# An algebraic approach to affine registration of point sets

Jeffrey Ho, Adrian Peter, Anand Rangarajan, Ming Hsuan Yang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

38 Citations (Scopus)

### Abstract

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.

Original language English 2009 IEEE 12th International Conference on Computer Vision, ICCV 2009 1335-1340 6 https://doi.org/10.1109/ICCV.2009.5459309 Published - 2009 Dec 1 12th International Conference on Computer Vision, ICCV 2009 - Kyoto, JapanDuration: 2009 Sep 29 → 2009 Oct 2

### Publication series

Name Proceedings of the IEEE International Conference on Computer Vision

### Conference

Conference 12th International Conference on Computer Vision, ICCV 2009 Japan Kyoto 09/9/29 → 09/10/2

### Fingerprint

Polynomials
Cost functions
Probability density function
Tensors
Geometry

### All Science Journal Classification (ASJC) codes

• Software
• Computer Vision and Pattern Recognition

### Cite this

Ho, J., Peter, A., Rangarajan, A., & Yang, M. H. (2009). An algebraic approach to affine registration of point sets. In 2009 IEEE 12th International Conference on Computer Vision, ICCV 2009 (pp. 1335-1340).  (Proceedings of the IEEE International Conference on Computer Vision). https://doi.org/10.1109/ICCV.2009.5459309
Ho, Jeffrey ; Peter, Adrian ; Rangarajan, Anand ; Yang, Ming Hsuan. / An algebraic approach to affine registration of point sets. 2009 IEEE 12th International Conference on Computer Vision, ICCV 2009. 2009. pp. 1335-1340 (Proceedings of the IEEE International Conference on Computer Vision).
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Ho, J, Peter, A, Rangarajan, A & Yang, MH 2009, An algebraic approach to affine registration of point sets. in 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.

2009 IEEE 12th International Conference on Computer Vision, ICCV 2009. 2009. p. 1335-1340 5459309 (Proceedings of the IEEE International Conference on Computer Vision).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Ho J, Peter A, Rangarajan A, Yang MH. An algebraic approach to affine registration of point sets. In 2009 IEEE 12th International Conference on Computer Vision, ICCV 2009. 2009. p. 1335-1340. 5459309. (Proceedings of the IEEE International Conference on Computer Vision). https://doi.org/10.1109/ICCV.2009.5459309