TY - GEN

T1 - Symmetry factored embedding and distance

AU - Lipman, Yaron

AU - Chen, Xiaobai

AU - Daubechies, Ingrid

AU - Funkhouser, Thomas

PY - 2010/7/26

Y1 - 2010/7/26

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84872237862&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872237862&partnerID=8YFLogxK

U2 - 10.1145/1778765.1778840

DO - 10.1145/1778765.1778840

M3 - Conference contribution

AN - SCOPUS:84872237862

T3 - ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010

BT - ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010

A2 - Hoppe, Hugues

PB - Association for Computing Machinery, Inc

T2 - 37th International Conference and Exhibition on Computer Graphics and Interactive Techniques, SIGGRAPH 2010

Y2 - 26 July 2010 through 30 July 2010

ER -