Andrew J. Hanson

Andrew J. Hanson, Sonya M. Hanson

Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geometry of the relationship between quaternions and rotation matrices by exploiting the adjugate matrix of the characteristic equation of a related eigenvalue problem to obtain the manifold of the space of a quaternion eigenvector. We argue that quaternions parameterized by their corresponding rotation matrices cannot be expressed, for example, in machine learning tasks, as single-valued functions: the quaternion solution must instead be treated as a manifold, with different algebraic solutions for each of several single-valued sectors represented by the adjugate matrix. We conclude with novel constructions exploiting the quaternion adjugate variables to revisit several classic pose estimation applications: 2D point-cloud matching, 2D point-cloud-to-projection matching, 3D point-cloud matching, 3D orthographic point-cloud-to-projection matching, and 3D perspective point-cloud-to-projection matching. We find an exact solution to the 3D orthographic least squares pose extraction problem, and apply it successfully also to the perspective pose extraction problem with results that improve on existing methods.

Andrew J. Hanson, Sonya M. Hanson

Pose estimation is a general problem in computer vision with wide applications. The relative orientation of a 3D reference object can be determined from a 3D rotated version of that object, or from a projection of the rotated object to a 2D planar image. This projection can be a perspective projection (the PnP problem) or an orthographic projection (the OnP problem). We restrict our attention here to the OnP problem and the full 3D pose estimation task (the EnP problem). Here we solve the least squares systems for both the error-free EnP and OnP problems in terms of the determinant ratio matrix (DRaM) approach. The noisy-data case can be addressed with a straightforward rotation correction scheme. While the SVD and optimal quaternion eigensystem methods solve the noisy EnP 3D-3D alignment exactly, the noisy 3D-2D orthographic (OnP) task has no known comparable closed form, and can be solved by DRaM-class methods. We note that while previous similar work has been presented in the literature exploiting both the QR decomposition and the Moore-Penrose pseudoinverse transformations, here we place these methods in a larger context that has not previously been fully recognized in the absence of the corresponding DRaM solution. We term this class of solutions as the DRaM family, and conduct comparisons of the behavior of the families of solutions for the EnP and OnP rotation estimation problems. Overall, this work presents both a new solution to the 3D and 2D orthographic pose estimation problems and provides valuable insight into these classes of problems. With hindsight, we are able to show that our DRaM solutions to the exact EnP and OnP problems possess derivations that could have been discovered in the time of Gauss, and in fact generalize to all analogous N-dimensional Euclidean pose estimation problems.