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originally published online 11 April 2014 2014 228: 523Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine

Samer AdeebShannon Hill, Esteban Franco-Sepulveda, Amin Komeili, Alexandra Trovato, Eric Parent, Doug Hill, Edmond Lou and

Assessing asymmetry using reflection and rotoinversion in biomedical engineering applications  

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Technical Note

Proc IMechE Part H:J Engineering in Medicine2014, Vol. 228(5) 523–529� IMechE 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0954411914531115pih.sagepub.com

Assessing asymmetry using reflectionand rotoinversion in biomedicalengineering applications

Shannon Hill1, Esteban Franco-Sepulveda2, Amin Komeili2,Alexandra Trovato2, Eric Parent3, Doug Hill4, Edmond Lou4

and Samer Adeeb2

AbstractSymmetry is a trait that has been extensively reviewed, especially clinically, as an indication of ideal geometry and health.Many geometric symmetry assessment techniques rely on two-dimensional measurements that do not account for thethree-dimensional nature of the object. In this article, two methods, reflection and a process termed rotoinversion, acombination of reflection and rotation, are presented as potential methods to assess an object’s deviation from symme-try. With reflection, the geometric models are reflected about a calculated best plane of symmetry. With rotoinversion,the models are reflected about an arbitrary plane and then rigidly translated and rotated to best align the translated andoriginal models. The methods give the same results for bilaterally symmetric objects, but different results for bilaterallyand rotationally symmetric objects. The two methods are applied to assess asymmetry in adolescent idiopathic scoliosistorso geometric models, producing similar results. There was an angle of 0.408� between the normal to the plane ofreflection from the reflection process and the normal from the rotoinversion process and average rotation of 0.067�from rotoinversion. The most appropriate method depends on the purpose of the symmetry assessment and must bedetermined on a case-by-case basis.

KeywordsAsymmetry, scoliosis, reflection, rotoinversion

Date received: 19 November 2013; accepted: 19 March 2014

Introduction

Symmetry is a trait that the human brain is highlyadept at discerning.1 Bilateral symmetry has beenwidely examined to assess clinical and aesthetic con-cerns. Asymmetry has been viewed as less appealingand is often perceived as an imperfection or defect.2

Various analysis techniques have been implemented toassess the symmetry of an object. Euclidian distancematrix analysis (EDMA) has been used to quantify facesymmetry.2 Three-dimensional (3D) models of femalebreasts have been used to analyse symmetry by distanceparameters, volume and by mirroring and overlayingleft and right breasts.3 Jaremko et al.4 used the outlineand area of axial slices from 3D torso models to derivetorso asymmetry indices to assess scoliosis progression.Many of these methods rely on markers and interpreta-tion from the operator; thus, the reliability and abilityto implement these topic-specific methods to a moreglobal application of symmetry analysis are limited.

In this article, we demonstrate two simple techniquesbased on reflection and rotoinversion to assess the torsoasymmetry of 10 patients affected with adolescent idio-pathic scoliosis (AIS). AIS is the most common form ofscoliosis, a condition that results in lateral curvatureand axial rotation of the spine and ultimately

1Department of Mechanical Engineering, University of Alberta,

Edmonton, AB, Canada2Department of Civil and Environmental Engineering, University of

Alberta, Edmonton, AB, Canada3Department of Physical Therapy, University of Alberta, Edmonton, AB,

Canada4Department of Rehabilitation Research and Technology Development,

Glenrose Rehabilitation Hospital, Alberta Health Services, Edmonton,

AB, Canada

Corresponding author:

Samer Adeeb, Department of Civil and Environmental Engineering,

University of Alberta, 3-133 NREF, Edmonton, AB T6G 2W2, Canada.

Email: adeeb@ualberta.ca

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asymmetry of the torso. AIS presents during adoles-cence and is present in 2%–4% of the population,appearing predominantly in females.5 Surface topogra-phy (ST) is a non-invasive technique that is used in theasymmetry analysis of patients with AIS. ST creates a3D image that examines the external effects of scoliosis,a major concern of many patients, harming their qual-ity of life.6 ST images are used to examine torso asym-metry caused by AIS.

For a purely bilateral symmetrical object (an objectin which one plane will divide an object into mirrorimage halves), the reflection of the object around itsplane of symmetry and the original object are identicaland perfectly aligned. If an object is not bilaterally sym-metrical, the ‘best plane of symmetry’ can be obtainedby finding the plane of reflection that would minimize ameasure of the distance between the object and itsreflection; in this article, this will be referred to as thereflection method. Alternatively, a reflection across anarbitrary plane can be performed, and then the reflectedobject may need to be rotated to be best aligned withthe original; this will be referred to as the rotoinversionmethod. One of the asymmetrical clinical examples is

adolescents who have scoliosis.Two methods are explored to find an object and its

reflection image or rotoinversion image. A deviationcolour map showing the contours of the differencebetween the object and its reflection is used to assessthe symmetry or lack thereof of the object. In eachmethod, the ‘best plane of symmetry’ is obtained andthe results from both methods have been compared.Different study cases demonstrate that the two meth-ods can provide the same results or deviate from eachother. These two techniques based on reflection androtoinversion are used to assess the torso asymmetry of10 patients affected with AIS.

Methods/results

Method 1: reflection

Given a cloud of points, the ‘best plane of symmetry’(Psym) is obtained from a subroutine developed inWolfram Mathematica (Mathematica, Version 8.0.4.0;Wolfram Research, Inc.). This subroutine calculatesthe Psym by reflecting the coordinates of the pointcloud over a plane and calculating the distance betweenthe original cloud of points and the correspondingpoints on the reflected cloud of points. For each pointon the original cloud, the corresponding point was theclosest point on the reflected cloud. The best plane ofsymmetry was taken as the plane that minimized thesum of the distances between the original cloud and thereflected cloud. The equation of the plane is recordedfor analysis. This method results in pure reflection.

Method 2: rotoinversion

For the rotoinversion process, the 3D point cloud ofthe original model is reflected across an arbitrary sagit-tal plane (a vertical plane that passes from the front tothe rear of an object dividing it into right and lefthalves). The reflected cloud of points is then aligned tothe original model as accurately as possible usingGeomagic’s ‘Best Fit’ function. This function mini-mizes the deviation between the two point clouds by aniterative least-squares method. The reflected model isrigidly transformed until the sum of squares of dis-tances between the reflected model and the closestpoint on the original model is minimized over the entiremodel. (For each point on the original cloud, thecorresponding point was the closest point on thereflected cloud.) The resulting transformation matrixthat accounts for the reflection and alignment of thereflected model is then output from Geomagic andrecorded for analysis.

The normal to the angle of rotation can be obtainedfrom the Geomagic transformation matrix by compar-ing the resulting matrix with the general form of arotoinversion matrix given by

Q ¼�p21ð1þ cos uÞ þ cos u �p1p2ð1þ cos uÞ þ p3 sin u �p1p3ð1þ cos uÞ � p2 sin u

�p1p2ð1þ cos uÞ � p3 sin u �p22ð1þ cos uÞ þ cos u �p2p3ð1þ cos uÞ þ p1 sin u

�p1p3ð1þ cos uÞ þ p2 sin u �p2p3ð1þ cos uÞ � p1 sin u �p23ð1þ cos uÞ þ cos u

24

35

where p1, p2 and p3 are the components of the normalto the plane of reflection, and u is the angle of rotationaround the normal to the plane of reflection. Note thatp21 þ p22 þ p23 ¼ 1.

A simple excel spreadsheet was used to calculate p1,p2, p3 and u from the Geomagic transformation matrix.These values were calculated such that a comparisoncould be made between the rotoinversion method planeof symmetry and the reflection plane of symmetry anddetermine how large an angle of rotation was requiredto get an optimal result.

In both methods, the 3D compare function onGeomagic can be used to project a colour contour mapon the original model to illustrate the magnitude of theasymmetry at each point on the model. The map repre-sents both the positive and negative displacementsbetween the image obtained under the transformationand the original model. Green represents the closestmatch, shades of blue represent the original modelbeing above the new model and shades of red representthe original model being inside the new. The reference(original) model must be wrapped by converting thecloud of points into a triangulated mesh, while the test(transformed) model can remain as a cloud of points.

Study cases

The two methods can either produce similar or dissimi-lar results depending on the type of symmetry presentin the object. Method 1, reflection, assesses only bilat-eral or mirror symmetry, whereas method 2,

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rotoinversion, can assess pure bilateral symmetry or acombination of bilateral and rotational symmetry.

Figure 1 shows a bilaterally symmetric model, cre-ated by mirroring a half object across the XZ plane.For a bilaterally symmetric model, the methods pro-duce essentially the same results since both assess bilat-eral symmetry. Method 1, reflection, created a Psymwith the normal {0.00, 1.00, 0.00}, the Y-axis. This nor-mal is concurrent with the normal of the mirroringplane when creating the model. As shown in Figure1(a), the reflection is almost a perfect match to the orig-inal since the colour contour map is mostly green. Themaximum deviation is 0.14% of the smallest dimensionof the bounding box (height). Small deviations betweenthe models can be expected since the displacement ismeasured between a triangulated mesh7 and a cloud ofpoints. Method 2, rotoinversion, resulted in 0.00� ofrotation and a normal of {0.00, 1.00, 0.00} to the planeof reflection. A bilaterally symmetric object does notrequire a rotation to align the reflected and originalmodel. As shown in Figure 1(b), the rotoinversion andthe original model are also almost a perfect match asthey both have the normal to the plane of symmetry as{0.00, 1.00, 0.00}. The angle between the normal frommethod 1 and the normal from method 2 is 0.00�. Asshown in this example, and by definition, for a perfectlybilaterally symmetric object, reflection and rotoinver-sion transformations produce the same results.

Figure 2 shows a bilaterally and rotationally sym-metric object, created by mirroring half the objectacross the XZ plane and rotating the mirrored half by180�. The resulting model contains both bilateral androtational symmetry. Method 1, reflection, onlyassesses bilateral symmetry and cannot assess rota-tional symmetry, obvious when reflection is used toassess the model. Figure 2(a) conveys the results of a3D compare between the reflected model and the origi-nal. The normal to the Psym was {20.04, 0.55, 0.83}(Figure 3). Method 2, rotoinversion, assesses both thebilateral and rotational symmetry of the model (Figure2(b)). The model was reflected across the plane withnormal {0.00, 1.00, 0.00}, concurrent with the normalof the mirrored plane. The model was rotated an angleof 179.99� about the normal, which corresponds withthe rotation used to create the model. The rotoinvertedmodel and the original are closely aligned using method2 (root mean square (RMS) of deviation 0.00 mm).

The angle between the normal from method 1 andthe normal from method 2 is 56.29�. Method 1 pro-duces a maximum deviation between the reflected andoriginal model of 45.07% of the smallest dimension ofthe bounding box, whereas method 2 produces a maxi-mum deviation of 0.17%. For the bilaterally and rota-tionally symmetric object, the two methods produceddifferent results. While method 2 can assess the rota-tional and bilateral symmetry present in a bilaterally

Figure 1. 3D colour contour map of a bilaterally symmetric object assessed with (a) method 1, reflection (top) and (b) method 2,rotoinversion (bottom).

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and rotationally symmetric object, method 1 can onlyassess the bilateral symmetry as there is no rotationalcomponent to account for the rotational symmetry(Figure 3). Nevertheless, the purpose of both methodsis to assess symmetry. For a bilaterally symmetricobject, the methods equally gauge the symmetry

present; the same cannot be said if the object containsrotational symmetry. Viewing the contour map of thebilaterally and rotationally symmetric model, method 1shows the model as an asymmetric object, whereasmethod 2 shows the model being symmetric.

Application: scoliosis

ST images of the back of 10 randomly selected AISpatients involved in an ongoing study were obtainedfrom the regional Scoliosis Clinic database and mergedinto a 3D cloud point torso, in the order of 10,000points. ST images had been collected following themethod outlined by Emrani et al.8 and Parent et al.9

The maximum Cobb10 angle (the frontal plane anglemeasured between the vertebrae at the upper and lowerlimits of the curve) in the patients was 36 6 12�, rang-ing from 15� to 53�. This represents mild to severescoliosis.

Both methods 1 and 2 were implemented on the STimages from AIS patients. In each case, the methodproduced a transformed (test) torso on which the ‘3DCompare’ function was used to assess asymmetry in theoriginal torso. This is possible because the human torso

Figure 2. 3D colour contour map of a bilaterally and rotationally symmetric object assessed with (a) reflection (top) and(b) rotoinversion (bottom).

Figure 3. Reflecting the cloud of points of an object whoseone half is a rotoinverted copy of its other half. The reflection isperformed around the best plane of symmetry.

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is not perfectly symmetric, and as such, there are differ-ences between the original ST image and the trans-formed torso points. A contour colour map projectedonto the original torso represents asymmetry by a col-our spectrum that represents the distance between thenew and the original torso (Figure 4). In the spectrum,green (23.0 to 3.0 mm deviation) represents deviationswithin the normal range, present subjects without AIS;shades of blue represent the original torso being above/outside the new torso and shades of red represent theoriginal torso being below/inside the new torso.

Over 10 subjects, the torsos were subjected to anaverage of 0.067� rotation after inversion with a stan-dard deviation (SD) of 0.058� (Table 1). The minimumrotation experienced was 0.026� and the maximumangle of rotation was 0.165�.

Once the normal of the planes of reflection was cal-culated, the angle between the normal obtained from

the reflection and the normal obtained from the rotoin-version was calculated (Table 1). This angle wasobtained to determine the extent to which the rotoin-version method relied on the rotational component tobetter align the transformed torso to the original torso.There was an average of 0.408� between the normalswith an SD of 0.313�. The angles between the normalranged from 0.048� to 1.04�.

The reflected–original torso pair and therotoinverted–original torso pair were analysed forasymmetry using the 3D compare function inGeomagic. The colour contour maps of the reflected–original torso pair and the rotoinverted–original torsopair were visually similar (Figure 4). The average posi-tive and negative deviation (6dev), maximum positiveand negative deviation (6maximum deviation), SD (s)and RMS were calculated by the 3D compare functionin Geomagic. In all 10 cases, the rotoinversion methodproduced results that were the same, or within 5.2%deviations lower than the reflection method in the s

and RMS categories. The added ability to rotate andtranslate the new torso to best align it with the originaltorso allowed for a fit that minimized displacementbetween the two torsos over the entire model.

The 3D compare function in Geomagic was used tocompare the reflected torso created from method 1 andthe rotoinverted torso created from method 2 (Table 2).The deviations are low, considering the size of theobjects being compared. A maximum deviation of7.9 mm over an entire 3D to scale torso is small for thetorsos that fit into an average bounded box of350 mm 3 452 mm 3 235 mm.

The asymmetry analysis for patients with AIS isbeing investigated for several uses. Assessing asymme-try through ST images to monitor scoliosis progressionmay allow for a reduction in X-rays required for

Figure 4. 3D colour contour map of a patient’s torso displaying asymmetry caused by a moderate case of AIS created using(a) method 1, reflection (left) and (b) method 2, rotoinversion (right).

Table 1. Angle of rotation from rotoinversion and anglebetween the normal of plane of reflection from reflection androtoinversion.

Subject Angle of rotation (�) Angle between normals (�)

1 0.033 0.1062 0.031 0.2353 0.165 0.3884 0.117 0.0925 0.029 1.0386 0.038 0.3767 0.165 0.6588 0.026 0.4709 0.032 0.04810 0.032 0.670Average 0.067 0.408SD 0.058 0.313

SD: standard deviation.

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patient follow-up. The external features of AIS, such asshoulder asymmetry, scapular asymmetry, waist asym-metry and lateral deviation, may help to determine theinternal deformities present and how these are affectedby treatment. Patients with AIS can use spinal bracesto prevent progression of spinal curvature. The colourcontour maps of the torso can assist in identifying areasmost affected by AIS and requiring more attentionfrom a brace. Asymmetry analysis can also be used inplastic surgery to target asymmetrical areas of thebody, for example, in breast reconstruction.

Conclusion

Two methods were examined to assess asymmetry in anobject. Both methods require minimal time and exper-tise or training, only requiring the appropriate softwareand code for the reflection method. Depending on thegeometric composition of the object and type of sym-metry present, the methods can produce the same ordifferent results. The rotoinversion method can allowfor better alignment since it assesses both bilateral androtational symmetry. The rotoinversion method alsohas a greater ease of use, for it only requires one soft-ware application for transformation and analysis. Therotoinversion method uses built-in functions inGeomagic, making it available for use by anyone withthe software. The reflection method requires customcodes to calculate the best plane of symmetry and com-pare the models. This method does not allow rotationor translation to align the reflected and original model.

Both methods have shown to accurately assess sym-metry of a 3D model. It is to the discretion of the userif they want to assess bilateral symmetry or a combina-tion of bilateral and rotational symmetry. If the user islooking at a skew symmetric object, rotoinversion isable to assess this symmetry, whereas if the user is tolook at bilateral symmetry, either method is sufficient,as the angle of rotation produced by the rotoinversionmethod will be 0 and as such the methods will producethe same result. The complication in method choicearises when the user needs to determine whether or notthe rotational component between the two objects con-tributes to or is irrelevant to the symmetry. As such,depending on the task at hand, one method could bebeneficial over the other, or a combination of the twomethods.

Although the rotoinversion method allows for betteralignment of the transformed and original torsothrough rotation, this appears to have little to no effecton the results of AIS torso asymmetry analysis. Thereis less than 0.2� of rotation in the rotoinversion methodand the normal created from each method is within1.04� of each other.

While the rotoinversion method can produce betteralignment between the rotoinverted and original torso,it cannot be concluded that this method more accu-rately assesses torso asymmetry, for rotation of thetorso is often a strong contributor to external deformitycaused by AIS, and as such when the rotoinversionmethod is used, it may be important to look at the angleat which the torso is rotated to see whether this corre-lates to an internal rotation caused by the scoliosis.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This study was partially supported by funding from NaturalSciences and Engineering Research Council of Canada(NSERC) and from the Scoliosis Research Society (SRS).

References

1. Wagemans J. Characteristics and models of human sym-

metry detection. Trends Cogn Sci 1997; 1: 346–352.2. Tomkinson GR and Olds TS. Physiological correlates of

bilateral symmetry in humans. Int J Sports Med 2000;

21: 545–550.3. Sun S-P, Hsu K-W and Chen J-S. The evaluation of

female breast symmetry by calculating related physics

parameters. J Med Biol Eng 2007; 27: 87–90.4. Jaremko JL, Poncet P, Ronsky J, et al. Indices of torso

asymmetry related to spinal deformity in scoliosis. Clin

Biomech 2002; 17: 559–568.5. Rogala EJ, Drummond DS and Gurr J. Scoliosis: inci-

dence and natural history. A prospective epidemiological

study. J Bone Joint Surg Am 1978; 60: 173–176.6. Lai SM, Asher M and Burton D. Estimating SRS-22

quality of life measures with SF-36: application in idio-

pathic scoliosis. Spine 2006; 31: 473–478.7. Geomagic Support Center. Glossary, http://support1.geoma-

gic.com/ics/support/glossary.asp (accessed 8 October 2013).8. Emrani M, Kirdeikis R, Igwe P, et al. Surface recon-

struction of torsos with and without scoliosis. J Biomech

2009; 42: 2200–2204.

Table 2. Geomagic 3D compare results by comparing new torsos from reflection and rotoinversion.

Average + (mm) Average 2 (mm) Maximum (mm) Minimum (mm) SD (mm) RMS (mm)

Average 1.0 21.0 3.5 23.8 1.2 1.2Maximum 1.6 21.7 7.9 27.3 2.3 2.3Minimum 0.3 20.3 0.7 20.7 0.4 0.4SD 0.6 0.6 2.4 2.4 0.7 0.7

SD: standard deviation; RMS: root mean square.

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at UNIVERSITY OF ALBERTA LIBRARY on August 14, 2014pih.sagepub.comDownloaded from

9. Parent EC, Zhang PQ, Hill D, et al. Sensitivity-to-changeof full torso surface topography measurements in adoles-cents with idiopathic scoliosis and a main thoracic curve.Stud Health Technol Inform 2012; 176: 484.

10. Cobb RJ. Outline for study of scoliosis. In: Edwards JW(ed) Instructional course lectures, the American Academy

of Orthopaedic Surgeons. St Louis, MO: C.V. MosbyPublishing Company, 1948, pp.261–275.

Hill et al. 529

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