A Power Differentiation Method of Fractal Dimension Estimation for 2-D Signals

other fractal-based features as descriptors of the texture of images [2–6]. Applications of the fractal theory in image Fractal dimension has been used for texture analysis as it is highly correlated with the human perception of surface analysis also include image segmentation [2, 6–8], shape roughness. Several methods have been proposed for the estimadescription [9], object characterization [10], and surface tion of the fractal dimension of an image. One of the most reconstruction [11], while there are several nice results on popular is via its power spectrum density, provided that it is the fractal dimension estimator using wavelets [12]. The modeled as a fractional Brownian function. In this paper, a fractal model has been used in medical imaging for analysis new method, called the power differentiation method (PDM), of bone X-rays [13, 14], classification of ultrasonic liver for estimating the fractal dimension of a two-variable signal images [15], edge enhancement [15, 16], and mammogram from its power spectrum density is presented. The method is analysis [17]. first applied to noise-free data of known fractal dimension. It In this paper, a new method, called the power differentiais also tested with noise-corrupted and quantized data. Particution method (PDM), for estimating the fractal dimension larly, in the case of noise-corrupted data, the modified power differentiation method (MPDM) is developed, resulting in more of a two-variable signal from its power spectrum density accurate estimation of the fractal dimension. The results obis presented. Along with the PDM a robust fitting techtained by the PDM and the MPDM are compared directly to nique for obtaining the fractal dimension from the resulting those obtained using four other well-known methods of fractal log–log plot is described. The method is first applied to dimension. Finally, preliminary results for the classification of noise-free data of known fractal dimension. Then it is ultrasonic liver images, obtained by applying the new method, tested with noise-corrupted data and quantized data (grayare presented.  1998 Academic Press level images). Particularly, in the case of noise-corrupted data, a modification of the method, called the modified power differentiation method (MPDM), is proposed, re