Abstract
This paper derives formulas for least-squares transformations between point-sets in ℝd. We consider affine transformations, with optional constraints for linearity, scaling, and orientation. We base the derivations hierarchically on reductions, and use trace manipulation to achieve short derivations. For the unconstrained problems, we provide a new formula which maximizes the orthogonality of the transform matrix.
Chapter PDF
Similar content being viewed by others
References
Horn, B.K.P., Hilden, H.M., Negahdaripour, S.: Closed-Form Solution of Absolute Orientation using Orthonormal Matrices. Journal of the Optical Society America 5(7), 1127–1135 (1988)
Green, B.: The orthogonal approximation of an oblique structure in factor analysis. Psychometrika 17(4), 429–440 (1952)
Chen, Y., Medioni, G.: Object modelling by registration of multiple range images. Image Vision Comput. 10(3), 145–155 (1992)
Besl, P.J., McKay, N.D.: A Method for Registration of 3-D Shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14(2), 239–256 (1992)
Rusinkiewicz, S., Levoy, M.: Efficient Variants of the ICP Algorithm. In: Proceedings of the Third Intl. Conf. on 3D Digital Imaging and Modeling, pp. 145–152 (2001)
Zhang, L., Choi, S., Park, S.: Robust ICP Registration Using Biunique Correspondence. In: International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission, 3DIMPVT 2011, Hangzhou, China, May 16-19, pp. 80–85 (2011)
Levoy, M., Pulli, K., Curless, B., Rusinkiewicz, S., Koller, D., Pereira, L., Ginzton, M., Anderson, S., Davis, J., Ginsberg, J., Shade, J., Fulk, D.: The digital Michelangelo project: 3D scanning of large statues. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2000, pp. 131–144 (2000)
Bartels, R.H., Stewart, G.W.: Solution of the matrix equation AX + XB = C. Commun. ACM 15(9), 820–826 (1972)
Walker, M.W., Shao, L., Volz, R.A.: Estimating 3-D location parameters using dual number quaternions. CVGIP: Image Underst. 54(3), 358–367 (1991)
Myronenko, A., Song, X.: Point Set Registration: Coherent Point Drift. IEEE Trans. Pattern Anal. Mach. Intell. 32(12), 2262–2275 (2010)
Schönemann, P.: A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)
Eggert, D.W., Lorusso, A., Fisher, R.B.: Estimating 3-D rigid body transformations: a comparison of four major algorithms. Mach. Vision Appl. 9(5-6), 272–290 (1997)
Arun, K.S., Huang, T.S., Blostein, S.D.: Least-Squares Fitting of Two 3-D Point Sets. IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987)
Horn, B.K.P.: Closed-form solution of absolute orientation using unit quaternions. Journal of the Optical Society of America 4, 629–642 (1987)
Umeyama, S.: Least-Squares Estimation of Transformation Parameters Between Two Point Patterns. IEEE Trans. Pattern Anal. Mach. Intell. 13(4), 376–380 (1991)
Goodrich, M.T., Mitchell, J.S.B., Orletsky, M.W.: Approximate Geometric Pattern Matching Under Rigid Motions. IEEE Trans. Pattern Anal. Mach. Intell. 21(4), 371–379 (1999)
van Wamelen, P.B., Li, Z., Iyengar, S.S.: A fast expected time algorithm for the 2-D point pattern matching problem. Pattern Recognition 37(8), 1699–1711 (2004)
Paige, C.C., Saunders, M.A.: Towards a Generalized Singular Value Decomposition. SIAM Journal on Numerical Analysis 18(3), 398–405 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rutanen, K., Gómez-Herrero, G., Eriksson, SL., Egiazarian, K. (2013). Least-Squares Transformations between Point-Sets. In: Kämäräinen, JK., Koskela, M. (eds) Image Analysis. SCIA 2013. Lecture Notes in Computer Science, vol 7944. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38886-6_47
Download citation
DOI: https://doi.org/10.1007/978-3-642-38886-6_47
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38885-9
Online ISBN: 978-3-642-38886-6
eBook Packages: Computer ScienceComputer Science (R0)