Estimation of fractal dimension of images using a ®xed mass approach P. Asvestas, G.K. Matsopoulos * , K.S. Nikita Department of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou Str., 15773 Zografos, Athens, Greece Received 12 May 1998; received in revised form 4 December 1998 Abstract A method from the ®eld of chaotic dynamics is applied for the estimation of fractal dimension of images. The method is compared with other well-known algorithms on a set of computer generated images of known fractal di- mension. The results con®rm the superiority of the method in terms of accuracy, dynamic range and computational time. Ó 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Fractal dimension; Box-counting method; Correlation algorithm; kth nearest neighbour method 1. Introduction Fractal geometry has received much attention as a useful tool for image analysis. The intensity surface of an image can be considered as a fractal object whose properties are quanti®ed numerically by the use of the fractal dimension. For an image, the fractal dimension is a non-integer number be- tween 2 and 3 and it is a measure of the roughness of its intensity surface. Experiments have demon- strated that the fractal dimension is highly corre- lated with the human perception of image texture; the rougher the texture appears the larger is the fractal dimension. Several methods have been proposed for the estimation of the fractal dimension of images. The most widely used method is box-counting, which is based on the covering of the intensity surface with cubes of ®xed size. The fractal dimension is ob- tained by the scaling of the number of non-empty cubes with the size of the cubes. However, this method underestimates the true fractal dimension for relatively high values (e.g. above 2.6), mainly due to the discretisation of the image domain and the quantisation of the grey levels. Keller et al. (1989) proposed a modi®cation of the box-count- ing method based on linear interpolation slightly improving its performance. However, this modi- ®ed method still underestimates the true fractal dimension for very high values (e.g. above 2.8), while it increases the computational time. The correlation algorithm (Theiler, 1990) provides a very elegant formulation for estimating fractal dimension. According to this algorithm the di- mension is obtained by the scaling of the mass of spheres (or boxes) with the size of the spheres. In this paper, an alternative method, from the ®eld of chaotic dynamics, is proposed for the Pattern Recognition Letters 20 (1999) 347±354 * Corresponding author. Tel.: 30 1 772 2285; fax: 30 1 772 3557; e-mail: gmatso@naxos.esd.ece.ntua.gr 0167-8655/99/$ ± see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 6 5 5 ( 9 9 ) 0 0 0 0 4 - 5