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

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.


INTRODUCTION
In human movement analysis, bony segments are generally considered as non-deformable.
However, due to the presence of soft tissues surrounding the bone, the related body segments behave as deformable bodies. When non-invasive techniques such as stereophotogrammetry are used, the accurate estimation of the position and orientation (pose) of a rigid bony segment from the measured position of markers attached to the skin is limited mostly by the relative motion between the markers and the bone, which is known as the Soft Tissue Artifact (STA) 26 . In the literature, several methods have been proposed that employ clusters composed of a redundant number of skin markers (>3) and which exploit this redundancy to minimize the effect of the STA on the estimation of the underlying bone pose, but no satisfactory solution yet exists 26,29 .
Since there is no unique separation of the kinematics of a deformable body into rigid and non-rigid motion, any estimation of the rigid motion of the underlying bone pose necessarily depends on non-unique definitions. Nevertheless, attempts have been made to separate the effect of the STA into a rigid motion (RM) of the cluster relative to the bone and non-rigid motion (NRM) of the cluster. Various different representations of these two contributions of the STA have been suggested 2-4, 19, 20, 23 . Most of the studies have shown that the STA's RM component has a similar 23 or larger magnitude than the STA's NRM component 2,4,5,19,21 . Therefore, commonly used techniques that compensate only for cluster NRM 15-17, 25, 32, 37 are insufficient to fully compensate for the STA effects, while the RM component, considered only in few compensation methods 10,19 , has been recently claimed as the only relevant component to be incorporated in pose estimators that can effectively compensate for the STA 21 .
The most recent studies show that the RM component represents the main portion of the STA, using different representations for the cluster movement and transformation. Barré et al. 4 assessed the RM component of the STA, during treadmill walking, using the Procrustes Superimposition (PS) approach 11 applied on the entire cluster of markers and showed that it represents approximately 80-100% of the STA amplitude. A similar conclusion was reached in three other works: de Rosario et al. 19 , who defined the NRM and RM components of the STA as the symmetric and skew-symmetric components of the vector field representing the relative 3 marker displacements; Dumas et al. 21 , who used the modal approach 20 to split the STA into additive components representing the RM and NRM cluster geometrical transformations in running; and Benoit et al. 5 , who modelled the cluster as resizable and deformable and described the STA components for walking, cutting maneuver, and one-legged hops. However, all cited STA quantifications analyze the kinematics of the entire cluster, and do not account for the fact that different sub-clusters within the large cluster of markers may demonstrate different kinematics 22,23 .
Attempts have been made to exploit the different kinematics of multiple sub-clusters to estimate the STA; Camomilla et al. 9 identified a subset of the latter clusters endowed with uncorrelated local movements and used the coherent average of the vectors reconstructed using them to estimate the STA; however, this estimate is affected by a phase indeterminacy that does not allow using the method to compensate for STA. Two studies investigated the possibility of using the NRM component, accessible from non-invasive measurements, to predict the RM components, or the pose errors. Stagni and Fantozzi 35  However, both studies obtained mostly poor correlations between RM and NRM, and concluded that the selection of clusters allowing for NRM minimization would not entail minimization of the RM component and, therefore, cannot be used for STA compensation. Nevertheless, these results are not conclusive, since both analyses are based upon maximum variations over time, and do not assess the possible instantaneous correlations between the different components.
In a previous work by some of the authors 33 , a new approach for the description of the deformation of clusters of skin markers was presented. The cluster of markers was characterized with Triangular Cosserat Point Elements (TCPEs) defined by all combinations of three markers and the kinematics of each TCPE was analysed using the nonlinear continuum mechanics theory of a Cosserat point with homogeneous deformations 30 . The main idea of the TCPE method is to use physical deformation parameters to identify instantaneous subsets of TCPEs which predict the best possible bone pose in the presence of STA. In the previous work, the orientation of a 4 rigid pendulum was estimated from the trajectories of markers fixed to a deformable silicone implant attached to its distal part, as a simulation of soft tissues around a bone. It was shown that partial compensation for the STA can be achieved by selecting subset groups of TCPEs having small strains and small relative rotations between them. However, the relative translations between TCPEs and the estimation of the position of the body were not taken into account.
The current work stems from these premises and both expands the method to characterize relative differences in the translations of TCPEs and further analyzes the ability of the physical deformation parameters to identify characteristics of those instantaneous subsets of TCPEs which most accurately estimate the bone pose.
The original aspects of the current study, including the TCPE developments, are the following: 1. The motion is quantified using physical parameters by analyzing the markers as Lagrangian material points in a continuum, rather than by applying purely mathematical or geometrical approximations.
2. The effects of the STA on sub-clusters of three markers (characterized by TCPEs) are described in terms of the strain in each TCPE and the relative rotation and relative translation between TCPEs.
3. The parameters are instantaneous, allowing for the selection of different subsets at each time step, rather than using a single marker cluster for the entire duration of the trials.
In this work, the developed method is applied to a set of previously collected ex-vivo data on the motion of the lower limbs of three cadavers with accurate measurements of the underlying bone pose 14,23 .

MATERIALS AND METHODS
The proposed method determines the pose of a bone segment from markers placed on the skin of the segment. These markers are combined into all possible clusters of three markers which define a group of Ntri TCPEs. A minimum number of four markers is needed to have more than one TCPE in a cluster. The non-rigid kinematics of each TCPE is presented here.

A. Kinematics of a TCPE
The present configuration of a given TCPE at time t is characterized by the position vectors The present and reference configurations are characterized by the director vectors A is the TCPE's reference area. Then, the reference reciprocal vectors where i j  is the Kronecker delta symbol. Furthermore, the deformation gradient tensor F of the TCPE is defined using the tensor product (outer product) operator  in the expression By definition, F is a nonsingular second order tensor so that the polar decomposition theorem 27 is used to determine its unique proper orthogonal rotation tensor R and the unique positive definite symmetric stretch tensor M . Moreover, the Lagrangian strain tensor E of the TCPE is defined by More details regarding the computation of M are given in Appendix 1. Moreover, the difference between R and the rotation tensor obtained by applying the PS approach on the same cluster is discussed in Appendix 2.
6 Furthermore, the centroids of the TCPE   , Xx in its reference and present configurations respectively, are given by

B. Physical Scalar TCPE Parameters
For ideal rigid body motion, all TCPEs have the same rotation tensor R , their strain tensors AB is the inner product of two second order tensors   , AB . In addition, at time t, pairs of TCPEs can rotate relative to each other and the positive relative angle J/K  between them can be defined using the rotation tensors J R and K R estimated by the J th and K th TCPEs, respectively: Moreover, the average rotational parameter J  of the J th TCPE relative to all other TCPEs at time t is defined as: Similarly, the translational difference / JK T between the J th and K th TCPE is defined using the (11) and the average translational parameter J T of the J th TCPE relative to all other TCPEs at time t is defined as: is the true translation vector of the reference point B and True R is the true rotation tensor of the bone with respect to the fixed laboratory frame.

D. Experimental Evaluation
To evaluate the effectiveness of the proposed method, the following experimental setup and data analysis were used. Complete details of the experimental data and procedures were described in previous works 8,14,23 . In brief, data were obtained from three intact cadavers (S1, S2, S3). Steel pins equipped with four markers were inserted into their right pelvis, femur, and tibia. Twelve skin markers were placed along three antero-lateral aspects of the right thigh, as shown in Fig. 2. An anatomical calibration procedure was performed to define the anatomical coordinate systems of the pelvis, femur and tibia 12    suggesting that the strain magnitude in a TCPE may not be the best parameter for selection of quality TCPEs to estimate bone pose. The strongest correlations were found between the rotational parameter  and the orientation error and between the translational parameter T and the position error for all specimens. This suggests that at each instant, TCPEs with smaller  are more likely to accurately represent the bone orientation and that TCPEs with smaller T are more likely to accurately represent the bone position. However, it was noted that the values of the correlation coefficients showed large variability over time, becoming poor in numerous instances. These findings seem to partially differ from those reported in previous studies 23,35 , that the NRM metrics for clusters of four markers were not markedly correlated with the pose errors. Nevertheless, these previous conclusions were based on inherently different metrics for the STA description than the TCPE parameters.
Moreover, it is important to stress that both  15 . When accuracy is required, additional markers can be used since their placement depends only on technical requirements (i.e. marker inter-distance and visibility) and not on specific anatomical placement locations. Consequently, placement of these markers should not add considerable amount of time. Furthermore, it is worth noting that the proposed continuum model approach for the description of the skin markers' kinematic variables could also be applied to data recorded using markerless motion capture techniques. In the near future, markerless techniques will become more accessible and accurate 28,31 and, in contrast with marker-based stereophotogrammetry, a very large number of observation points (a few hundreds per segment) will be available at minimal additional cost, either monetary or in subject preparation time 18 .
The data used to examine the TCPE method here suffer from a number of limitations. Firstly, the STA measured in this study does not include all sources of the STA phenomenon, due to the 16 lack of muscular activity and inertial effects normally associated with voluntary in-vivo motions.
Nevertheless, previous work 7 has shown that for tasks not entailing abrupt accelerations, such as slow running, soft tissue wobbling and deformations due to muscle contraction and inertial effects (in non-obese subjects), have a smaller impact with respect to the STA associated with the passive joint movement. This suggests that the kinematic data used in this study exhibit comparable amplitudes to those observed in previous in-vivo studies 1,6,7,13,22,36 . Secondly, the small number of specimens and the ex-vivo scenario prevent a generalization of the experimental results. However, the conceptual methods presented here remain valid.
In summary, the TCPE method proposes a number of physical parameters that can be used to describe, at each instant, the deformation of the body segment under analysis when the trajectories of several physical points of observation are available. It has been shown that some of these parameters correlate well with femur pose errors measured ex-vivo, which suggests that they can be used to select, at each instant, subsets of TCPEs which are more likely to accurately estimate the underlying bone pose. It has been shown that subsets of TCPEs exist which predict bone position and orientation more accurately than predicted by the PS approach applied on the entire cluster of markers, and that physical parameters of the TCPEs could be used as indicators to identify these subsets. It is expected that the operational framework provided by the TCPE method can be used to describe the body segment deformation and to lay the foundation for the compensation of the STA in in-vivo measurements of different populations, motor tasks and body segments when a high number of observation points are recorded. Future work will attempt to develop an advanced STA compensation procedure based on the TCPE parameters, using more comprehensive in-vivo data. This compensation procedure requires criteria for the instantaneous selection of subsets of TCPEs and an optimal technique for averaging the position and orientation of these subsets. Nevertheless, our preliminary analysis of the TCPE method in the presence of STA demonstrates its potential for improving the bone pose estimates with respect to those predicted using the entire cluster of markers.

APPENDIX 1
The objective of this appendix is to give more details regarding the computation of the kinematics of the TCPEs. The symmetric right Cauchy-Green deformation tensor C associated with the deformation gradient tensor F defined in (3) is used to calculate the stretch tensor M , as follows: APPENDIX 2 The objective of this appendix is to compare the rotation matrix of a cluster containing three markers obtained by the TCPE method with that obtained by the PS approach. Let   where Q is constrained to be an orthogonal rotation matrix that possess the properties , det( ) 1 T  Q Q I Q (16) The centroids   Then, the positive polar decomposition theorem can be used to decompose F into its rotation tensor R and stretch tensor M which is a positive-definite symmetric tensor  F RM (18) Different PS methods used for pose estimation 32,34,37 demonstrated that the rotation matrix Q which satisfies (15) under the constraint condition in (16) is  QR (19) 18 The PS methods mentioned above predict the same rotation tensor R and differ only by their algorithms for deriving it, as demonstrated on Appendix 3. On the other hand, the deformation gradient tensor F defined by the TCPE method in (3) predicts the rotation tensor R . In this regard, it is noted that F transforms any material line in the analyzed triangular body from the reference to the present configuration so that ii    x F X (20) Then, with the help of (4), (17), (18), and (20) In general, M does not commute with H , which proves that R does not equal to R whenever MH is not a symmetric tensor. Moreover, it is noted that since the rotation obtained by the PS approach is used in several other methods 5,9,23 for defining different metrics for defining the STA components, these definitions are also affected by the differ rotation tensor obtained by the PS method and the TCPE method.

APPENDIX 3
The objective of this appendix is to demonstrate that the rotation matrices obtained by eigenvalue decomposition 34 , polar decomposition 37 and singular value decomposition 32 are identical. The polar decomposition of the tensor F defined in (17) can be used to obtain R by  (22) Similarly, the singular value decomposition of F can be used to obtain the same R :