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

Y1 - 2009

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

T2 - 12th International Conference on Computer Vision, ICCV 2009

Y2 - 29 September 2009 through 2 October 2009

ER -