Browsing by Author "Yu, Zihan"

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    Construction of Confidence Regions for Uncertain Spatial Displacements With Dual Rodrigues Parameters
    (ASME, 2024) Yu, Zihan; Ge, Qiaode Jeffrey; Langer, Mark P.; Radiation Oncology, School of Medicine
    This paper follows our recent work on the computation of kinematic confidence regions from a given set of uncertain spatial displacements with specified confidence levels. Dual quaternion algebra is used to compute the mean displacement as well as relative displacements from the mean. In constructing a 6D confidence ellipsoid, however, we use dual Rodrigue parameters resulting from dual quaternions. The advantages of using dual quaternions and dual Rodrigues parameters are discussed in comparison with those of three translation parameters and three Euler angles, which were used for the development of the socalled the Rotational and Translational Confidence Limit (RTCL) method. The set of six dual Rodrigue parameters are used to define a parametric space in which a 6 × 6 covariance matrix and a 6D confidence ellipsoid are obtained. An inverse operation is then applied to first obtain dual quaternions and then to recover the rotation matrix and translation vector for each point on the 6D ellipsoid. Through examples, we demonstrate the efficacy of our approach by comparing it with the RTCL method known in literature. Our findings indicate that our method, based on the dual-Rodrigues formulation, yields more compact and effective swept volumes than the RTCL method, particularly in cases involving screw displacements.
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    On the Computation of Mean and Variance of Spatial Displacements
    (ASME, 2024) Ge, Qiaode Jeffrey; Yu, Zihan; Arbab, Mona; Langer, Mark P.; Radiation Oncology, School of Medicine
    This paper studies the problem of computing an average (or mean) 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 is shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the mean for each quaternion component separately and independently. While the resulting algorithm is simple, a change in 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 displacements. This kinematic measure is used to define the variance of a set of displacements, which is then used to formulate a constrained least squares minimization problem. It is 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 translation vectors and the latter is obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. In addition, 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.
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    On the Computation of the Average of Spatial Displacements
    (ASME, 2022) Ge, Q. J.; Yu, Zihan; Arbab, Mona; Langer, Mark; Radiation Oncology, School of Medicine
    Many 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.
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    On the Construction of Confidence Regions for Uncertain Planar Displacements
    (ASME, 2024) Yu, Zihan; Ge, Qiaode Jeffrey; Langer, Mark P.; Arbab, Mona; Radiation Oncology, School of Medicine
    This paper studies the statistical concept of confidence region for a set of uncertain planar displacements with a certain level of confidence or probabilities. Three different representations of planar displacements are compared in this context and it is shown that the most commonly used representation based on the coordinates of the moving frame is the least effective. The other two methods, namely the exponential coordinates and planar quaternions, are equally effective in capturing the group structure of SE(2). However, the former relies on the exponential map to parameterize an element of SE(2), while the latter uses a quadratic map, which is often more advantageous computationally. This paper focus on the use of planar quaternions to develop a method for computing the confidence region for a given set of uncertain planar displacements. Principal component analysis (PCA) is another tool used in our study to capture the dominant direction of movements. To demonstrate the effectiveness of our approach, we compare it to an existing method called rotational and translational confidence limit (RTCL). Our examples show that the planar quaternion formulation leads to a swept volume that is more compact and more effective than the RTCL method, especially in cases when off-axis rotation is present.
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    On the Construction of Kinematic Confidence Ellipsoids for Uncertain Spatial Displacements
    (Springer, 2023) Yu, Zihan; Ge, Qiaode Jeffrey; Langer, Mark P.; Arbab, Mona; Radiation Oncology, School of Medicine
    This paper deals with the problem of estimating confidence regions of a set of uncertain spatial displacements for a given level of confidence or probabilities. While a direct application of the commonly used statistic methods to the coordinates of the moving frame is straight-forward, it is also the least effective in that it grossly overestimate the confidence region. Based on the dual-quaternion representation, this paper introduces the notion of the kinematic confidence ellipsoids as an alternative to the existing method called rotation and translation confidence limit (RTCL). An example is provided to demonstrate how the kinematic confidence ellipsoids can be computed.