Lacunarity of Rough Surfaces From the Wavelet Analysis of Scattering Data

Many natural and man-made rough surfaces exhibit multiscale features and may be advantageously described by fractal models. The surface roughness is then characterized by the fractal dimension while the distribution of gap or scale sizes (or gapiness) is measured by the lacunarity. Fractal dimension and lacunarity are two independent descriptors, that is, two surfaces having the same fractal dimension may look different and differ according to the distribution of gap or scale sizes. The remote analysis of the scaling properties of such multiscale surfaces may be achieved from interrogation by an electromagnetic pulse. Numerous approaches have been developed to describe the relationship between the fractal dimension of rough surfaces and the properties of the scattered waves. But, despite its potential importance in engineering, the remote extraction of the lacunarity from scattering data has not received the same attention. This paper is devoted to the extraction of the lacunarity of rough (fractal) surface from the wavelet analysis of the near-field impulse response using a heuristic method. It is shown that the skeleton of the wavelet-transform modulus-maxima of the scattered signal allows the straightforward estimation of the surface lacunarity for the examples studied here. Finally the wavelet analyze is applied for studying rough surfaces of different lacunarities and various fractal dimensions.