Rigid and non-rigid geometrical transformations of a marker-cluster and their impact on bone-pose estimation

Journal of Biomechanics 48 (2015) 4166–4172

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Rigid and non-rigid geometrical transformations of a marker-cluster and their impact on bone-pose estimation T. Bonci a,b,c,d,e, V. Camomilla a,e,n, R. Dumas b,c,d,e, L. Chèze b,c,d,e, A. Cappozzo a,e a

Department of Movement, Human and Health Sciences, Università degli Studi di Roma ‘‘Foro Italico’’, Rome, Italy Université de Lyon, F-69622 Lyon, France c Université Claude Bernard Lyon 1, Villeurbanne, France d IFSTTAR, UMR_T9406, Laboratoire de Biomécanique et Mécanique des Chocs (LBMC), F-69675 Bron, France e Interuniversity Centre of Bioengineering of the Human Neuromusculoskeletal System, Università degli Studi di Roma “Foro Italico”, Rome, Italy b

art ic l e i nf o

a b s t r a c t

Article history: Accepted 21 October 2015

When stereophotogrammetry and skin-markers are used, bone-pose estimation is jeopardised by the soft tissue artefact (STA). At marker-cluster level, this can be represented using a modal series of rigid (RT; translation and rotation) and non-rigid (NRT; homothety and scaling) geometrical transformations. The NRT has been found to be smaller than the RT and claimed to have a limited impact on bone-pose estimation. This study aims to investigate this matter and comparatively assessing the propagation of both STA components to bone-pose estimate, using different numbers of markers. Twelve skin-markers distributed over the anterior aspect of a thigh were considered and STA time functions were generated for each of them, as plausibly occurs during walking, using an ad hoc model and represented through the geometrical transformations. Using marker-clusters made of four to 12 markers affected by these STAs, and a Procrustes superimposition approach, bone-pose and the relevant accuracy were estimated. This was done also for a selected four marker-cluster affected by STAs randomly simulated by modifying the original STA NRT component, so that its energy fell in the range 30–90% of total STA energy. The pose error, which slightly decreased while increasing the number of markers in the markercluster, was independent from the NRT amplitude, and was always null when the RT component was removed. It was thus demonstrated that only the RT component impacts pose estimation accuracy and should thus be accounted for when designing algorithms aimed at compensating for STA. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Human movement analysis Modelling Soft tissue artefact Soft tissue deformation Rigid motion

1. Introduction When bone-pose is estimated using non-invasive stereophotogrammetry, skin markers move with respect to the underlying bone, generating the soft tissue artefact (STA). This source of error is regarded as a major issue in movement analysis (Leardini et al., 2005; Peters et al., 2010). Given four or more markers arranged in a cluster, a sequence of four independent geometrical transformations (GTs) can be used to describe the STA that affects it (Benoit et al., 2015; Dumas et al., 2014; Grimpampi et al., 2014): a translation and a rotation, representing the rigid transformation (RT), and a change of size and shape, representing the non-rigid transformation (NRT). These n Correspondence to: University of Rome "Foro Italico", piazza Lauro De Bosis 15, 00135 Rome, Italy. Tel.: þ39 06 36733522; fax: þ 39 06 36733517. E-mail address: [email protected] (V. Camomilla).

http://dx.doi.org/10.1016/j.jbiomech.2015.10.031 0021-9290/& 2015 Elsevier Ltd. All rights reserved.

components were quantified on humans while performing various motor tasks (Andersen et al., 2012; Barré et al., 2013; Grimpampi et al., 2014; Benoit et al., 2015; de Rosario et al., 2012; Dumas et al., 2015) and on a sheep (Taylor et al., 2005), showing that the magnitude of the RT is most often greater than that of the NRT. Based on this observation it has been concluded, either explicitly or implicitly, that the STA NRT component has a limited impact on bone-pose estimation and that STA compensation should concentrate on the marker-cluster RT. Bone-pose estimators described in the past do not reduce the propagation of the cluster RT to their end results (Alexander and Andriacchi, 2001; Andriacchi et al., 1998; Challis, 1995; Chèze et al., 1995; Dryden and Mardia, 1998; Heller et al., 2011; Taylor et al., 2005; Veldpaus et al., 1988), and are considered totally unsatisfactory for reconstructing all bone displacements that occur during function (Carman and Milburn, 2006; Cereatti et al., 2006; de Rosario et al., 2013). For this reason, using plate-mounted

T. Bonci et al. / Journal of Biomechanics 48 (2015) 4166–4172

markers that remove the NRT component while enhancing applicability, would not improve accuracy (Garling et al., 2007). In order to solve these issues, advanced bone-pose estimators must be designed that embed mathematical models to estimate the STA during the motor act being analysed (Bonci et al., 2014; Camomilla et al., 2009, 2013, 2015). This estimate of the STA would be used to correct the recorded marker trajectories. In the perspective of designing these bone-poses, so that the inherent optimisation problem converges to the correct solution, it is important to mathematically represent the STA using the minimum number of parameters. This complexity reduction entails accounting only for that portion of the artefact which has a major impact on the end results (Dumas et al., 2015). In addition, since it may be supposed that a redundancy in the number of markers forming the cluster would be beneficial, the question arises as to whether and to what extent increasing this number affects mathematical complexity. Dumas et al. (2014) showed that a modal approach mathematically represents the STA at marker-cluster level as a series of twelve independent modes, six of which describe the RT and six the NRT. The STA is fully described by this series only when the cluster is formed by four markers. The present study tackles the hypothesis that the loss of information caused by this mathematical representation of the STA with clusters of more than four

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markers would have a minimal impact on STA-compensated bonepose estimation, and is thus sustainable. The other objective of this study was to prove the hypothesis that, when the pose estimator is based on a Procrustes superimposition approach (Dryden and Mardia, 1998; Söderkvist and Wedin, 1993; Spoor and Veldpaus, 1980), as is normally the case, the marker-cluster NRT does not have a limited effect on bonepose accuracy, as claimed in the literature, but, rather, it has no effect whatsoever. This is the case independently from the number of markers included in the cluster and the relative magnitude between NRT and RT. In order to demonstrate this hypotheses, the paradigmatic case of the thigh STA generated during walking was used in association with a simulation approach, to explore the impact on bone-pose estimation error caused by STAs characterized by RT and NRT in different proportions, and the use of marker-clusters formed by different numbers of markers.

2. Materials and method 2.1. Generation of reference and artefact-affected data The time functions of pelvic-bone, femur and tibia anatomical frame (AFs; Fig. 1) pose, acquired during a level walking cycle of an able-bodied adult in a

Fig. 1. Skin-marker locations on the thigh segment are shown. The pelvic-bone, femur and tibia anatomical frames are also indicated. The time-histories of hip (α – flexion/ extension, β – ab/abduction, γ – internal/external rotation) and knee (δ – flexion/extension) angles, generated during gait and used as input for the soft tissue artefact model, are shown. The model used to estimate the STA for the skin marker indicated with j* is also reported. The circled skin-markers are those used in the Monte Carlo simulation described in Section 2.4.

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previous study (Dumas and Chèze, 2009) at a rate of 100 frames/s, were used to calculate reference hip and knee joint kinematics (Wu and Cavanagh, 1995; Wu et al., 2002), assumed to be STA-free. Twelve skin-markers arranged in a matrix-like fashion over the anterior aspect of a thigh, in positions similar to those used in Bonci et al. (2014) and illustrated in Fig. 1, were considered. For each jth (j¼ 1,…,m) skin marker, at each sampled instant of time k (k ¼1,…,n), realistic STA vectors, vj ðkÞ, representing the displacement of the marker relative to its reference position (k ¼ 1), were modelled as described in Bonci et al. (2014). We used as model parameters those calibrated using ex-vivo data (specimen S1, Table 1s in Bonci et al., 2014) relative to joint movements similar to those during walking. The input variables of the model were the hip and knee joint angles, determined as illustrated above, the range of which was sufficiently large to cause skin stretching and, thus, STA (Fig. 1). A thigh STA field, V ðkÞ was built as: 2

v1 ðkÞ 6 6 ⋮ 6 j 6 V ðkÞ ¼ 6 v ðkÞ 6 6 ⋮ 4 vm ðkÞ

3 7 7 7 7 7 7 7 5

ð1Þ

2.2. Marker-cluster geometrical transformation The h STA field V iðkÞ can be decomposed along the 3*m unit vectors of a selected 1 3m basis Φ ; …; Φ and represented as a series of independent 3*m components (modes) (Dumas et al., 2014): V ðk Þ ¼

3m X

al ðkÞΦ

l

ð2Þ

l¼1

where al ðkÞ is the amplitude of the l-th mode. If appropriate basis unit vectors are used, V ðkÞ may be represented as the sum of 12 modes that describe a marker-cluster geometrical transformation (GT) Dumas et al. (2014). The first six modes ðl ¼ 1; …; 6Þ describe the RT component (i.e., translation and rotation), while the others ðl ¼ 7; …; 12Þ describe the NRT component (i.e., homothety and stretch): V ðkÞ ffi

6 X

al ðkÞΦ þ l

l¼1

12 X

al ðkÞΦ

l

ð3Þ

l¼7

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

rigid

nonrigid

The energy carried by the two STA components can be calculated as the sum of the mean (over the n sampled instants of time) of the squared amplitudes al ðkÞ of the relevant modes and is usually expressed in percentage of the energy carried by all 3*m modes.

 The energy of each geometrical transformation component.  The difference between STA field, V ðkÞ, and its GT representation 2 3 e 1 ðk Þ 6 7 6 ⋮ 7 12 6 j 7 X 6 7 l E ðkÞ ¼ 6 e ðkÞ 7 ¼ V ðkÞ  al ðkÞΦ 6 7 l¼1 6 ⋮ 7 4 5 em ðkÞ

ð4Þ

 The e value, defined as the mean value, over the m markers, of the norm of ej ðkÞ.  The pose of the femoral AF during the walking cycle and associated inaccuracy. The marker positions in the femoral AF at time k ¼ 1 were used to define a marker-cluster model. In each subsequent instant of time, the cluster model was superimposed on the current cluster and the instantaneous AF pose relative to its reference pose at k ¼ 1, was determined using the Procrustes method. This pose was represented using the attitude and the position vectors, and the relevant moduli (θ and p, respectively) were assumed to be a measure of the bone-pose error obtained as a result of the STA. To evaluate the impact of the NRT and of E ðkÞ on the end results, the root mean square of the orientation (rmseθ) and position (rmsep) error components was computed before and after removing the RT component from both relevant STA field and its GT representation. 2.4. Assessment of the impact of NRT on bone-pose estimation To isolate the effect of the RT and NRT components on bone-pose estimation from the role of the marker number, we chose to analyse an STA field for which E ðkÞ (Eq. (4)) was null, i.e. we selected a four marker-cluster to obtain an STA field V 4 with the same DOFs as the GT basis (circled markers in Fig. 1). The energy content of the NRT was modified using a Monte Carlo simulation (MCS) (Mahadevan, 1997). The following STA fields were simulated: V MCS ðk; mcÞ ¼

6 X

al ðkÞΦ þ l

l¼1

12 X

r l ðmcÞal ðkÞΦ mc ¼ 1; …; 1000 l

0:5o r l o 5

ð5Þ

l¼7

The factor r l was chosen to either reduce or amplify the NRT, with more emphasis on the amplification. The energy of the RT and NRT components of V 4 and V MCS were calculated and a factor S was defined as the ratio between RT and NRT energy. During the simulation, S changed from 0.1 to 2.4 (Fig. 2), and the NRT energy ranged from 30% to 90% of the total energy. The inaccuracy associated to the artefact-affected femur pose was calculated with the Procrustes superimposition approach as illustrated in Section 2.3 using the cluster model of the selected four markers and the STA fields V 4 and V MCS . The rmseθ and rmsep errors were calculated before and after removing the RT component from the relevant STA field.

2.3. Dependency of the portion of STA not represented by GT representation on the number of skin markers Since the GT representation is based on 12 DOFs, if m 44, the number of DOFs of the STA field is larger than that of the GT basis, thus representing only a portion of the STA field. The importance of the portion not represented was assessed for an increasing number of skin markers (m¼ 4,...,12), considering, for each m, all the possible marker-clusters (i.e., 12!/(m!(12  m)!)). A 2-D isotropy index was calculated for each cluster (its value ranging from 0 to 1, assuming value 0 when the markers are collinear, and 1 when they are uniformly distributed according to Cappozzo et al. (1997)). Only clusters having a 2-D isotropy index greater than 0.5 were included in the following analysis (their number is reported in Table 1) and, for each of them, we calculated:

3. Results 3.1. Dependency of the portion of STA not represented by GT representation on the number of skin markers When the number of the skin markers used in the cluster increases from 4 to 12, the mean STA energy represented with 12 GT modes decreases from 100% to 91% of the total STA energy; the mean e values associated with the STA portion not represented in

Table 1 Median, lower and upper quartile of the root mean square position (rmsep) and orientation (rmseθ) errors with respect to the reference femur pose. The Procrustes superimposition approach was applied to markers affected by the STA field generated as described in Fig. 1. Statistics were obtained for all selected marker-clusters. The number of all possible cluster combinations for the m markers (i.e., 12!/(m!(12 m)!)) and for those selected is also shown. m

4

5

6

7

8

9

10

11

12

Cluster combinations Selected clusters rmsep [mm] Lower quartile Median Upper quartile rmseθ [deg] Lower quartile Median Upper quartile

495 331 11.6 18.2 30.5 2.8 4.4 7.4

792 543 10.6 17.1 26.9 2.4 3.7 5.8

924 684 10.8 16.7 23.9 2.1 3.2 5.0

792 627 10.9 16.7 23.0 1.7 2.7 4.5

495 410 11.7 17.0 22.0 1.3 2.3 4.0

220 188 12.1 16.6 21.0 1.1 1.9 3.5

66 58 13.4 16.2 19.9 1.0 1.7 3.2

12 11 13.9 17.2 18.8 1.2 1.5 2.1

1 1 16.8 16.8 16.8 1.8 1.8 1.8

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3.2. Assessment of the impact of NRT on bone-pose estimation The amplitudes along each mode, al ðkÞ [mm], calculated using the GT representation for all STAs, V 4 ðkÞ and V MCS ðk; mcÞ, relative to the selected four marker-cluster, are shown in Figs. 4 and 5. The rmsep and rmseθ values did not change for the different NRT values: the median error was 5.7 mm and 1.6 deg and the maximum error was 13.2 mm and 4.1 deg, for position and orientation respectively (Fig. 6). Removal of the cluster RT eliminates pose estimation errors, irrespective of the cluster NRT amplification (Fig. 6).

4. Discussion Fig. 2. Histogram of occurrences of ratio S between RT and NRT energy obtained from the Monte Carlo simulation. For the original NRT component the S value was 2.2.

Fig. 3. Upper panel: STA energy carried by the rigid (in white, RT) and the non-rigid modes (in grey, NRT), calculated in percentage of the entire STA energy, as a function of the number of markers in the cluster (m¼ 4,…,12), mean and standard deviation values obtained over all selected marker-clusters, the number of which is provided in Table 1. Lower panel: mean and standard deviation values of the e (defined in Section 2.3), calculated over all selected marker-clusters.

the GT subspace increases from 0 to 3.7 mm (Fig. 3). The average RT energy decreased as the marker number increased; its peak to peak variation was higher than 20% for a marker number up to 7 (e.g. from 57% to 90% for m ¼ 7), because the various possible clusters were located in different areas of the thigh and were, therefore, affected by STAs with different energy. When analysing the impact of the entire STA field, the bonepose estimation errors proved to be dependent on the number of skin markers (Table 1): the median and interquartile range (defined as the difference between upper and lower quartile of the distribution) error values decreased while m increased, for both position and orientation. When considering only the 12 GT modes, although the STA energy they represented was as low as 86% for m ¼6, the pose estimation errors were the same as in Table 1. The residual error after removing the RT component from both STA field and its GT representation was approximately zero (orders of magnitude: 10  8), irrespective to m.

This study has confirmed that, when a Procrustes superimposition approach is used, the non-rigid transformation of a skin cluster has no effect on bone-pose estimation accuracy, which depends solely on the cluster rigid transformation, independently from the number of markers in the cluster, location of the markercluster on the segment, and magnitude of the NRT component. This conclusion was attained through mathematical simulation, generating thigh artefacts during walking, representing them using a marker-cluster geometrical transformation for markerclusters which differed in marker number, size and location and then, modifying the relative amplitude of the rigid and non-rigid components of a paradigmatic four marker-cluster using a Monte Carlo simulation approach. It must be kept in mind that these results apply only when the modes used to describe homothety, stretch, and changes of size and shape, are obtained after orthogonalisation (Gram–Schmidt). If homothety or stretch are applied to the marker-cluster without performing this procedure, they may have a non-negligible effect on the Procrustes superimposition approach. The modelled thigh cluster RT (translation and rotation modes in the range 11–37 mm and 3–9 mm, respectively) had the same magnitude as the RT components measured during different motor tasks: level walking (4–25 mm, Barré et al., 2015; 20–26 mm, Benoit et al., 2015; and 19 mm (maximum), Gao and Zheng, 2008), hopping (11–19 mm, Benoit et al., 2015), cutting (16–26 mm, Benoit et al., 2015), and ex-vivo hip and knee flexion/extension (5– 21 mm, Grimpampi et al., 2014). Higher thigh RT amplitudes were reported during running support phase (50 mm and 25 mm, maximum value for translation and rotation modes respectively, Camomilla et al., 2015). Increasing the number of markers that form the cluster caused only a slight decrease in the median pose errors, as already evidenced in Cereatti et al., (2006). The error variability decreased when increasing m, possibly due to an increase in cluster size and, as a consequence, the markers further apart are affected by STAs less mutually correlated (Akbarshahi et al., 2010; Barré et al., 2013, 2015; Camomilla et al., 2009; Stagni et al., 2005). Although a number of markers higher than four is in conflict with practical applicability, these results seem to suggest that increasing the number is, theoretically, a valid criterion when designing a good marker-cluster. However, we believe that focusing on the number of markers or on marker weightings (Begon et al., 2015), may divert attention from the main problem represented by the STA rigid component. In fact, irrespective of the marker number and the marker-cluster location and size, when the RT is removed, an error-free pose can be obtained. Similarly, the cluster RT effect on bone-pose error was shown to be independent from the relative magnitude of the cluster RT and NRT (Fig. 6) components. This independence generalises the results reported by Dumas et al. (2015) who, using in-vivo experimental data, obtained virtually null residual joint kinematics errors while

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Fig. 4. RT mode amplitudes, al ðkÞ, for the thigh segment during the gait cycle, calculated on the GT representation of the selected four marker-cluster (V 4 ). These amplitudes remain unchanged for the simulated STA fields.

Fig. 5. NRT mode amplitudes, al ðkÞ, for the thigh segment during the gait cycle, calculated on the GT representation of the selected four marker-cluster (V 4 ) (black continuous line) and on all simulated STA fields (V MCS ) (average value: dashed line, 7 2 standard deviations: grey continuous lines).

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Fig. 6. Amplitude of the artefact-affected femur position (p) and orientation (θ) errors obtained during simulated gait, using V 4 , V MCS (dotted line) and removing the cluster RT component from the same STA fields (continuous line).

compensating for the thigh and shank STA RT component which accounted for approximately 90% of the STA total energy. The results obtained in the present study indicate that the cluster RT can be considered the only cause of error in bone-pose estimation. This minimises the complexity of STA mathematical representation and of relevant mathematical models, such as those proposed in Camomilla et al. (2015). This simplification allows such a STA mathematical model to be embedded in an optimal bone-pose estimator and makes its convergence to the correct solution feasible (Camomilla et al., 2013; Richard et al., 2012).

Conflict of interest statement The authors do not have any financial or personal relationships with other people or organisations that could inappropriately influence the manuscript.

Acknowledgements This research was funded by the project “Fall risk estimation and prevention in the elderly using a quantitative multifactorial approach” (Project ID number 2010R277FT) awarded by the Italian Ministry of Education, University and Research (Ministero dell'Istruzione, dell'Università e della Ricerca). The travel fund granted to Tecla Bonci by the Mediterranean Office for Youth (OMJ/MOY), No. 2011026/100, is gratefully acknowledged.

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