Quaternion

Abstract: The algorithm for traditional absolute orientation is an iterative algorithm which needs relatively accurate iterative initial valueBased on quaternion,a close-form solution is put forward in the paper which is used to calculate absolute orientation parameters without iteration method by way of strict theoretical deductionThe principal theories are as followsFirstly,the unit quaternion is used to describe the rotational conversion relation of the coordinatesThen the problem about absolute orientation is transformed into a problem about optimization to solveFinally,a simulation test using analogue data is carried out to validate the correctness and reliability of the algorithm

Abstract: Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the 19th century - a feat he celebrated by carving the founding equations into a stone bridge - mathematicians and engineers have been fascinated by these mathematical objects. They are used in applications as various as describing the geometry of space-time, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J.B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations.

Abstract: Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the real case, under the restriction that all eigenvalues are real). We also determine, in the complex case, all the correlation functions of the eigenvalues, as well as their limits when the order N of the matrices becomes infinite. In particular, the limit of the eigenvalue density as N → ∞ is constant over the whole complex plane.

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. A4, 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.