Möbius Transformations for Global Intrinsic Symmetry Analysis

Abstract

The goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are self-isometries of surfaces and as such are contained in the low dimensional group of Mobius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Mobius transformations,and 3) selects among those candidate Mobius transformation(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Mobius group for enumerating potential self-mappings. During experiments with a wide variety of meshes augmented with human-specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries in large collection of shape classes and under


Möbius Transformations for Global Intrinsic Symmetry Analysis
     Vladimir G. Kim, Yaron Lipman, Xiaobai Chen, and Thomas Funkhouser
     SGP 2010 (Symposium on Geometry Processing)

Paper: high-res (6.7Mb) low-res (0.4Mb)
BibTex: bib
Notes: read
Slides: pdf

Sourcecode and Data


Results

Non-Rigid World Dataset:

ModelCorrespondence
Rate (%)
Mesh
Rate (%)
Meshes
succeeded
Geodesic
Error [*]
Cat66%54%6/113.72725
Centaur92%100%6/62.99434
David82%57%4/74.40918
Dog91%89%8/91.40498
Horse92%100%8/81.5807
Michael87%75%15/204.23893
Victoria83%63%7/114.18347
Wolf100%100%3/30.461238
Total
75 Models
8 classes
85%76%57/753.29735

Scape Dataset:

ModelCorrespondence
Rate (%)
Mesh
Rate (%)
Meshes
succeeded
Geodesic
Error [*]
Human82%72%51/714.20537

Watertight'07 Dataset:

ModelCorrespondence
Rate (%)
Mesh
Rate (%)
Meshes
succeeded
Geodesic
Error [*]
Humans89%90%18/203.10448
Glasses88%80%16/201.17799
Planes91%80%16/200.935129
Ants98%100%20/200.629374
Teddy86%85%17/202.30933
Pliers61%40%8/202.84148
Fish93%95%19/200.426383
Birds99%100%20/200.291926
Armadillos63%35%7/206.4765
Busts59%50%10/201.53278
Animals83%75%15/201.53278
Total
220 Models
11 classes
83%75%166/2201.93256

Comparison (on Scape).

NOTE: 128 samples used (for consistency, we reduced number of samples so that Lipman'09 method performed faster, about 30min per model). That's why results are different for the first column from the scape experiment (there 256 samples were used).
Ours
(Mutually Closest Neighbors)
Ours
(Linear Assignment)
Lipman and Funkhouser'09 Lipman and Funkhouser'09
with prunned set
Ours
(Euclidean Distances
in Conformal Space)
Ours
(Voting)
Ours
(No Symmetric Set S2)
Correspondence Rate (%) 86%86%70%67%81%64%85%
Mesh Rate (%) 72%75%51%42%59%23%69%
Meshes succeeded 51/7153/7136/7130/7142/7118/7149/71
Geodesic Error [*] 3.486593.450216.781027.520494.383337.843633.50660