Despite many advances, frameworks for geometric morphometry still rely on point-to-point correspondences between shapes, either explicitly in form of homologous landmarks or implicitly in terms of diffeomorphisms of the ambient space. Point-to-point correspondences, however, have fundamental limitations that prohibit the analysis of shape collections with incomplete or topologically varying objects. This is a major problem for the analysis of empirically given sets of shapes, since these often contain topological variations (“real” ones as well as those caused by noise or reconstruction errors) or are incomplete (e.g. due to spatial limitations in tomographic reconstructions or due to destruction and decay). The goal of this project is to extend the scope of shape analysis methodology in order to overcome these limitations. To this end, we will generalize approaches defined in shape spaces based on explicit representations by adapting and refining the concept of soft correspondences.
The methodological developments will be driven by applications from archaeology and biology. In archaeology, we will consider ancient sundials, most of which are only partially preserved. A particular focus in this application will be the inference of shape differences depending on the geographic location, allowing for identification of correlating design and construction principles. 
In biology one has to deal with topologically varying shapes, e.g., when studying anatomical changes, as well as with incomplete shapes, e.g. when studying fossil records.
