Möbius Transformations
For Global Intrinsic Symmetry Analysis
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 strong deviations from perfect symmetry.
Möbius Transformations For Global Intrinsic Symmetry Analysis Vladimir G. Kim, Yaron Lipman, Xiaobai Chen, Thomas Funkhouser Computer Graphics Forum (Symposium On Geometry Processing) 2010 Paper: [PDF] BibTex: [Bib] Presentation: [PPT] |
Non-Rigid World Dataset:
Scape Dataset:
Watertight'07 Dataset:
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).
QualitativeReflectional Symmetry
Rotational Symmetry
Partial Symmetry
|