# Möbius voting for surface correspondence

@article{Lipman2009MbiusVF, title={M{\"o}bius voting for surface correspondence}, author={Yaron Lipman and Thomas A. Funkhouser}, journal={ACM Trans. Graph.}, year={2009}, volume={28}, pages={72} }

The goal of our work is to develop an efficient, automatic algorithm for discovering point correspondences between surfaces that are approximately and/or partially isometric.
Our approach is based on three observations. First, isometries are a subset of the Möbius group, which has low-dimensionality -- six degrees of freedom for topological spheres, and three for topological discs. Second, computing the Möbius transformation that interpolates any three points can be computed in closed-form…

## 360 Citations

### 3D Shape correspondence by isometry-driven greedy optimization

- Computer Science, Mathematics2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition
- 2010

An automatic method is presented that establishes 3D correspondence between isometric shapes in a fast and robust manner at a relatively coarse level as imposed by the patch radius and explicitly minimize the isometry cost via an iterative greedy algorithm in the original 3D Euclidean space.

### Möbius Transformations For Global Intrinsic Symmetry Analysis

- Computer ScienceComput. Graph. Forum
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An automatic algorithm for finding a correspondence map between two 3D surfaces is proposed, finding that it is superior to previous methods in cases where two surfaces have different overall shapes but similar reflective symmetry axes, a common case in computer graphics.

### Stable Region Correspondences Between Non‐Isometric Shapes

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This paper considers the problem of finding meaningful correspondences between 3D models that are related but not necessarily very similar, and provides an algorithm to solve this problem that mimics the power method.

### Isometric Shape Correspondence Based on the Geodesic Structure

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A novel approach to find the correspondence between two nearly isometric shapes by solving the correspondence problem as some minimum cost max flow problems MCFP with shortest path faster algorithm SPFA, and combining the results of these MCFP as the last correspondence result.

### One Point Isometric Matching with the Heat Kernel

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It is shown that under mild genericity conditions, a single correspondence can be used to recover an isometry defined on entire shapes, and thus the space of all isometries can be parameterized by one correspondence between a pair of points.

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A novel approach to represent maps between two discrete surfaces of the same genus and to minimize intrinsic mapping distortion, built upon the fact that such metrics exist on surfaces of arbitrary topology, without the need for any cuts or cones.

### Compatible intrinsic triangulations

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This work proposes a simple method utilizing intrinsic triangulations, operating directly on the original surfaces without going through any intermediate domains such as a plane or a sphere, which is robust enough to be used within a similar second-order optimization framework as was used previously in the literature.

### Semantic isometry for 3-D shape correspondence

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The semantic isometry to three-dimensional shape correspondence is introduced and a new framework called detection–recognition–correspondence is proposed that can greatly improve the corresponding accuracy over the existing state-of-the-art algorithms.

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