Int J Interact Des Manuf (2012) 6:233–240 DOI 10.1007/s12008-012-0152-1
ORIGINAL PAPER
An iterative method to detect symmetry line falling far outside the sagittal plane L. Di Angelo · P. Di Stefano · A. Spezzaneve
Received: 24 January 2012 / Accepted: 7 March 2012 / Published online: 31 March 2012 © Springer-Verlag 2012
Abstract In this work, a new technique for symmetry line detection for asymmetric postures, which can not be investigated with the other methods presented in the literature, is proposed. It evaluates the symmetry line by means an adaptive process in which a first attempt is modified step by step until the solution converges to the best estimation. The method here proposed is validated by analysing four different asymmetric postures in which the spine lies far outside the sagittal plane, having as reference the cutaneous marking. Results are analysed and critically discussed. Keywords Raster stereography · Back shape analysis · Symmetry line · Posture prediction · Anatomical landmarks 1 Introduction and related works Studies, described in published literature, demonstrate that the curvature of the spine is one of the most important characteristics for determining posture, intervertebral disc loads and stresses and musculoskeletal injuries [1]. Such considerations have led ergonomics researchers to pay particular attention to the posture analysis and prediction for different worker’s typologies [2]. Nowadays, posture and vertebral column detection can be based on modern techniques which perform the 3D geometric scanning of the back [3–5]. These techniques are not L. Di Angelo · P. Di Stefano (B) · A. Spezzaneve Department of Mechanical Engineering, Energy and Management, University of L’Aquila, L’Aquila, Italy e-mail:
[email protected] L. Di Angelo e-mail:
[email protected] A. Spezzaneve e-mail:
[email protected]
invasive, give accurate measurements with acceptable costs, perform posture detection not conditioned by the instrument and offer new prospective for posture analysis. Nevertheless, the geometric data relating to a subject’s back, obtained by means of a 3D geometric scanning (point cloud), cannot be directly used for posture analysis. In order to recognize the characteristic elements of the spatial configuration of the vertebral column, it is necessary a proper data processing. Typically, this configuration is evaluated by the symmetry line, that is the 3D curve joining the external position of the vertebral apophyses. The geometric feature that is associated with the position of the vertebral apophyses is the bilateral symmetry; the position of the spine can be deduced from the symmetry line taking properly into account the thickness of the soft tissues [4]. The first approach for the determination of symmetry line based on the three-dimensional scanning of the back, was proposed by Turner-Smith [6]. Their method performs the acquisition of the 3D position of landmarks located on the vertebrae spinal apophyses by means of manual palpation. The symmetry line is determined as the broken line which joins the barycentre of each marker. A similar approach was used by Pazos et al. [7], and by Bergeron et al. [8]. In order to make a 3D parametric description of the symmetry line, Sotoca et. al [9] approximate the acquired locations of the markers with a polynomial curve. The principal limit of these methods is due to the necessity of applying markers. The other methodology, which has been already validated in clinical field, was proposed by Drerup and Hierholzer [4]. The back surface is sliced with planes perpendicular to vertical axis. For each slice curve the point representative of the position of the spine is associated to the minimum value of lateral asymmetry function based on the evaluation of the Mean and Gaussian curvatures.
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Huysmans et al. [10] propose a complex methodology based on the analysis of curves that are obtained sectioning with horizontal planes the surface of the acquired back. This methodology requires some coefficients (weight factors) evaluated on average individuals and that cannot take into account the variety of real subjects. Thériault et al. [11] associate the spinal apophyses to the back valley which is identified as those areas having a strong negative mean curvature of the back’s surface. This back valley cannot identify the symmetry line in any case. For example in the spine-cervical tract the symmetry line is associated with regions characterized by a strong positive mean curvature. In order to analyse the back sectioning profiles, Santiesteban et al. [12] define the asymmetry function in terms of the Gaussian first derivative of the principal directions of slope angle tangent. In a previous work [5] the authors introduced an original method (in the following HSA: Horizontal Sectioning Analysis), based on the analysis of the symmetry in the orientation of the normal vectors of horizontal sections of the back surface. Since the symmetry line representation gives information about the posture correctness assumed by the subject, it becomes important displaying postures that are closer to those of workers in the workplace. All the above mentioned methods evaluate the symmetry line by analysing horizontal planar sections. The horizontal slicing methods perform good results only for symmetrical upright or sitting postures, but workers postures can be identified in a wider set of spine configurations. In order to analyse postures or movements which give rise spine configurations outside sagittal plane, the methods presented in the literature are not adequate. In order to overcome the limitations of the previously mentioned methods, a new method (called APA: Asymmetric Posture Analysis), based on an adaptive process is presented in this paper. The method performs the evaluation of the best set of sectioning planes suited to identify symmetry line. It is tested in some real cases and compared with the HSA, having as reference the cutaneous marking. In particular test cases refer to the analysis of twisted and laterally flexed postures.
2 Definitions Three basic reference planes are typically used in human anatomy: sagittal, coronal, and transversal. In the case of symmetrical upright posture sagittal plane divides the human body into two specular portions; coronal plane divides the body into dorsal and ventral portions; transversal is a plane orthogonal to the others and, it is a horizontal plane. By the intersection of these planes it is possible to identify the three reference axes: longitudinal axis (obtained by the inter-
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section of the sagittal and the coronal planes), coronal axis (formed by the intersection of the sagittal and the transversal planes) and sagittal axis (resulting by the intersection of the coronal and the transversal planes). In the case of asymmetric postures such as twisted and laterally flexed, it is not possible to identify the previous mentioned reference frame. So that a new reference system is used. This global reference system {O, x,y,z} is defined by quasi-coronal, quasi-sagittal and quasi-transversal planes. The quasi-coronal plane is the plane best fitting the point cloud of the acquired back. The origin of {O, x,y,z} is the barycentre O (Fig. 1) of the orthogonally projected points on the quasi-coronal plane. The direction of the quasi-longitudinal axis z is calculated as the principal axis of inertia, associated to lower inertia moment, of the projected points. The x axis, called quasi-coronal axis, is associated to the higher inertia moment. The third axis y, called quasi-sagittal axis, is perpendicular to the previous two axes. The global reference system is affected by the acquisition process, the extension of the acquired area of the back and its location around the symmetry line of the subject. In this work in order to analyze asymmetric postures, also a local reference system {OL (t), ξ L (t), L (t), ζ L (t)} is introduced. This local frame is defined by the local coronal axis (ξ L ), the local sagittal axis ( L ), and by the local longitudinal axis (ζ L ). Local reference system can be associated to the Frenet – Serret frame of the symmetry line: ζ L (t) is the tangent of the symmetry line in OL (t), L (t) is the normal of the symmetry line in OL (t) and ξ L (t) is perpendicular to the other two. So, ξ L (t) and L (t) define the local transversal plane (π (t)), ζ L (t) and L (t) define the local sagittal plane, ζ L (t) and ξ L (t) define the local coronal plane. The origin (OL ) is a point of the symmetry line. Since the symmetry line is defined by a parametric curve, the local frame is a function of the same parameter of the curve (Fig. 2): (1) OL , ξ L , L , ζ L = Φ(t).
3 The method for symmetry line detection for asymmetric postures The method here proposed consists of the following steps: 1. 3D acquisition of subject’s back; 2. Estimation of the symmetry line of the first attempt (C0 ); 3. Refinement of the symmetry line estimation (C f ). 3.1 3D acquisition of subject’s back The geometry of the back is acquired using a 3D geometric optical scanner (www.scansystems.it) which gives the
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Fig. 1 Source point cloud and global reference system (colour figure online)
3.2 Estimation of the symmetry line of the first attempt
ζL(t3) ΟL(t3)
ζL(t2) ΟL(t2)
ψL(t3) ξL(t3)
ΠL(t3)
ψL(t2) ξL(t2)
ΠL(t2)
The first attempt of the symmetry line evaluation (C0 ) is performed by the HSA which analyzes the slice profiles obtained by sectioning the tessellated back by a set of planes perpendicular to the quasi-coronal axis z. For each slice profile the symmetric point is evaluated as that point having the maximal value of the index SI [5]: S I = e−AF(ν,L 0 )
ζL(t1) ΟL(t1)
ψL(t1) ξL(t1)
(2)
where ΠL(t1)
• • Fig. 2 Local reference systems (colour figure online)
• •
possibility to acquire the whole geometry of the back by a unique scanning. The back acquisition must be performed completely and symmetrically to the vertebral column as possible. After the acquisition, the point cloud is smoothed with a Gaussian filter and, then, it is tessellated and transformed into Eulerian geometric models connecting each point by triangles. This filtering is not a critical phase of the method but it is useful to reduce outliers and large noise.
•
L 0 /2 ˆ ¯ AF(ν, L 0 ) = L10 u=0 N(ν, u) − N(v, L 0 )2 du; L 0 is the length of reference, that is the measure of neighbour of p explored to check the symmetry; n(ν + u) + n(ν − u) ˆ ; N(ν, u) = n(ν + u) + n(ν − u) L 0 /2 1 L 0 /2 1 ¯ N(v,L 0 ) = L 0 u=0 N x (ν, u)du; L 0 u=0 N y (ν, u)du; L 0 /2 1 N (ν, u)du ; z L 0 u=0 n(ν + u) and n(ν − u) are the normal unit vectors evaluated at opposite points of the slice profile (p(ν + u) and p(ν − u)), lying symmetrically with respect to the point p(ν) (see Fig. 3).
To estimate the normal unit vectors, the back surface is locally approximated by a biquadratic Bézier surface by analyzing, for each point p of the slice profile, the 3-ring neighbourhood
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• the most symmetric point P s(i,k) of Γi,k , evaluated by using the index SI [5]; • if wi ,k > wi ,k−1 → PS∗ (i) = PS(i,k) , being wi ,k the symmetry index value of P S(i,k) .
Fig. 3 Terms definition of the symmetry index (colour figure online)
around p. This approximation provides a local smoothing of the data, reducing the effect of noise in differential geometrical properties evaluation. The symmetric points are then approximated by a 3D parametric curve typically used in the spinal process approximation [8]: x=
3
ai t i ; y =
i=0
3
(1) the symmetry line passes through the most symmetric points of the back; (2) Π (t) sections the back in the most symmetric profiles. These hypotheses could not be verified in any real cases, but, however, the presented method can just evaluate the geometrical symmetric feature of the scanned back surface.
[bi cos (it) + ci sin (it)]; z = t. (3)
i=0
The coefficients ai , bi and ci are calculated by a least square method weighted by the symmetry index SI [5], so that the curve (3) fits the best estimation of the back symmetry. In this way local errors in symmetry line identification are filtered. In some cases, the position of the symmetry line could be associated to relative and not to absolute maximum values of the symmetric index. In order to find an appropriate sequence of symmetry points, all the possible combinations of the sequences of candidate points (one point for each slice) are approximated by (3) and, then, analysed. From all the symmetry line candidates, the symmetry line of the first attempt Co is that one which minimises the sum of its distances from the approximated points. 3.3 Refinement of the symmetry line estimation Once obtained the initial estimation of the symmetry line Co the refinement algorithm, shown in Fig. 4, is applied. Let be TB the tessellated back and Ps∗ the array of symmetry points obtained by the k-1-th attempt of the symmetry line estimation (Ck−1 (Ps∗ )). For each Ps∗ (i) (where i = 1, . . ., number of slices) at the k-th step, the refinement algorithm determines, in sequence: • the plane Πi,k : P S(i,k−1) ⊂ Πi,k and Πi,k ⊥Ck−1 ; • the profile Γi,k : Γi,k = Πi,k ∩ T B;
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At the end of the k-th step, the symmetry line Ck (Ps∗ ) is reevaluated. The process stops when all the symmetry indexes of Ps(i,k) (where i = 1,…, number of slices) are lower than the corresponding values of Ps∗ or when the number of iterations is equal to i max . The latest evaluation of the symmetry line is assumed as C f . This refinement algorithm can be considered as an optimization process that finds the set of planes Π (t) for which the obtained profiles Γ (t) have the maximum symmetry of the back. This process converges to the best symmetry line estimation under the following double hypothesis:
4 Experimentation and results The APA has been implemented in an original software, coded in C++. The performances of the method have been evaluated in the symmetry line detection of 20 subjects, students aged from 20 to 22. In order to identify a reference evaluation of the symmetry line, markers (white adhesive disks with diameter of 10 mm) have been applied in correspondence of the vertebrae spinal apophyses, by mean of tactile individualization. The markers have been applied in each analysed posture, so that the errors, due to slipping of the skin on apophyses and the rotation of the vertebrae are avoided. For each analysed subject, the postures reported in the Table 1 are considered. The vertebral column has been divided in two tracts: lumbar and thoracic. In each tract the errors have been evaluated as distances between the estimated symmetry line and the barycentre of the markers. In Table 2 the errors obtained in the presented experimentation are reported for HSA and APA. By analyzing the results is possible to highlight that in the case of laterally flexed postures, APA improves the symmetry line estimation of about 8÷10 %. In Fig. 5 the HSA method is compared with the APA in three cases of laterally flexed postures. The improvement introduced with APA is due to the refinement which chooses sectioning planes that better supports the back symmetry. Instead, there is not significant difference in the case of
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Fig. 4 Refinement algorithm (colour figure online)
twisted postures in which the better slicing planes are substantially perpendicular to the quasi-longitudinal axis.
5 Conclusions In this paper a new method for symmetry line detection in asymmetric postures is presented. APA is based on an iterative process that, starting from a symmetry line of the first attempt, step by step converges to the best estimation. The method is here validated analysing the twisted postures (left, right) and laterally flexed postures (left, right) of 20 subjects. In the analysed cases, it is observed that the method converges after not more than 3 refinement steps. APA is compared with
a previous method (HSA), also developed by these authors, to symmetry line detection, having as a reference the cutaneous marking. In particular, for both analysed methods, the errors are measured as the distances between the estimated symmetry line and the barycentre of the markers. APA satisfactorily performs estimation of the symmetry line furnishing the capability to analyse asymmetric postures. Furthermore, it shows promising characteristics to analyse subject’s backs affected by deforming pathologies of the spine and it is also suited to analyse athletic gestures which involve torsion–flexion of the torso. As it was expectable, the method is strongly affected by the body morphology of the subject: gibbosities or other alterations can generate false symmetries that affect the symmetry line detection process. In addition, the refinement
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Table 1 Protocols and explicative figures of the postures here considered
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Posture Twisted Left
Right
Laterally flexed Left
Right
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Protocol The posture is obtained by the maximum torsion of the trunk done left (in the following PTL ). The head follows the twist of the torso. The subject keeps his feet firmly fixed on the ground to prevent a rotation of the stool.
The posture is obtained by the maximum torsion of the trunk done right (in the following PTR ). The head follows the twist of the torso. The subject keeps his feet firmly fixed on the ground to prevent a rotation of the stool.
The posture is performed left (in the following PFL ). The limit of the inclination is determined by the condition of detachment of the buttocks from sitting. The head follows the movement without forcing the slope of the spine.
The posture is performed left (in the following PFL ). The limit of the inclination is determined by the condition of detachment of the buttocks from sitting. The head follows the movement without forcing the slope of the spine.
Exemplificative figure
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Table 2 Mean, standard deviations and maximum error values and gain percentage after refinement
Posture PTL
PTR
PFL
PFR
mean std maximum mean std maximum mean std maximum mean std maximum
overall 6,97 3,16 11,55 5,95 3,37 12,92 4,98 3,24 8,18 5,86 2,29 9,75
markers
HSA lumbar 7,15 3,37 14,80 5,20 2,91 9,07 5,28 3,45 14,34 5,60 2,91 10,11
thoracic 6,86 3,79 10,56 6,45 3,18 18,14 4,78 3,79 9,60 6,04 3,18 11,23
slices of 1st attempt
APA lumbar 7,26 3,19 14,55 5,17 2,06 8,37 4,86 3,19 9,23 5,12 2,06 10,75
overall 7,01 3,37 13,38 5,87 2,74 12,53 4,59 3,21 8,08 5,25 2,16 8,71
final slices
Gain percentage thoracic overall lumbar thoracic 6,84 -0,57% -1,53% 0,29% 3,79 -6,65% 5,34% 0,00% 12,59 -15,86% 1,69% -19,31% 6,34 1,35% 0,62% 1,76% 2,91 18,69% 29,07% 8,49% 17,04 3,06% 7,75% 6,08% 4,41 7,78% 8,03% 7,59% 3,50 0,93% 7,79% 7,65% 9,21 1,29% 35,60% 4,09% 5,33 10,49% 8,61% 11,65% 2,91 5,68% 29,07% 8,24% 9,47 10,64% -6,34% 15,66%
symmetry line by HSA
symmetry line by APA
Fig. 5 Results for HSA and APA methods for laterally flexed postures (colour figure online)
process here proposed needs a good initial estimation of the symmetry line to converge to a good estimation.
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