# 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…

## 364 Citations

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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.

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## References

SHOWING 1-10 OF 51 REFERENCES

### A Method for Registration of 3-D Shapes

- MathematicsIEEE Trans. Pattern Anal. Mach. Intell.
- 1992

A general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces based on the iterative closest point (ICP) algorithm.

### The Correlated Correspondence Algorithm for Unsupervised Registration of Nonrigid Surfaces

- Computer ScienceNIPS
- 2004

An unsupervised algorithm for registering 3D surface scans of an object undergoing significant deformations that can be used for compelling computer graphics tasks such as interpolation between two scans of a non-rigid object and automatic recovery of articulated object models.

### Robust global registration

- Computer ScienceSGP '05
- 2005

This work presents an algorithm for the automatic alignment of two 3D shapes ( data and model), without any assumptions about their initial positions, and develops a fast branch-and-bound algorithm based on distance matrix comparisons to select the optimal correspondence set and bring the two shapes into a coarse alignment.

### 3D Non-rigid Surface Matching and Registration Based on Holomorphic Differentials

- MathematicsECCV
- 2008

A novel approach for generating an efficient and optimal combined matching from multiple boundary-constrained conformal parameterizations for multiply connected domains, especially for 3D non-rigid surfaces with significantly inconsistent boundaries is described.

### Shape matching and object recognition using low distortion correspondences

- Computer Science2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05)
- 2005

This work approaches recognition in the framework of deformable shape matching, relying on a new algorithm for finding correspondences between feature points, and shows results for localizing frontal and profile faces that are comparable to special purpose approaches tuned to faces.

### A new point matching algorithm for non-rigid registration

- Computer ScienceComput. Vis. Image Underst.
- 2003

### On Bending Invariant Signatures for Surfaces

- MathematicsIEEE Trans. Pattern Anal. Mach. Intell.
- 2003

This work presents a method to construct a bending invariant signature for isometric surfaces, an embedding of the geometric structure of the surface in a small dimensional Euclidesan space in which geodesic distances are approximated by Euclidean ones.

### Generalizing the Hough transform to detect arbitrary shapes

- MathematicsPattern Recognit.
- 1981

### Non-Rigid Spectral Correspondence of Triangle Meshes

- Computer ScienceInt. J. Shape Model.
- 2007

An algorithm for finding a meaningful vertex-to-vertex correspondence between two triangle meshes, which is designed to handle general non-rigid transformations, outperforms previous spectral methods, as well as schemes that compute correspondence in the spatial domain via non- Rigid iterative closest points or the use of local shape descriptors.

### New Algorithms for 2D and 3D Point Matching: Pose Estimation and Correspondence

- Computer ScienceNIPS
- 1994