A least-squares algorithm for the equiform transformation from spatial marker co-ordinates

Veldpaus, FE Frans; Woltring, Herman J.; Dortmans, L.J.M.G.

doi:10.1016/0021-9290(88)90190-x

Cited by 337 publications

(190 citation statements)

References 11 publications

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“…The closer the markers are to the centre of rotation, the smaller the errors are for rotations and translations [7]. This effect also applies to the calculation of helical axes [18,23]. Thus, in order to minimize the calculation error, the distance between the markers and the specimen should be kept as small as possible.…”

Section: Error Estimationmentioning

confidence: 99%

“…This validation experiment includes all competing error sources caused by data acquisition and axes calculation during a real experiment. Consequently, it directly depicts the error of the actual test setup, whereas mathematical error estimations, as proposed by Woltring et al [23] or Veldpaus et al [18], would only provide theoretical errors for single error sources. The measurement error caused by the third of the three points, the registration, depends on many factors, such as the kind of X-ray apparatus or the position of the specimen with respect to film and focus.…”

Section: Error Estimationmentioning

confidence: 99%

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Finite helical axes of motion are a useful tool to describe the three-dimensional in vitro kinematics of the intact, injured and stabilised spine

et al. 2004

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IntroductionKinematical methods are widely used to describe the function of the healthy spine, the effect of degeneration or trauma, or the quality of treatment procedures. For in vitro experiments on the spine, usually three angles are reported, typically Cardanic/Eulerian angles or projection angles. Prerequisite for a description of these is a predefined, anatomical coordinate system. The coordinate system generally used for the study of the spine spans the frontal, sagittal and transverse planes. The three Cardanic/ Abstract The finite helical-axes method can be used to describe the three-dimensional in vitro kinematics of the spine. However, this method still suffers from large stochastic calculation errors and poorly conceived visualisation techniques. The aim of the present study, therefore, was to improve the currently used finite helical axes description, by use of a less error-prone calculation algorithm and a new visualisation technique, and to apply this improved method to the study of the threedimensional in vitro kinematics of the spine. Three-dimensional, continuous motion data of spinal motion segments were used to calculate the position and orientation of the finite helical axes (FHAs). The axes were then projected on plane antero-posterior, lateral and axial radiographs in order to depict the relation to the anatomy of each individual specimen. A hinge joint was used to estimate the measurement error of data collection and axes calculation. In an exemplary in vitro experiment, this method was used to demonstrate the ability of a prosthetic disc nucleus to restore the three-dimensional motion pattern of lumbar motion segments.In the validation experiment with the hinge joint, the calculated FHAs were lying within ±2.5 mm of the actual joint axis and were inclined relative to this axis at up to ±1.5°. In the exemplary in vitro experiment, the position and orientation of the FHAs of the intact specimens were subject to large inter-individual differences in all loading directions. Nucleotomy of the lumbar segments caused the axes to spread out, indicating complex coupled motions. The implantation of the prosthetic disc nucleus, for the most part, more than reversed this effect: the axes became oriented almost parallel to each other. The experiments showed that the present improved description of finite helical axes is a valid and useful tool to characterise the three-dimensional in vitro kinematics of the intact, injured and stabilised spine. The main advantage of this new method is the comprehensive visualisation of joint function with respect to the individual anatomy.

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Section: Error Estimationmentioning

confidence: 99%

Section: Error Estimationmentioning

confidence: 99%

Finite helical axes of motion are a useful tool to describe the three-dimensional in vitro kinematics of the intact, injured and stabilised spine

et al. 2004

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“…where   (Veldpaus et al, 1988), singular value decomposition (Söderkvist and Wedin, 1993), or eigenvalue decomposition (Spoor and Veldpaus, 1980). All three options analytically provide the same result , although small numerical differences may occur if the markers are badly configured.…”

Section: Pose Estimation By Least Squares (Ls) Methodsmentioning

confidence: 88%

“…Moreover, the rigid component has been demonstrated to be the only one impacting pose estimation accuracy when using RBLS Dumas et al, 2015;Grimpampi et al, 2014). This explains why commonly used BPEs based on RBLS, which compensate only for the non-rigid component (Cappozzo et al, 1997;Grimpampi et al, 2014;Heller et al, 2011;Söderkvist and Wedin, 1993;Veldpaus et al, 1988), are insufficient to fully compensate for the STA effects (Cereatti et al, 2006;Stagni and Fantozzi, 2009).…”

Section: Introductionmentioning

confidence: 99%

Bone orientation and position estimation errors using Cosserat point elements and least squares methods: Application to gait

Solav

Camomilla

Cereatti

et al. 2017

Journal of Biomechanics

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The aim of this study was to analyze the accuracy of bone pose estimation based on sub-clusters of three skinmarkers characterized by triangular Cosserat point elements (TCPEs) and to evaluate the capability of four instantaneous physical parameters, which can be measured non-invasively in-vivo, to identify the most accurate TCPEs. Moreover, TCPE pose estimations were compared with the estimations of two least squares minimization methods applied to the cluster of all markers, using rigid body (RBLS) and homogeneous deformation (HDLS) assumptions. Analysis was performed on previously collected in-vivo treadmill gait data composed of simultaneous measurements of the gold-standard bone pose by bi-plane fluoroscopy tracking the subjects' knee prosthesis and a stereophotogrammetric system tracking skin-markers affected by soft tissue artifact. Femur orientation and position errors estimated from skin-marker clusters were computed for 18 subjects using clusters of up to 35 markers. Results based on gold-standard data revealed that instantaneous subsets of TCPEs exist which estimate the femur pose with reasonable accuracy (median root mean square error during stance/swing: 1.4/2.8 deg for orientation, 1.5/4.2 mm for position). A non-invasive and instantaneous criteria to select accurate TCPEs for pose estimation (4.8/7.3 deg, 5.8/12.3 mm), was compared with RBLS (4.3/6.6 deg, 6.9/16.6 mm) and HDLS (4.6/7.6 deg, 6.7/12.5 mm). Accounting for homogeneous deformation, using HDLS or selected TCPEs, yielded more accurate position estimations than RBLS method, which, conversely, yielded more accurate orientation estimations. Further investigation is required to devise effective criteria for cluster selection that could represent a significant improvement in bone pose estimation accuracy.

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“…The objective of this appendix is to demonstrate that the rotation matrices obtained by eigenvalue decomposition 34 , polar decomposition 37 …”

Section: Appendixmentioning

confidence: 99%

Bone Pose Estimation in the Presence of Soft Tissue Artifact Using Triangular Cosserat Point Elements

Solav¹,

Rubin²,

Cereatti³

et al. 2017

Preprint

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Accurate estimation of the position and orientation (pose) of a bone from a cluster of skin markers is limited mostly by the relative motion between the bone and the markers, which is known as the Soft Tissue Artifact (STA). This work presents a method, based on continuum mechanics, to describe the kinematics of a cluster affected by STA. The cluster is characterized by Triangular Cosserat Point Elements (TCPEs) defined by all combinations of three markers.The effects of the STA on the TCPEs are quantified using three parameters describing the strain in each TCPE and the relative rotation and translation between TCPEs. The method was evaluated using previously collected ex-vivo kinematic data. Femur pose was estimated from 12 skin markers on the thigh, while its reference pose was measured using bone pins. Analysis revealed that instantaneous subsets of TCPEs exist which estimate bone position and orientation more accurately than the Procrustes Superimposition applied to the cluster of all markers. It has been shown that some of these parameters correlate well with femur pose errors, which suggests that they can be used to select, at each instant, subsets of TCPEs leading an improved estimation of the underlying bone pose.2

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A least-squares algorithm for the equiform transformation from spatial marker co-ordinates

Cited by 337 publications

References 11 publications

Finite helical axes of motion are a useful tool to describe the three-dimensional in vitro kinematics of the intact, injured and stabilised spine

Finite helical axes of motion are a useful tool to describe the three-dimensional in vitro kinematics of the intact, injured and stabilised spine

Bone orientation and position estimation errors using Cosserat point elements and least squares methods: Application to gait

Bone Pose Estimation in the Presence of Soft Tissue Artifact Using Triangular Cosserat Point Elements

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