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 |
|---|---|
| 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 |
|---|
Conference
| Conference | 12th International Conference on Computer Vision, ICCV 2009 |
|---|---|
| Country | Japan |
| City | Kyoto |
| Period | 09/9/29 → 09/10/2 |
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All Science Journal Classification (ASJC) codes
- Software
- Computer Vision and Pattern Recognition
Cite this
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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 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 -