ON THE MATHEMATICAL FORMULATION OF THE SAR OIL-SPILL OBSERVATION PROBLEM Attilio Gambardella 1 , Giorgio Giacinto 2 , and Maurizio Migliaccio 1 1 Università degli Studi di Napoli “Parthenope”, Dipartimento per le Tecnologie Centro Direzionale, Isola C4, 80143 Napoli, Italy - e-mail: attilio.gambardella@uniparthenope.it 2 Università degli Studi di Cagliari, Dipartimento di Ingegneria Elettrica ed Elettronica Piazza d’Armi 19, 09123 Cagliari, Italy 1. INTRODUCTION Sea oil pollution causes marine ecological disasters that damage the quality and productivity of the marine environment, and produce severe financial consequences associated both to clear-up operations, and to the decrease of coastal tourism and the related economy [1], [2]. Remote sensing is an important part of oil spill countermeasures. The satellite microwave high resolution active sensor known as Synthetic Aperture Radar (SAR) is the key tool to accomplish such a task because of its wide field-of-view, full weather independence and day/night capabilities [3]. The physical phenomenon that makes possible the detection of oil spills from SAR measurements can be described as follows. Radar backscatter is due to the roughness of the ocean surface, where roughness is determined by the directional spectrum of the waves at the wavelengths of Bragg resonance [1]. As oil slicks damp such short waves, they appear as dark patches in SAR images that can be detected by image processing techniques [1]-[3]. Unfortunately, several natural and atmospheric phenomena produce dark areas in SAR images that are similar to oil spills. These dark areas are usually referred to as look-alikes, whose presence makes the detection of oil spills a challenging task [1]. Phenomena giving rise to look- alikes may include biogenic films, areas of low wind (<3ms-1), areas of wind-shadow near the coasts, rain cells, zones of upwelling, internal waves, and oceanic or atmospheric fronts [3]. Nevertheless, the contrast between a spill and the surrounding water depends both on the amount and type of oil, and on various environmental factors such as the wind speed, the wave height, and the sea current [3]. Usually, oil spill detection is framed into three fundamental phases: detection of dark areas over SAR images, feature extraction and dark area classification [3]. The detection part is meant to locate all dark areas which are associated to a set of features. Then, by means of the contrast, geometrical, textural, etc. features, a classifier assign to the dark area a level of confidence exploiting a predictive probability. A large number of automatic and semi-automatic oil spill detection procedures based on SAR images have been presented in literature [4]-[10]. The analysis of the state-of-the-art clearly emphasizes two aspects: i) the feature space to perform classification phase is empirically defined, i.e. there is no agreement among researchers on the features more apt to discriminate between oil-spills and look-alikes, and ii) classification of dark patches is always formulated in terms of a two-class classification problems, where examples of the two classes, i.e. oil-spills and look-alikes, have to be provided to train the classification model. Although this approach often allows attaining acceptable operational results, there is still room for improving both the comprehension of the physical phenomenon, and the performance of classification techniques. 2. THE APPROACH In this study, a novel approach for the definition of a mathematical objective framework of the oil spill observation problem on the paradigm of one-class classification [11] is proposed. Such a fundamental mathematical problem has looked for the objective definition of the feature space and of the nature of the classification problem. Basically, a classifier is trained using only examples of oil-spills, instead of using oil-spills and look-alikes as in two-class approaches [12]-[14]. We claim that this approach is suited to the oil-spill classification problem as one-class classification is suited for those problems where reliable examples can be provided for one of the classes. In the case of the oil-spill detection