We use segmentations to match images by shape. The new matching technique does not require point-to-point edge correspondence and is robust to small shape variations and spatial shifts. To address the unreliability of segmentations computed bottom-up, we give a closed form approximation to an average over all segmentations. Our method has many extension, 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. For smoothing, instead of smoothing images based on local structures, we smooth based on the global optimal image structures. Our methods for segmentation, smoothing, and object detection perform competitively, and we also show promising results in shape-based tracking.