Comparing shapes through multi-scale approximations of the matching distance

• This paper deals with the concepts of persistence diagram and matching distance.
• We present multi-scale approaches to approximate the matching distance.
• Experiments show the capability of the proposed methodologies for shape retrieval.

Two of the main ingredients of topological persistence for shape comparison are persistence diagrams and the matching distance. Persistence diagrams are signatures capturing meaningful properties of shapes, while the matching distance can be used to stably compare them. From the application viewpoint, one drawback of these tools is the computational cost for evaluating the matching distance. In this paper we introduce a new framework for the matching distance estimation: It preserves the reliability of the entire approach in comparing shapes, extremely reducing the computational cost. Theoretical results are supported by experiments on 3D-models.