Shape analysis and similarity assessment using functions

Mathematics provides formal settings which are really interesting to approach shape analysis and un- derstanding: shape descriptors, and the invariants they carry, are indeed the tools for abstracting shape properties that can be then linked to semantic interpretation. Characterizing a shape means constructing a computational description of the most representative features of the shape, usually a few basic types, along with their relationships (structural decomposition) [1]. Assessing and quantifying the similarity between shapes is necessary, e.g, to explore large dataset of shapes, and tune the analysis framework to the users needs [2]. In this contribution, we present our recent research activities on shape reasoning, with an emphasis on a shape analysis and comparison pipeline based on a collection of functions defined on the objects to be studied. These functions are supposed to represent meaningful shape properties and drive the extraction of a geometric/topological shape descriptor. The approach we discuss works in two steps: first, one or more real functions defined on the same shape are considered. Since there is not a method to select the functions that better describe a 3D object, we discuss how to automatically group functions defined on the same shape and select a subset of functions that are as much as possible independent each other. Given a distance between two real, continuous functions defined on the shape, we adopt a clustering technique and a voting strategy to deduce the functions that better characterize a set of shapes. Second, we will present recent advancements on the generalization of these approaches to the case of multivariate functions, allowing us to capture properties of shapes that are intrinsically multidimensional, such as in the case of textured 3D models. Finally, we will sketch how these tools may be applied to the analysis of protein docking, by modelling electrostatic potential and hydrofobicity sing multivariate functions.