Möbius voting for surface correspondence
Abstract
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 after a mid-edge flattening to the complex plane. Third, deviations from isometry can be modeled by a transportation-type distance between corresponding points in that plane. Motivated by these observations, we have developed a Möbius Voting algorithm that iteratively: 1) samples a triplet of three random points from each of two point sets, 2) uses the Möbius transformations defined by those triplets to map both point sets into a canonical coordinate frame on the complex plane, and 3) produces "votes" for predicted correspondences between the mutually closest points with magnitude representing their estimated deviation from isometry. The result of this process is a fuzzy correspondence matrix, which is converted to a permutation matrix with simple matrix operations and output as a discrete set of point correspondences with confidence values. The main advantage of this algorithm is that it can find intrinsic point correspondences in cases of extreme deformation. During experiments with a variety of data sets, we find that it is able to find dozens of point correspondences between different object types in different poses fully automatically.
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Recent intrinsic techniques [36], [38], [39] use low dimensional embedding to by-pass the high-dimensionality problem by assuming a strong inelastic isometry constraint.
“Recent intrinsic techniques [36], [38], [39] use low dimensional embedding to by-pass the high-dimensionality problem by assuming a strong inelastic isometry constraint.”
From Registration of 3D Point Clouds and Meshes: A Survey from Rigid to Nonrigid (2013) · cites this paper as [39]
These transformations are similar to the domain theory in analytical learning-a global rule, one reason why these techniques can be cast as optimization problems and solved using the Hough transform [17], [39] or RANSAC technique [4], [71] efficiently.
“These transformations are similar to the domain theory in analytical learning-a global rule, one reason why these techniques can be cast as optimization problems and solved using the Hough transform [17], [39] or RANSAC technique [4], [71] efficiently.”
From Registration of 3D Point Clouds and Meshes: A Survey from Rigid to Nonrigid (2013) · cites this paper as [39]
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- Authors
- Lipman, Y., & Funkhouser, T.
- Affiliations
- Princeton University (US)
- Venue
- ACM Transactions on Graphics
- Published
- 2009-07-27
- Volume
- 28
- Issue
- 3
- Pages
- 1-12
- ISSN
- 0730-0301 (ISSN-L)
- 1557-7368
- OpenAlex
- W2105106232
- Cited by
- 332
- References
- 49
- Citations by year
- 2026: 6 · 2025: 7 · 2024: 4 · 2023: 8 · 2022: 15 · 2021: 3 · 2020: 15 · 2019: 13 · 2018: 27 · 2017: 25 · 2016: 34 · 2015: 18 · 2014: 37 · 2013: 38 · 2012: 30
- FWCI
- 27.26
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- ACM Transactions on Graphics (publishedVersion)
- Funders
- Division of Information and Intelligent Systems (CNFS-0406415IIS-0612231CCF-0702672)
- National Science Foundation (CNFS-0406415IIS-0612231CCF-0702672)
- Division of Computing and Communication Foundations (CNFS-0406415IIS-0612231CCF-0702672)
- Topics
- Physical Sciences › Engineering › Computational Mechanics › 3D Shape Modeling and Analysis
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › Computer Graphics and Visualization Techniques
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image Processing and 3D Reconstruction
- Keywords
- point matching, surface matching, Möbius transformations, isometries, transport distance, partial isometries
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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