Fractal contours: Order, chaos, and art

Abstract

Over the recent decades, a variety of indices, such as the fractal dimension, Hurst exponent, or Betti numbers, have been used to characterize structural or topological properties of art via a singular parameter, which could then help to classify artworks. A single fractal dimension, in particular, has been commonly interpreted as characteristic of the entire image, such as an abstract painting, whether binary, gray-scale, or in color, and whether self-similar or not. There is now ample evidence, however, that fractal exponents obtained using the standard box-counting are strongly dependent on the details of the method adopted, and on fitting straight lines to the entire scaling plots, which are typically nonlinear. Here, we propose a more discriminating approach with the aim of obtaining robust scaling plots and extracting relevant information encoded in them without any fitting routines. To this goal, we carefully average over all possible grid locations at each scale, rendering scaling plots independent of any particular choice of grids and, crucially, of the orientation of images. We then calculate the derivatives of the scaling plots, so that an image is described by a continuous function, its fractal contour, rather than a single scaling exponent valid over a limited range of scales. We test this method on synthetic examples, ordered and random, then on images of algorithmically defined fractals, and finally, examine selected abstract paintings and prints by acknowledged masters of modern art.