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The integral image is typically used for fast integrating a function over a rectangular region in an image. We propose a method that extends the integral image to do fast integration over the interior of any polygon that is not necessarily rectilinear. The integration time of the method is fast, independent of the image resolution, and only linear to the polygon's number of vertices. We apply the method to Viola and Jones' object detection framework, in which we propose to improve classical Haar-like features with polygonal Haar-like features. We show that the extended feature set improves object detection's performance. The experiments are conducted in three domains: frontal face detection, fixed-pose hand detection, and rock detection for Mars' surface terrain assessment.