ARTICLE IN PRESS Journal of Biomechanics 42 (2009) 2214–2217

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Short Communication

Soft tissue artifact compensation by linear 3D interpolation and approximation methods R. Dumas , L. Cheze Universite de Lyon, F-69622, Lyon, France Universite Lyon 1, F-69622, Villeurbanne, France INRETS, UMR_T9406 Laboratoire de Biome canique et Me canique des Chocs, F-69675, Bron, France

a r t i c l e in f o

a b s t r a c t

Article history: Accepted 2 June 2009

Several compensation methods estimate bone pose from a cluster of skin-mounted makers, each influenced by soft tissue artifact (STA). In this study, linear 3D interpolation and approximation methods (affine mapping, Kriging and radial basis function (RBF)) and the conventional singular value decomposition (SVD) method were examined to determine their suitability for STA compensation. The ability of these four methods to estimate knee angles and displacements was compared using simulated gait data with and without added STA. The knee angle and the displacement estimates of all four methods were similar with root-mean-square errors (RMSEs) near 1.51 and 4 mm, respectively. The 3D interpolation and approximation methods were more complicated to implement than the conventional SVD method. However, these non-standard methods provided additional geometric (homothety, stretch) and time functions that model the deformation of the cluster of markers. This additional information may be useful to model and compensate the STA. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Knee kinematics Soft tissue artifact Gait simulation Bone pose Affine mapping Kriging Radial basis function

1. Introduction

2. Material and methods

Motion analysis has application in animation, sports, ergonomics, and health care. However, when accurate assessment of joint kinematics is required, the motion of a marker on the skin relative to the underlying bone (soft tissue artifact, STA) is problematic. Several compensation methods have been proposed (Leardini et al., 2005). The motion of the skin makers can be minimized by least square methods (Spoor and Veldpaus, 1980; Veldpaus et al., 1988; Soderkvist and Wedin, 1993; Challis, 1995) and can be specifically modeled (Ball and Pierrynowski, 1998; Alexander and Andriacchi, 2001; Cappello et al., 2005; Camomilla et al., 2009; Ryu et al., 2009). STA compensation is analogous to non-rigid body registration (Holden, 2008) and anthropometric scaling (Lewis et al., 1980; Sommer et al., 1982) as they include spatial transformations of a set of 3D points. Linear interpolations and approximations such as affine mapping, Kriging, and radial basis function (RBF) are examples of these methods. The objective of this study is to briefly introduce and evaluate affine mapping, Kriging, and RBF, compared to the conventional singular value decomposition (SVD) method, using simulated gait data with and without added STA.

The different theoretical aspects of STA compensation and modeling using affine mapping, Kriging, RBF, and SVD are presented in Table 1.

2.1. Affine mapping For each coordinate of a point p of the cluster from image i to image j, affine mapping corresponds to a linear polynomial: xjp ¼ ax0 þ ax1 xip þ ax2 yip þ ax3 zip yjp ¼ ay0 þ ay1 xip þ ay2 yip þ ay3 zip zjp ¼ az0 þ az1 xip þ az2 yip þ az3 zip

The 12 coefficients a can be found by resolving a system of 3  n linear equations for the n points of the cluster (nZ4): 2 3 2 3 3 1 xi1 yi1 zi1 2 x yj1 zj1 xj a ay0 az0 6 1 7 7 0 6... ... ...7 6 . . . . . . . . . . . . 76 ax ay az 7 6 7 6 6 76 1 17 1 6 j j j 7 6 7 1 xip yip zip 7 ð2Þ 6 xp yp zp 7 ¼ 6 y z 7 76 x 6 7 6 6 . . . . . . . . . . . . 74 a2 a2 a2 5 6... ... ...7 4 5 y 4 5 x z a3 a3 a3 1 xin yin zin |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} xjn yjn zjn |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Xjðn3Þ

 Corresponding author. Tel.: +33 4 72 44 85 75; fax: +33 4 72 44 80 54.

E-mail address: [email protected] (R. Dumas). 0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2009.06.006

ð1Þ

i

Að43Þ

X ðn4Þ

ij The coefficients ax0, ay0, az0 represent the translation ~ t O of the origin of the coordinate system in which the points of the cluster are expressed (considered as fixed with respect to the moving segment). The 9 others represent, as a whole, a

ARTICLE IN PRESS R. Dumas, L. Cheze / Journal of Biomechanics 42 (2009) 2214–2217

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Table 1 Theoretical aspects of STA compensation and modeling using affine mapping, Kriging, RBF, and SVD. Methods Bone pose computation

Points of the cluster

STA deformation model

Model parameter selection

Affine (Pseudo-) inverse mapping and positivea polar decomposition Kriging Inverse and positivea polar decomposition RBF (Pseudo-) inverse and positivea polar decomposition SVD Singular value decomposition

4 non-coplanar points minimum: Homothety, stretch (and least square interpolation if n ¼ 4 (approximation residual) otherwise)

None

5 non-coplanar points minimum: interpolation in any case

Stochastic selection of the generalized covariance function K: minimal Kriging variance

Homothety, stretch, and function of 3D distances between points of the cluster

5 non-coplanar points minimum: Homothety, stretch, basis of radial functions Deterministic selectionb of the basis of radial interpolation if all points selected as of 3D distances between centers and points of functions hf by regularized forward selection: lowest interpolation errors for a controlled smoothing effect centers (approximation otherwise) the cluster (and least square residual) 3 non-aligned points minimum: approximation in any case

Least square residual

None

a

Positive polar decomposition (i.e., restricting the determinant of the rotation to be +1) avoids a supplementary reflection in the STA deformation model. As a result of the model selection, RBF is the same as Kriging if all the points of the cluster are selected as centers and if a unique basis function is chosen for all centers or else RBF is the same as affine mapping if no center is selected. b

ij

rotation, a homothety, and a stretch. The rotation Rij can be extracted by polar decomposition (Sommer et al., 1982).

The translation ~ t O and the rotation Rij can be computed from the coefficients of A as for affine mapping. Furthermore, the coefficients b are the weights of each 3D distance between the points of the cluster.

2.2. Kriging

2.3. Radial basis function (RBF)

For each coordinate of a point p of the cluster from image i to image j, Kriging corresponds to a drift (e.g., a linear polynomial as for affine mapping) plus a fluctuation (i.e., a function of Euclidean distances between the point p and the other points q):

For each coordinate of a point p of the cluster from image i to image j, RBF corresponds to the linear decomposition on a basis of m functions (of Euclidean distances between the point p and some centers) plus a linear polynomial (Buhmann, 2000):

xjp ¼ ax0 þ ax1 xip þ ax2 yip þ ax3 zip þ

n X

i

i

bxq KðJ~ xp  ~ x q JÞ

xjp ¼

q¼1

yjp ¼ ay0 þ ay1 xip þ ay2 yip þ ay3 zip þ

n X

n X

i

wxf hf ðJ~ xp  ~ c f JÞ þ ax0 þ ax1 xip þ ax2 yip þ ax3 zip

f ¼1 i

i

byq KðJ~ xp  ~ x q JÞ

yjp ¼

m X

i

wyf hf ðJ~ xp  ~ c f JÞ þ ay0 þ ay1 xip þ ay2 yip þ ay3 zip

f ¼1

q¼1

zjp ¼ az0 þ az1 xip þ az2 yip þ az3 zip þ

m X

i i bzq KðJ~ xp  ~ x q JÞ

ð3Þ

zjp ¼

m X

i wzf hf ðJ~ xp  ~ c f JÞ þ az0 þ az1 xip þ az2 yip þ az3 zip

ð6Þ

f ¼1

q¼1 i

i

where K is a function called generalized covariance chosen as: linear J~ xp  ~ x q J, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i 2 i i i i x q JÞ3 , Gaussian eðJ~x p ~x q JÞ , multiquadratic ðJ~ xp  ~ x q JÞ2 þ 1, inverse cubic ðJ~ xp  ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i i i i i multiquadratic 1= ðJ~ xp  ~ x q JÞ2 þ 1, or logarithmic ðJ~ xp  ~ x q JÞ2 logðJ~ xp  ~ x q JÞ. These functions are the most widely used in the fields of medical images and anthropometric models (Trochu, 1993; Holden, 2008). Given a selected function K, the 3  (n+4) coefficients a and b can be found altogether by resolving a square system of linear equations for the n points of the cluster (Trochu, 1993) (nZ5): " #  " j# i  B X K X ¼ ð4Þ i T A ðX Þ 0 0

where hf are chosen functions (similar to the Kriging generalized covariance K) and ~ c f are called the centers of these functions. The centers can be arbitrarily fixed but are more generally selected out of the n points of the cluster (Orr, 1995, 1996). The number m of the functions of the basis is dependent on these selected centers and therefore mrn. The chosen function can be the same (in the same way as for Kriging) or different for each center. Given a selected basis of m functions hf (with their m corresponding centers), the 3  (m+4) coefficients a and w can be found altogether by resolving a system of 3  (n+4) linear equations for the n points of the cluster (nZ5): " #  " j# i  W X H X ð7Þ ¼ A ðCÞT 0 0 with

with 2

Bðn3Þ

bx1 6... 6 6 x 6 bq ¼6 6... 4 bxn

by1 ... byq ... byn

2

3 bz1 ...7 7 7 7 bzq 7 ...7 5 bzn

C ðm4Þ

i

i

xp  ~ x q JÞ. and where the elements (p, q) of the matrix K(n  n) are KðJ~ ¯ i)TB ¼ 0 in Eq. (4) is called non-bias conditions and The additional relation (X deals with the zero expectation of the prediction error (Trochu, 1993). The result for A and B is dependent on the chosen generalized covariance function K. A way to select the most appropriate one is to inverse Eq. (4) for all possible functions and to estimate the Kriging variance (Martin and Simpson, 2005): 1 n

i

i

r2ð33Þ ¼ ðXj  X AÞT K1 ðXj  X AÞ

1 6... 6 6 ¼6 61 6... 4 1

xi1 ...

yi1 ...

xif

yif

... xim

... yim

2 x 3 w1 zi1 6... ...7 6 7 6 7 6 x zif 7 7; Wðm3Þ ¼ 6 wf 6... ...7 4 5 wxm zim

wy1 ... wyf ... wym

3 wz1 ... 7 7 7 wzf 7 7 ... 7 5 wzm

i xp  ~ c f JÞ. and where the elements (p, f) of the matrix H(n  m) are hf ðJ~ In Eq. (7), the non-bias conditions deal with the centers (C¯)T W ¼ 0. The result for A and W is dependent on the chosen basis of functions. These functions and their corresponding centers can be first selected by regularized forward selection (Orr, 1995, 1996), minimizing a cost function including the interpolation errors and a controlled smoothing effect. ij The translation ~ t O and the rotation Rij can be computed from the coefficients of A as for affine mapping and Kriging. Furthermore, the coefficients w are the weights of each 3D distance between the points of the cluster and the centers.

ð5Þ

Therefore, the function K that provides the minimal Kriging variance can be selected. In multivariate interpolations, especially in 3D, the variances on x, y, and z axes can be found in the diagonal of r2 and can be averaged.

2.4. Simulated gait data

 Nominal patterns without added STA: Considering a typical knee flexion–extension curve for a gait cycle, the 5 other degrees of freedom have been computed using a constrained knee model

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(Feikes et al., 2003). The femur and tibia coordinate systems were superimposed in full extension with the origins at the midpoint between the condyle centers. Without added STA, both coordinate systems allow computing directly the nominal knee joint angles and displacements (Grood and Suntay, 1983). Root-mean-square errors (RMSEs) of bone pose with added STA: The trajectories of 8 markers embedded in both coordinate systems (Fig. 1) were constructed, and the STA were simulated by adding a sinusoidal zeromean random noise (Cheze et al., 1995) to the markers’ trajectories (within 3 and 1.5 cm ranges for the thigh and the shank). For 500 sets with added STA, ij t O from full extension i to all the bone pose of the coordinate systems (Rij and ~

Simulated STA:  sin (t + ) GT T3

T1 T2

T5

other positions j) were estimated using Eqs. (2), (4), and (7) and the conventional SVD method (Soderkvist and Wedin, 1993). The knee joint angles and displacements with added STA were compared to the nominal patterns in terms of RMSEs.

3. Results Knee joint angles and displacements with and without added STA (nominal patterns 7 RMSEs) are presented in Fig. 2. The RMSEs were similar for affine mapping and RBF (values around 1.51 and 3 mm). The errors were slightly superior for Kriging (1.51 and 4 mm), and slightly inferior for SVD (1.21 and 3 mm). The typical selected models were linear for Kriging and cubic for RBF (only one basis function with the center at T3, T4, or T5 for the thigh and FH or S3 for the shank).

4. Discussion

T4 LFE

MFE

FH

TT

S1

S3 LM

S2

S4 MM

5-constraint knee model

Bone pose: → Rij and tOij Fig. 1. Illustrations of the simulated gait data (5-constraint knee model, embedded markers in femur and tibia, simulated STA) and bone pose estimation.

In this article, affine mapping, Kriging, and RBF were evaluated in their abilities to remove the STA when estimating knee joint kinematics during gait. On simulated gait data, all performed well compared to the conventional SVD method. The linear 3D interpolation and approximation methods are more complicated to implement. However, they provide additional geometric (homothety, stretch) and time functions that model the deformation of the cluster of markers. Conversely, the conventional SVD method only provides root-mean-square errors that do not assist with quantification of STAs. This additional information may be useful to model and compensate the STA, to help select an optimal marker set (Manal et al., 2000; Peters et al., 2009) and identify the position of an accelerometer or an inertial sensor with respect to the underlying bone. However, the validities and reliabilities of these methods must be evaluated using gold-standard knee kinematic data measured using embedded bone pins or medical imaging.

Fig. 2. Knee angles (extension/flexion, adduction/abduction, and internal/external rotation) and knee displacements (lateral/medial, anterior/posterior, and superior/ inferior): nominal patterns without added STA 7 RMSE of bone pose with added STA estimated by affine mapping, Kriging, RBF, and SVD.

ARTICLE IN PRESS R. Dumas, L. Cheze / Journal of Biomechanics 42 (2009) 2214–2217

Conflict of interest statement The authors have nothing to disclose.

Appendix 1. Supporting Information Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2009.06.006.

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