Fast algorithm for computing the fractal dimension of binary images

D = lim log(.iV(A,c)) (1) E-o log1 where .Af(A, e) denotes the smallest number of closed balls of radius > 0 needed to cover the set A. The aim of this paper is to introduce a new fast algorithm for computing the fractal dimension of binary images according to (1). Consider a binary image that is represented as a matrix of binary pixels stored in a rectangular array I(x, y). Let this image denote a subset of the set of all pixels in the plane. In order to compute the fractal dimension of a digitized image, it is necessary to cover the set with the "balls" of radius ( as given by the definition of the fractal dimension (1). When processing digitized binary images, it is convenient to use squares of side length instead of balls of radius c This simplification allows a considerable speedup of the set's covering. As an approximation to the covering of the set, we choose the "Minkowski sausages" 6 (see fig. 1). The fractal dimension can be computed by measuring the area covered by the Minkowski sausages with increasing radius . The results of each step and the corresponding numbers of pixels are collected in a Richardson-vlandelbrot-plot giving the fractal dimension of the original image 2,3,6 The main problem is the efficient computation of the Minkowski sausage with a certain radius. A naive solution is obtained by simple shift operations and logical OR of the original and the shifted image. This has to be done for all shift offsets contained in the square with given radius leading to a number of (2€ + 1)2 — 1 shiftand OR-operations per f-value. An improvement can be obtained by an inductive approach for the determination of the Minkowski sausage of radius 1 when the Minkowski sausage of radius is given. 1 Email: creutzhurg©fh-brandenburg.de; WWW: http://zeus.fli-bi'andenburg.de/creutzburg/ Telephone: +49-3381-355 442; Fax: +49-3381-201 608