Fractal Filtering Applied to SAR Images of Urban Areas Gerardo Di Martino, Antonio Iodice, Daniele Riccio, Giuseppe Ruello, Ivana Zinno Department of Biomedical, Electronic and Telecommunication Engineering University of Napoli "Federico II"  Naples, Italy gerardo.dimartino, iodice, daniele.riccio, ruello, ivana.zinno@unina.it Abstract                                               !    "        I. INTRODUCTION New generation high resolution sensors (e.g., TerraSARX, COSMOSkyMed) significantly changed the level of detail which can be appreciated in SAR images, in particular with reference to urban areas. A new set of previously masked features has emerged on these images: in fact, due to the metric resolution, many objects characterizing typical urban scenes present a strong radar return and appear on the images as strong intensity contributions. Hence, in order to retrieve valueadded information from these data, it is very important to extract and analyze these contributions, using, if possible, automatic unsupervised techniques. In this context, the filtering of urban SAR data has a key role. Natural surfaces can be effectively described using fractal models, which take into account in a simple way the irregularity and autoaffinity of this kind of surfaces [1], [2]. Conversely, manmade objects does not show a fractal behavior, at least at scales ranging from the sensor resolution one and that of the microwave electromagnetic field wavelenght. Thus, a geometric description in the context of Euclidean geometry is in order [3]. From a fractal point of view this means that they show a nonfractional dimension and, hence, at least in principle, they should be clearly separable from fractal objects in SAR images. In this paper the behavior of urban elements in fractal filtered SAR images is investigated. The filter of interest has been introduced by the authors for the estimation of the fractal dimension of natural surfaces from their amplitude SAR image [4]. The fractal dimension D is a physical parameter of the observed surface and brings key information about its composition and morphology, being a good candidate for image segmentation and classification purposes. The filter is based on a complete analytical model of the SAR imaging process and has been positively tested on simulated data relevant to canonical fractal surfaces [3]. The rationale of the model and the implementation of the filter is summarized in Section II. Significant applications on canonical fractal cases are also shown. In this paper the fractal filter is tested on simulated and actual SAR images relevant to urban areas, in order to assess its behavior in presence of manmade objects on the observed scene. This kind of structures determine the appearance on the image of very bright features, determining strong discontinuities in the radar signal. In fact, this is due to the presence of multiple scattering contributions, mainly generated by the dihedral configuration of soil and building walls [3]. The presence of discontinuities affects strongly the behavior of the fractal filter, whose rationale is based on the evaluation of the image spectrum. To analyze the behavior of the filter in this context, we apply it on simulated and actual SAR images . The simulated images are obtained by means of the SAR raw signal simulator SARAS [5], [3]. In the simulated scenario the presence of buildings is accounted for by means of bright points and lines playing the role of the multiple scattering contributions, which are typical signatures of the buildings on SAR images [3]. Thus, the analysis of the simulated case is the base for the interpretation of the behavior of the filter on a COSMOSkyMed SAR image relevant to L'Aquila, Italy. The description of the simulation setup, along with significant obtained results, is presented in Section III. Finally, in Section IV some concluding remarks are drawn and possible applications of the proposed filter in the context of classification of urban areas are outlined. II. FILTERING RATIONALE A. Theoretical Framework It is widely recognized that fractal models represent the best way to describe the irregularity of natural scenes [1], [2]. Among this kind of models, the regular stochastic fBm (fractional Brownian motion) process completely describes natural surfaces by means of two independent parameters: the Hurst coefficient, H (which is linked to the fractal dimension by the simple relation D=3H) and the standard deviation of surface increments at unitary distance, s [m 1H ]. The power density spectrum of the isotropic two dimensional fBm process exhibits a powerlaw behavior: α − = k S k S 0 ) ( (1) wherein S 0 and α are functions of the fractal parameters [1]. Besides the surface model, in order to retrieve the fractal dimension of a natural scene starting from its SAR image we need a direct model relating the surface to its amplitude image.