On the Computation of the Average of Spatial Displacements

dc.contributor.authorGe, Q. J.
dc.contributor.authorYu, Zihan
dc.contributor.authorArbab, Mona
dc.contributor.authorLanger, Mark
dc.contributor.departmentRadiation Oncology, School of Medicine
dc.date.accessioned2024-10-11T12:57:29Z
dc.date.available2024-10-11T12:57:29Z
dc.date.issued2022
dc.description.abstractMany applications in biomechanics and medical imaging call for the analysis of the kinematic errors in a group of patients statistically using the average displacement and the standard deviations from the average. This paper studies the problem of computing the average displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. It has been shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the average for each quaternion component separately and independently. While the resulting algorithm is simple, the change of the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacement. This kinematic measure is then used to formulate a constrained least squares minimization problem. It has been shown that the problem decomposes into that of finding the optimal translation vector and the optimal unit quaternion. The former is simply the centroid of the set of given translation vectors and the latter can be obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. It is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome. Examples are provided to show the comparisons of these methods.
dc.eprint.versionAuthor's manuscript
dc.identifier.citationGe QJ, Yu Z, Arbab M, Langer M. ON THE COMPUTATION OF THE AVERAGE OF SPATIAL DISPLACEMENTS. Proc ASME Des Eng Tech Conf. 2022;7(46):V007T07A078. doi:10.1115/detc2022-90156
dc.identifier.urihttps://hdl.handle.net/1805/43907
dc.language.isoen_US
dc.publisherASME
dc.relation.isversionof10.1115/detc2022-90156
dc.relation.journalProceedings of the ASME Design Engineering Technical Conferences
dc.rightsPublisher Policy
dc.sourcePMC
dc.subjectBiomechanics
dc.subjectMedical imaging
dc.subjectKinematic errors
dc.titleOn the Computation of the Average of Spatial Displacements
dc.typeArticle
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