A multi-scale topological shape model for single and multiple component shapes

Abstract

A novel shape model of multi-scale topological features is proposed which considers those features relating to connected components and holes. This is achieved by considering the persistent homology of a pair of sublevel set functions corresponding to a pair of distance functions defined on the ambient space. The model is applicable to both single and multiple component shapes and, to the authors knowledge, is the first shape model to consider multi-scale topological features of multiple component shapes. It is demonstrated, both qualitatively and quantitatively, that the proposed model models useful multi-scale topological features and outperforms a commonly used benchmark models with respect to the task of multiple component shape retrieval.

An affine moment invariant for multi-component shapes

“…We show that A(S) equals the integral of the squared triangle areas, i.e. the integral of Area_(∆ABC) 2 , where A, B, and C vary through a given shape S whose area is 1 3 . We give the following theorem.…”

“…The concept of multi-component shapes is a very generic concept [2,9,10,14,15]. It allows us to segment a single object onto components, or to group objects into a multi-component shape to suit a particular application.…”