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Although numerous pre-filtering and area sampling antialiasing techniques have been devised, few are capable of rendering complex, self-intersecting polygons. This paper presents a new antialiasing method capable of handling such polygons, according to any fill rule. The method is based on area sampling and is hence free from the visual aliasing artifacts of super sampling associated with near horizontal edges. It works by decomposing the coverage area of a polygon within each pixel into separate constituent regions, which are simple regions whose areas can be easily computed. The decomposition step does not attempt to generate non-overlapping regions, unlike that employed by existing methods, and hence operates very efficiently. This, together with other novel approximation techniques in the recombination stage allow pixel coverage areas to estimated with an efficiency approaching that of super sampling.