Imaging of Fractal Profiles

Abstract—In this paper, a model for radar images of fractal (topologically 1-D) profiles is introduced. A twofold approach is followed: on one hand, we analytically solve the problem whenever small-slope profiles are in order; on the other hand, we present a partly analytical and partly numerical setup to cope with the general-slope case. By means of the analytical approach, we eval-uate in closed form both the structure function and the power density spectrum of the radar signal. An appropriately smoothed (physical) fractional Brownian model (fBm) process is employed; its introduction is justified by the finite sensor resolution. A fractal scattering model is employed. It is shown that for a fractal profile modeled as an fBm stochastic process, the backscattered signal turns out to be strictly related to the associated fractional Gaussian noise process if a small-slope regime for the observed profile can be assumed. In the analytical–numerical framework, a profile with prescribed fractal parameters is first synthesized; then, fractal scattering methods (applicable to wider slope regimes with respect to the previous case) are employed to compute the signal backscat-tered toward the sensor. Finally, the power density spectrum of the acquired radar image is estimated. The obtained spectra are favorably compared with the theoretical results, and a parametric study is performed to assess the overall method behavior. Index Terms—Electromagnetic scattering, fractals, synthetic aperture radar, radar, radar imaging. I