Shape matching and moduli spaces

Shape matching represents a challenging problem in both information engineering and computer science, exhibiting not only a wide spectrum of multimedia applications, but also a deep relation with conformal geometry. After reviewing the theoretical foundations and the practical issues involved in this fashinating subject, we focus on two state-of-the-art approaches relying respectively on local features (landmark points) and on global properties (conformal parameterizations). Finally, we introduce the Teichmüller space Tg,n of n-pointed Riemann surfaces of genus g into the realm of multimedia, showing that its beautiful geometry provides a natural unified framework for three-dimensional shape matching. Admittedly, the purpose of the present contribution is not to prove any new mathematical result, but rather to suggest an intriguing connection between the area of three-dimensional shape matching and Teichmüller spaces of pointed Riemann surfaces. In order to address such an interdisciplinary topic in a hopefully transparent way, first of all we offer to the non-specialist reader a gentle introduction as well as an updated list of references to the technical literature on shape matching. As incisively summarized in the survey paper [1], shape matching deals with transforming a shape and measuring the resemblance with another one via some similarity measure, thus providing an essential ingredient in shape retrieval and registration. By the way, there seems to be no universal and standard definition of what a shape is (see [1] and [2]): [1] goes even back to Plato’s Meno, where Socrates claims in ”terms employed in geometrical problems” that ”figure is limit of solid” ([3]), while [2] relies on Kendall’s definition of shape as ”all the geometrical information that remains when location, scale, and rotational effects (Euclidean transformations) are filtered out from an object” ([4]). The authors of [2] also emphasize the progressive shift of interest in the last decade from 2D (where shapes are just silhouettes) to 3D (where shapes are embedded surfaces), motivated by spectacular advances in the field of three-dimensional computer graphics. Unluckily, as remarked in [5], most 2D methods do not generalize directly to 3D model matching, thus inducing a flurry of recent research to focus on the specific problem of 3D shape retrieval. It is worth stressing that, unlike text documents, 3D models do not allow the simplest form of searching by keyword and it is now a rather Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully acknowledged.