for complex real-world objects. Fractal analysis has excessive impact in digital image analyses. It is extended by many more concepts applicable to a broader class of fractals. Fractal geometry is one of the best areas for doing research. As this area is vast, many researchers have worked on different problems and given their contribution for estimating fractal dimension. After the term fractal geometry has been introduced, several researchers shared their knowledge on this field and because of their effortless job various methodologies have been introduced to calculate fractal dimension. In 1986 Gangepain and Roques Carmes stated a method called reticular cell counting technique (Gangepain, & Roques-Carmes, 1986) and this method is enhanced by Voss in 1986 (Ford & Roberts, 1998) discussed about the method of probability. Pentland suggested the Fourier power spectrum of image intensity surface for fractal dimension estimation (Pentland, 1984). Keller et al. also proposed Reticular cell counting method and this one is the improved version of the previous reticular cell counting (1989). In 1986, Sarkar and Chaudhuri suggested a well-organized methodology to estimate the fractal dimension known as the Differential Box Counting (DBC) (Sarker & Chaudhuri, 1994). Instead of using the process of cell counting method to count the number of boxes, they used minimum and maximum grays levels of the corresponding grid in the image for calculation known as Differential Box Count Method (DBC). Jian, Qia and Caixin jointly enounced the improved Differential Box Counting or improved DBC (2009). In the improved DBC the drawbacks of DBC, Box height calculation, Box number calculation, partition of image intensity surface was analyzed and improved. The mean and standard deviation of intensity of a block was used for calculation of box height. The minimum gray level was used in the box height calculation rather than gray level 0. The image intensity partition occurs resulting in zero distance of neighboring blocks.