Peter Gerl et al. / HarFA- Harmonic and Fractal Image Analysis (2004), pp. 70- 73 70 The Use of Fractal Dimension in Arts Analysis

The fractal dimension is a measure for the coarseness of objects and textures. This property can be assigned to graylevel images to get a measure for the coarseness of a texture in an image, [2]. The theme of this article is the application of fractal dimension to arts analysis. Considering 7 artists, everyone represented by about 18 images, we searched for qualified description parameters for the painters characteristics. We evaluated among other feature values 3 different types of fractal dimension, namely capacity (or box counting) dimension, information dimension and correlation dimension, [3]. With a special feature selection algorithm the best features for the classification of the images with respect to the painter were evaluated, [5]. The fractal dimension turned out to be under the best features. In the following the computation of the fractal dimension, our classification task and a few results are discussed. 1. Fractal Dimension We want to define the notion of a dimension of an image. From the mathematical point of view the best way would be to take the Hausdorff dimension [1], since this exists for all (bounded) subsets of a metric space, and an image is a bounded subset of Euclidean 3−dimensional space (at least a graylevel image). But the Hausdorff dimension has a complicated definition and it is by no means easy to calculate it. Therefore simpler versions for the notion of a dimension were suggested. A very useful notion goes back to A. Renyi [4], the generalized q−dimension. We will start with one version of this q{dimension and will specialise then to its calculation for images. 1.1. The generalized q−dimension of A. Renyi Let A be a bounded, measurable subset of the 3-dimensional space ℜ. For 3 0>ε we consider a lattice of cubes of side length ε in ℜ call these cubes C1; C2;.... Then let