Möbius Transformations For Global Intrinsic Symmetry Analysis

By Vladimir G. Kim, Yaron Lipman, Xiaobai Chen, Thomas A. Funkhouser. 2010

95 citations recorded by the source corpus.

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 Möbius 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 Möbius transformations, and 3) selects among those candidate Möbius transformations the one(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 Möbius group for enumerating potential self‐mappings. During experiments with a benchmark set of meshes augmented with human‐specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry.

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Michael M. Bronstein
2. Emanuele Rodolà
3. A. Bronstein
4. Vladimir G. Kim
5. Leonidas J. Guibas
6. Yaron Lipman
7. Thomas A. Funkhouser
8. Davide Boscaini

In Computer Graphics and Computer-Aided Design

1. Emanuele Rodolà
2. A. Bronstein
3. Daniel Cremers
4. Qi-Xing Huang
5. Yaron Lipman
6. Vladimir G. Kim
7. Thomas A. Funkhouser
8. Matthias Vestner

Citations in the corpus

• ShapeNet: An Information-Rich 3D Model Repository (2015, neutral): “Other commonly-used datasets contain segmentations [2], correspondences [13, 12], hierarchies [19], symmetries [11], salient features [3], semantic segmentations and labels [36], alignments of 3D models with images [35], semantic ontologies [5], and other functional annotations — but again only for…”
• Dense Human Body Correspondences Using Convolutional Networks (2015, methodology): “To address both challenges, we learn per-pixel descriptors for depth images by first training a network to solve a group of classification problems, using a single feature extraction tower shared by the different classification tasks.”
• Blended intrinsic maps (2011, methodology): “Please refer to [Lipman and Funkhouser 2009; Kim et al. 2010] for details.”
• Product Manifold Filter: Non-rigid Shape Correspondence via Kernel Density Estimation in the Product Space (2017, methodology): “[25, 20, 21] used conformal embeddings into disks and spheres to parametrize correspondences between homeomorphic surfaces as Möbius transformations.”
• Bayesian Inference of Bijective Non-Rigid Shape Correspondence (2016, methodology): “[24, 19, 20] used conformal embeddings into disks and spheres to parametrize correspondences between homeomorphic surfaces as Möbius transformations.”