We use segmentations to match images by shape. To address the unreliability of segmentations, we give a closed form approximation to an average over all segmentations. Our technique has many extensions, yielding new algorithms for tracking, object detection, segmentation, and edge preserving smoothing. For segmentation, instead of a maximum a posteriori approach, we compute the “central” segmentation minimizing the average distance to all segmentations of an image. Our methods for segmentation and object detection perform competitively, and we also show promising results in tracking and edge-preserving smoothing.