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We present computer vision algorithms that recognize and locate partially occluded objects. The scene may contain unknown objects that may touch or overlap giving rise to partial occlusion. The algorithms revolve around a generate-test paradigm. The paradigm iteratively generates and tests hypotheses for compatibility with the scene until it identifies all the scene objects. Polygon representations of the object's boundary guide the hypothesis generation scheme. Choosing the polygon representation turns out to have powerful consequences in all phases of hypothesis generation and verification. Special vertices of the polygon called ``corners'' help detect and locate the model in the scene. Polygon moment calculations lead to estimates of the dissimilarity between scene and model corners, and determine the model corner location in the scene. An association graph represents the matches and compatibility constraints. Extraction of the largest set of mutually compatible matches from the association graph forms a model hypothesis. Using a coordinate transform that maps the model onto the scene, the hypothesis gives the proposed model's location and orientation. Hypothesis verification requires checking for region consistency. The union of two polygons and other polygon operations combine to measure the consistency of the hypothesis with the scene. Experimental results give examples of all phases of recognizing and locating the objects.