We propose a fully automatic method for tooth detection and classification in CT or cone-beam CT image data. First we compute an accurate segmentation of the maxilla bone. Based on this segmentation, our method computes a complete and optimal separation of the row of teeth into 16 subregions and classifies the resulting regions as existing or missing teeth. This serves as a prerequisite for further individual tooth segmentation. We show the robustness of our approach by providing extensive validation on 43 clinical head CT scans.