Solution of Nine-Parameter Affine Transformation Based on Quaternions

UYGUR S. Ö., Akyılmaz O., AYDIN C.

JOURNAL OF SURVEYING ENGINEERING, vol.147, no.3, 2021 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 147 Issue: 3
  • Publication Date: 2021
  • Doi Number: 10.1061/(asce)su.1943-5428.0000364
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, Computer & Applied Sciences, INSPEC, Metadex, DIALNET, Civil Engineering Abstracts
  • Istanbul Technical University Affiliated: Yes


A least-squares solution for (a)symmetric nine-parameter three-dimensional affine coordinate transformation based on quaternions is presented. Retrieval of the nine transformation parameters (three translations, three scale factors, and three Euler rotation angles) and their covariance matrix from the quaternion-based estimates is given in detail. The proposed method can successfully handle all the cases regardless of the magnitude of the parameters and full covariance matrices of the control points. Efficiency and consistency of the solution are discussed within five different case studies consisting of many numerical examples.