Multifractal analysis and feature extraction in satellite imagery

In the last 20 years, the use of fractal dimension indices has found wide application in various disciplines, including remote sensing. Recent studies have improved our understanding of both scaling and statistical properties of natural images and it is now broadly accepted that the fractal dimension, used as a sole texture descriptor, does not provide complete textural information. Natural images are characterized by a set of fractal dimensions (i.e. multifractal indices) rather than by a single fractal dimension. This paper aims to give a brief introduction to the theory of fractals and multifractals and to confirm, by means of examples, the inadequacy of using a single fractal dimension in image analysis. In addition, it will show how multifractal indices can provide better textural description. The main purpose of this paper is to prove that multifractal descriptors can substitute or complement classical textural descriptors by presenting a method to calculate the multifractal spectrum of pixels inside images and to extract multifractal indices that can be used as descriptors of textural features. In particular, the paper will propose a multifractal index named Spectrum Range and will introduce some of its properties. To assess the efficacy of the method and of the derived multifractal indices, three examples will be presented. The first two images are artificial images generated by a pseudo-random generator and partly by a deterministic function. The third image is an AVHRR image taken above Scotland.