Fractals under the microscope or reaching beyond the dimensional formalism of fractals with the wavelet transform

Recently, there has been a tendency to see natural phenomena as resulting from the action of dynamical processes. These processes (and the objects to which they give rise) are surprisingly often well characterised using some sort of hierarchical approach. It is also one of the key features of fractal geometry that the action of dynamical systems can result in intrinsically hierarchical structures. In this paper we shortly outline the rapid progress which was made in analysis of fractals with the wavelet transform (often, but not exclusively, making use of the derivatives of the Gaussian kernel). We demonstrate that the natural ability of the wavelet transform to analyse objects using both position and scale localised filters proves ideal in the context of hierarchical formalism of fractal geometry. In addition to this, the inherent robustness of the transform provides reliable access to multi-scale representations.