Symmetry Factored Embedding and distance

Yaron Lipman, Xiaobai Chen, Ingrid Daubechies, Thomas Funkhouser

Research output: Contribution to journalArticle

76 Scopus citations

Abstract

We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is "factored out," and the SFD is the Euclidean distance in that space. These constructions characterize the space of symmetric correspondences between points - i.e., orbits. A key observation is that a set of points in the same orbit appears as a clique in a correspondence graph induced by pairwise similarities. As a result, the problem of finding approximate and partial symmetries in a point set reduces to the problem of measuring connectedness in the correspondence graph, a well-studied problem for which spectral methods provide a robust solution. We provide methods for computing the SFE and SFD for extrinsic global symmetries and then extend them to consider partial extrinsic and intrinsic cases. During experiments with difficult examples, we find that the proposed methods can characterize symmetries in inputs with noise, missing data, non-rigid deformations, and complex symmetries, without a priori knowledge of the symmetry group. As such, we believe that it provides a useful tool for automatic shape analysis in applications such as segmentation and stationary point detection.

Original languageEnglish (US)
Article number103
JournalACM Transactions on Graphics
Volume29
Issue number4
DOIs
StatePublished - Sep 13 2010

All Science Journal Classification (ASJC) codes

  • Computer Graphics and Computer-Aided Design

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