Modified Rodrigues Parameters: an efficient representation of orientation in 3D vision and graphics

George Terzakis, Manolis Lourakis, Djamel Ait-Boudaoud

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Modified Rodrigues Parameters (MRPs) are triplets in R3, bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-offreedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution,
we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates in unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their back-projections in 3D space. Finally, in the general topic of non-linear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.
Original languageEnglish
Number of pages21
JournalJournal of Mathematical Imaging and Vision
Early online date12 Oct 2017
Publication statusEarly online - 12 Oct 2017


  • orientation parameterization
  • quaternions
  • stereographic projection
  • Modified Rodrigues Parameters
  • orientation interpolation
  • quaternion


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