An Improved Fuzzy Fractal Dimension for Texture Analysis Nadia M.G. Al-Saidi University of Technology Branch of Applied Mathematics Applied Sciences Department Baghdad - Iraq nadiamg08@gmail.com Mohamad Rushdan Md. Said University Putra Malaysia Institute for Mathematical Research 43400, Serdang, Darul Ehsan Kualalumpor - Malaysia mrushdan@fsas.upm.edu.my Wael J. Abdulaal University of Technology Branch of Applied Mathematics Applied Sciences Department Baghdad - Iraq waeljbbar@gmail.com Abstract: In the real life, a lot of phenomena cannot be described by the traditional geometry; fractals are one of these objects. It serves to give a suitable description for these objects. Texture analysis plays an important role in image processing. Fractal dimension is utilized in texture segmentation and classification and prove to be an interesting parameter to characterize image roughness and extract image feature. In some complex and irregular scenes, it becomes not effective for feature extracting and classification. Therefore, a more general approach known as fuzzy fractal dimension can be used to model such types of scenes effectively. A new fuzzy fractal dimension method is proposed in this paper, it is verified by the experiment on a set of nature texture images to show its efficiency and accuracy, a satisfactory result is found. It also offers promising performance when it is tested among some types of noises to show a good robustness to them. Key–Words: Fractal dimension, Fuzzy fractal dimension, Texture, Box counting method 1 Introduction This work is motivated due to the great demands for new techniques to handle the complex and irregular scenes. Many areas of image description, recognition, and segmentation are based on fractal dimension that play a key role in these problem. Texture analysis is an important issue in many applications. It composes of sub-patterns that have some characteristic such as; slope, color, size, brightness, etc., which give rise to recognize the roughness, regularity, randomness, smoothness etc. It is believed to be a prosperous source of visual information. The Hausdorff- Besicovitch measure in (1) is one of the earliest definitions of fractal dimension which is considered as a basic for the definition of the fractal set [1]. H s δ (E) lim δ→0 [inf Σ | U i | s ] (1) Since this type of dimension is computationally hard, a new approach known as Box counting dimension to calculate the fractal dimension is emerged as the best estimator for self-similar images. It takes its popular- ity due to the ease of its numerical computation. In 1982, Mandelbrot [2, 3] was the first who describes an approach to find the fractal dimension while he tried to find the length of the coastlines. After that, many approaches had been proposed to estimate the FD. Since most real textures are in fact semi fractals; it cannot be precisely characterized by fractal dimen- sion. Many researchers have argued to extend Man- delbrot’s method to 2D, 3D and nD based on dif- ferent approaches; such as, fractal Brouwnian motion (fBm), wavelate transform, contourlet transform. S. Peleg [4] extended Mandelbrot’s method the 2D im- ages where the image can be represented as a hilly terrain surface; the height from the ground represents the gray value of the image. Therefore, all points which have a distance 2ε from both sides of the sur- face formed a blanket divided by 2ε. For a different ε the area of the blanket is estimated in (2) and the area of the fractal surface behaves according to the follow- ing expression: A(ε)= Fε 2−D (2) where D is the fractal dimension. The estimation of fractal dimension from power spectral density of fBm is introduced by Pentland [5], who considerd the im- age intensity as fBm. J.Gangepain and R.Carmes [6], proposed a method named the reticular cell counting approach using the Box counting method. In their method, they partitioned the 3D space into boxes of size L × L × L ′ for a given scale L and by calculat- ing the number of the sub cubes N L in (3) that at least one sample from the intensity surface, D represent the fractal dimension. N L = 1 r D = [ L max L ] D (3) Advances in Mathematics and Statistical Sciences ISBN: 978-1-61804-275-0 475