### Abstract

This paper proposes a new affine registration algorithm for matching two point sets in ℝ^{2} or ℝ^{3}. The input point sets are represented as probability density functions, using either Gaussian mixture models or discrete density models, and the problem of registering the point sets is treated as aligning the two distributions. Since polynomials transform as symmetric tensors under an affine transformation, the distributions' moments, which are the expected values of polynomials, also transform accordingly. Therefore, instead of solving the harder problem of aligning the two distributions directly, we solve the softer problem of matching the distributions' moments. By formulating a least-squares problem for matching moments of the two distributions up to degree three, the resulting cost function is a polynomial that can be efficiently optimized using techniques originated from algebraic geometry: the global minimum of this polynomial can be determined by solving a system of polynomial equations. The algorithm is robust in the presence of noises and outliers, and we validate the proposed algorithm on a variety of point sets with varying degrees of deformation and noise.

Original language | English |
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Title of host publication | 2009 IEEE 12th International Conference on Computer Vision, ICCV 2009 |

Pages | 1335-1340 |

Number of pages | 6 |

DOIs | |

Publication status | Published - 2009 Dec 1 |

Event | 12th International Conference on Computer Vision, ICCV 2009 - Kyoto, Japan Duration: 2009 Sep 29 → 2009 Oct 2 |

### Publication series

Name | Proceedings of the IEEE International Conference on Computer Vision |
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### Conference

Conference | 12th International Conference on Computer Vision, ICCV 2009 |
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Country | Japan |

City | Kyoto |

Period | 09/9/29 → 09/10/2 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Computer Vision and Pattern Recognition

### Cite this

*2009 IEEE 12th International Conference on Computer Vision, ICCV 2009*(pp. 1335-1340). [5459309] (Proceedings of the IEEE International Conference on Computer Vision). https://doi.org/10.1109/ICCV.2009.5459309

}

*2009 IEEE 12th International Conference on Computer Vision, ICCV 2009.*, 5459309, Proceedings of the IEEE International Conference on Computer Vision, pp. 1335-1340, 12th International Conference on Computer Vision, ICCV 2009, Kyoto, Japan, 09/9/29. https://doi.org/10.1109/ICCV.2009.5459309

**An algebraic approach to affine registration of point sets.** / Ho, Jeffrey; Peter, Adrian; Rangarajan, Anand; Yang, Ming Hsuan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - An algebraic approach to affine registration of point sets

AU - Ho, Jeffrey

AU - Peter, Adrian

AU - Rangarajan, Anand

AU - Yang, Ming Hsuan

PY - 2009/12/1

Y1 - 2009/12/1

N2 - This paper proposes a new affine registration algorithm for matching two point sets in ℝ2 or ℝ3. The input point sets are represented as probability density functions, using either Gaussian mixture models or discrete density models, and the problem of registering the point sets is treated as aligning the two distributions. Since polynomials transform as symmetric tensors under an affine transformation, the distributions' moments, which are the expected values of polynomials, also transform accordingly. Therefore, instead of solving the harder problem of aligning the two distributions directly, we solve the softer problem of matching the distributions' moments. By formulating a least-squares problem for matching moments of the two distributions up to degree three, the resulting cost function is a polynomial that can be efficiently optimized using techniques originated from algebraic geometry: the global minimum of this polynomial can be determined by solving a system of polynomial equations. The algorithm is robust in the presence of noises and outliers, and we validate the proposed algorithm on a variety of point sets with varying degrees of deformation and noise.

AB - This paper proposes a new affine registration algorithm for matching two point sets in ℝ2 or ℝ3. The input point sets are represented as probability density functions, using either Gaussian mixture models or discrete density models, and the problem of registering the point sets is treated as aligning the two distributions. Since polynomials transform as symmetric tensors under an affine transformation, the distributions' moments, which are the expected values of polynomials, also transform accordingly. Therefore, instead of solving the harder problem of aligning the two distributions directly, we solve the softer problem of matching the distributions' moments. By formulating a least-squares problem for matching moments of the two distributions up to degree three, the resulting cost function is a polynomial that can be efficiently optimized using techniques originated from algebraic geometry: the global minimum of this polynomial can be determined by solving a system of polynomial equations. The algorithm is robust in the presence of noises and outliers, and we validate the proposed algorithm on a variety of point sets with varying degrees of deformation and noise.

UR - http://www.scopus.com/inward/record.url?scp=77953193055&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953193055&partnerID=8YFLogxK

U2 - 10.1109/ICCV.2009.5459309

DO - 10.1109/ICCV.2009.5459309

M3 - Conference contribution

AN - SCOPUS:77953193055

SN - 9781424444205

T3 - Proceedings of the IEEE International Conference on Computer Vision

SP - 1335

EP - 1340

BT - 2009 IEEE 12th International Conference on Computer Vision, ICCV 2009

ER -