Partial 3D Shape Retrieval by Reeb Pattern Unfolding

This paper presents a novel approach for fast and efficient partial shape retrieval on a collection of 3D shapes. Each shape is represented by a Reeb graph associated with geometrical signatures. Partial similarity between two shapes is evaluated by computing a variant of their maximum common sub-graph.
By investigating Reeb graph theory, we take advantage of its intrinsic properties at two levels. First, we show that the segmentation of a shape by a Reeb graph provides charts with disk or annulus topology only. This topology control enables the computation of concise and efficient sub-part geometrical signatures based on parameterisation techniques. Secondly, we introduce the notion of Reeb pattern on a Reeb graph along with its structural signature. We show this information discards Reeb graph structural distortion and still depicts the topology of the related sub-parts. The number of combinations to evaluate in the matching process is then dramatically reduced by only considering the combinations of topology equivalent Reeb patterns.
The proposed framework is invariant against rigid transformations and robust against non-rigid transformations and surface noise. It queries the collection in interactive time (from 4 to 30 seconds for the largest queries). It outperforms the competing methods of the SHREC 2007 contest in term of NDCG vector and provides, respectively, a gain of 14.1% and 40.9% on the approaches by Biasotti et al. [ BMSF06 ] and Cornea et al. [ CDS*05 ].
As an application, we present an intelligent modelling-by-example system which enables a novice user to rapidly create new 3D shapes by composing shapes of a collection having similar sub-parts.