### Abstract

Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [J. Opt. Soc. Am. A 4, 629 (1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

Original language | English (US) |
---|---|

Pages (from-to) | 1127-1135 |

Number of pages | 9 |

Journal | Journal of the Optical Society of America A: Optics and Image Science, and Vision |

Volume | 5 |

Issue number | 7 |

DOIs | |

State | Published - 1988 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Computer Vision and Pattern Recognition
- Electronic, Optical and Magnetic Materials

### Cite this

*Journal of the Optical Society of America A: Optics and Image Science, and Vision*,

*5*(7), 1127-1135. https://doi.org/10.1364/JOSAA.5.001127

**Closed-form solution of absolute orientation using orthonormal matrices.** / Horn, Berthold K P; Hilden, Hugh M.; Negahdaripour, Shahriar.

Research output: Contribution to journal › Article

*Journal of the Optical Society of America A: Optics and Image Science, and Vision*, vol. 5, no. 7, pp. 1127-1135. https://doi.org/10.1364/JOSAA.5.001127

}

TY - JOUR

T1 - Closed-form solution of absolute orientation using orthonormal matrices

AU - Horn, Berthold K P

AU - Hilden, Hugh M.

AU - Negahdaripour, Shahriar

PY - 1988

Y1 - 1988

N2 - Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [J. Opt. Soc. Am. A 4, 629 (1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

AB - Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [J. Opt. Soc. Am. A 4, 629 (1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

UR - http://www.scopus.com/inward/record.url?scp=84975562075&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975562075&partnerID=8YFLogxK

U2 - 10.1364/JOSAA.5.001127

DO - 10.1364/JOSAA.5.001127

M3 - Article

AN - SCOPUS:84975562075

VL - 5

SP - 1127

EP - 1135

JO - Journal of the Optical Society of America. A, Optics and image science

JF - Journal of the Optical Society of America. A, Optics and image science

SN - 1084-7529

IS - 7

ER -