Abstract
Modified Rodrigues parameters (MRPs) are triplets in {mathbb {R}}^3 bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom 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 to 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 nonlinear 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.
Highlights
Orientation or attitude is a prominent facet of problems pertaining to disciplines such as computer graphics, computer vision, photogrammetry, robotics and augmented reality
Modified Rodrigues parameters is a formalism for the representation of orientation based on stereographic projection, originally introduced in the field of aerospace engineering by Wiener [68] in 1962
This paper has advocated the use of modified Rodrigues parameters (MRPs) for parameterizing rotations in problems arising in the fields of computer graphics and vision
Summary
Orientation or attitude is a prominent facet of problems pertaining to disciplines such as computer graphics, computer vision, photogrammetry, robotics and augmented reality. 3D computer vision deals with 3D reconstruction, often referred to as structure from motion estimation (SfM) This consists in using sets of images depicting an unknown scene and captured from unknown locations, in order to automatically extract a 3D geometric representation of the imaged scene plus the camera intrinsic parameters and their poses, i.e., positions and orientations [26,45]. This paper studies the representation of orientation via MRPs. An important finding is that the Jacobian of a quaternion is a polynomial function of its scalar and vector parts, thereby yielding simple expressions in rotation derivatives. The paper demonstrates the applicability of MRPs in problems related to orientation interpolation and pose estimation and provides experimental evidence that their use leads to new solutions or the simplification of existing ones and, in most cases, the improvement of performance. Experimental results comparing the performance of MRPs against different parameterizations of rotations are given in Sects. 8 and 9 summarizes the contributions of the paper
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