Topological Shape Matching using Multi-Dimensional Reeb Graphs

Abstract

Shape matching or retrieval is an important problem in computer graphics and data analysis. Topological techniques based on Reeb graphs and persistence diagrams have been employed to obtain an effective solution in this problem. In the current paper, we propose an improved technique based on the multi-dimensional Reeb graph (MDRG) that captures the topology of a multi-field through a hierarchy of Reeb graphs in different dimensions. To capture the persistent features in a multi-field, a hierarchy of persistence diagrams is then constructed by computing a persistence diagram corresponding to each Reeb graph of the MDRG. Based on this representation, we propose a novel distance measure between two MDRGs by extending the bottleneck distance between two Reeb graphs. We show that the proposed measure satisfies the pseudo-metric and stability properties. The effectiveness of the proposed multi-field topology based measure is tested on the shape data as compared to scalar topology based measures. We use normalized eigenfunctions of the Laplace-Beltrami operator, in pairs, as the bivariate descriptors of the shapes. The performance of the proposed measure is compared with the well-known topology based measures in shape matching using Heat Kernel Signature, Wave Kernel Signature and Scale-Invariant Heat Kernel Signature.