JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION Vol. 9, No. 4, December, pp. 392–400, 1998 ARTICLE NO. VC980394 A Power Differentiation Method of Fractal Dimension Estimation for 2-D Signals P. Asvestas, G. K. Matsopoulos, and K. S. Nikita Department of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece Received February 16, 1998; accepted August 7, 1998 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 estima- description [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 differentia- is also tested with noise-corrupted and quantized data. Particu- tion 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 ob- is presented. Along with the PDM a robust fitting tech- tained 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 (gray- are 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- 1. INTRODUCTION sulting in more accurate estimation of the fractal dimen- sion. Results obtained by the PDM and the MPDM are Fractal geometry was introduced and developed by Man- compared directly to results obtained using four other well- delbrot [1] as a means for describing and analyzing the known methods of fractal dimension estimation. Finally, properties of objects with irregular and complex structure preliminary results for the classification of ultrasonic liver (fractals), such as coastlines and surfaces of mountains. images, obtained by applying the new method, are pre- The characteristic property of a fractal is that it is self- sented. similar for every scale of analysis. This fact implies that any part of a fractal object is a scaled-down copy of the 2. FRACTAL DIMENSION: DEFINITION AND original. However, for natural objects the self-similarity is ESTIMATION METHODS observed only for a limited range of scales and it appears in a statistical sense. In this case, a part of the object, There are several definitions of the fractal dimension, magnified to the size of the original, exhibits statistical FD, of a set. The most popular of them is the box-counting properties similar to those of the original. The numerical dimension, which is an upper limit of the Hausdorff– quantification of self-similarity is obtained by the fractal di- Besicovich dimension [1]. The box-counting dimension of mension. a set S  R n is defined as The fractal dimension is a measure of the roughness of the surface represented by the fractal set: the larger the fractal dimension is, the rougher the surface appears. This FD = lim r0 log N(r) log(1/ r) , (1) fact has led to the utilization of the fractal dimension and 392 1047-3203/98 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.