Tensorial analysis of a Fourier-transform profilometric setup devoted to the evaluation of muscular...

Tensorial analysis of a Fourier-transformprofilometric setup devoted to the evaluation ofmuscular contractions

Abdelmalek Hanafi, Tijani Gharbi, and Jean-Yves Cornu

We explore the potential use of the Fourier-transform profilometry technique in in vivo studies ofmuscular contractions through the variation of muscle-group cross sections. Thanks to a tensorialanalysis of the technique, a general expression of its sensitivity vector is established. It allows derivationof the expression of the resolution and the limit condition imposed by the spatial sampling of the fringepattern. Key parameters that maximize the sensitivity are then simulated. A measurement system isaccordingly built up and characterized. It is then successfully applied to the evaluation of the deformationof the forearm muscles during grasping exertions. © 2005 Optical Society of America

OCIS codes: 120.3890, 070.2580, 120.2650, 120.3930, 120.4120.

1. Introduction

The contraction of a human muscle group generates avariation of its cross section. This variation consti-tutes a physiological parameter that is used for thestudy of any evolution, weakening, or degeneration ofthe muscles,1–3 as in the case of myopathic diseases.It also reflects the growth of skeletal muscle thatresults from sport activities or physiotherapy treat-ments. The variation of the muscular cross sectioncauses displacements on the surface of the body thatcan be measured using a three-dimensional profilo-metric technique. In addition to their high accuracyand precision, optical fringe-projection techniquesproved to be efficient in measuring shapes of smalland large surfaces without being invasive ortraumatic.4–8 This implies that the strategies devel-oped by the central nervous system during the con-tractions will not be affected by the measurementconstraints. This characteristic suits the in vivo stud-ies of the human muscle evolution.

Fringe-projection techniques that are based on in-coherent illumination are particularly of interest asthey are versatile and inexpensive. Moiré projection,for example, has already been used for the profilereconstruction of the human back.9 The present studyis directed toward exploring the implementation ofFourier-transform profilometry (FTP) technique10 forthe in vivo quantification of the out-of-plane displace-ments of different muscle groups. These displace-ments are derived from the three-dimensionalreconstructed profiles of the muscles before and afterloading. The FTP technique has been widely exploredand used for different applications since its introduc-tion by Takeda et al.10 Yet, the design of the appro-priate and optimum FTP-based measurement setupfor a living muscular group depends not only on theentity to be measured but also on the amount andrate of changes on its highly absorbing, scatteringskin. As a result, the maximization of the setup’ssensitivity appears to be valuable to offset the ef-fects of the absorbing and scattering properties ofthe skin. By doing so, a very good resolution of thesystem could be reached, leading to the measure-ment of unwanted entities such as the vibration ofthe muscular fibers and�or the deformations causedby skin stretches. Therefore, a clear formulation ofthe setup’s sensitivity and its limit are required toselect the entity to be measured through the controlof key parameters of the setup. Tensor calculusproved its efficiency in providing accurate analysisof complicated setup configurations for holographicinterferometry11–13 and projection moiré systems.14

A. Hanafi and T. Gharbi are with Laboratoire d’Optique P.M.Duffieux, Institute of Microtechniques of Franche-Comté, Univer-sité de Franche-Comté, Route de Gray, F-25030 Besançon,France. J. Y. Cornu is with Laboratoire d’Explorations Fonction-nelles et Biomécanique Appliquée, Centre Hospitalier Universi-taire, Université de Franche-Comté, F-25000 Besançon, France. A.Hanafi can be reached by e-mail at [email protected].

Received 6 July 2004; revised manuscript received 4 February2005; accepted 20 February 2005.

0003-6935/05/214501-09$15.00/0© 2005 Optical Society of America

20 July 2005 � Vol. 44, No. 21 � APPLIED OPTICS 4501

It allows a clear formulation of the setup’s sensitiv-ity vector.14

In this paper, we first introduce a detailed tensorialanalysis from which the expression of the sensitivityvector of the measurement system is derived. Its res-olution is then investigated, and the expression of themaximal sensitivity is outlined. Based on this inter-pretation, we present the simulated data that lead tothe design of an FTP experimental setup. Finally, weshow its characteristics and how it is applied to theevaluation of the contractions of the forearm muscles.

2. Tensorial Analysis of the Fourier-TransformProfilometry

A. General Expression of the Sensitivity Vector

The FTP technique relies on a triangular system,which consists mainly of projection and observationarms along with the object under investigation, asdepicted in Fig. 1. Due to the geometry of a curvedsurface, the image of the projected fringe pattern isspatially modulated. It results in a variation of itsspatial frequency around a central frequency thatcorresponds to the projection of the same fringe pat-tern on a flat surface. This flat surface is consideredto be positioned parallel to the grating and located inthe vicinity of the object. Thanks to this variation, theprofile of the object is coded.

Tensor calculus represents a valuable mathemati-cal tool that conveys an intrinsic geometrical mean-ing to complex physical configurations.11 It will beused in the following to fully describe the curved-surface coding by a fringe-projection system. Thetransmittance Tg of a linear sinusoidal grating illu-minated by a polychromatic incoherent source can bewritten as follows:

Tg(Pg) �12 {1 � cos[2�Dp(Pg)]}, (1)

where Dp is the projected fringe-order function atpoint Pg taken on the grating, as shown in Figs. 2 and3. This expression applies also to the case of a linear

square-wave grating. Indeed, the transmitted lightthrough the grating is periodically structured. Whenit propagates through the projection optical system,its higher spatial harmonic components are filteredout as a result of the effect of the exit pupil’s opticaltransfer function.15–17 Therefore, the transmittanceof this type of grating can be expressed as a functionof the low-frequency components of its Fourier cosineseries expansion. The diffused light from the object’ssurface is imaged onto the camera grid plane. Theintensity in an arbitrary point Pc of the camera grid is

I(Pc) � I(P) � Is(P)Tg(P)R(P)

�A(P)

2 {1 � cos[2�Do(P)]}, (2)

where P is its counterpart on the object’s surface (seeFig. 3). The point P is also located in the field of viewof both the projection and observation optical sys-tems. Do is the observed fringe-order function, whileR is the reflectance function of the diffuse object,which is the human skin in the present case. Is is theilluminating intensity of the source. Note that theobserved fringe-order function has the same value atpoints P, P1, and Pc. Similarly, the values of the pro-jected fringe-order function are the same at P, P2, and

Fig. 1. Spatial modulation of a fringe pattern in a triangular,nontelecentric system with cross-optical axes configuration.

Fig. 2. Geometrical elements of the projection grating and thearray of sensors.14 The fringe order in Pg is D�Pg� � ��rg · gg��pconsidering that D�P0g� � 0.

Fig. 3. Geometrical configuration of a nontelecentric fringe pro-jection system; S, R are, respectively, the projection centers of theprojection and observation systems, which are located in the exitpupils.

4502 APPLIED OPTICS � Vol. 44, No. 21 � 20 July 2005

Pg. P2 and P1 are the projection of P on the referenceplane along the illuminating and observing direc-tions, respectively. Assuming that the projectiongrating is regular with equidistant lines, the pro-jected fringe order at point P can then be written as

Dp(P) � Dp(Pg) �1p gg · OgPg � constant, (3)

considering that P0g is a reference point on the grat-ing. Og represents the intersection of the grating andthe optical axis of the projection optical system. As itappears in Fig. 2, gg is a unit vector perpendicular tothe lines in the grating plane. p is the spacing be-tween its lines. The expression with boldface letterssuch as OgPg stands for a first-order tensor (i.e., vec-tor), whose tail is Og and head is Pg. The extension ofEq. (3) in the space relative to the projection center Sderives from the fact that triangles �SOgPg� and�SOP3� are homothetic. Indeed, this leads to OgPg

� �lg��lg � Lg��OP3. As shown in Fig. 3, P3 is theoblique projection of points P and P2 on the plane thatcontains point O and is parallel to the grating. OP3can be decomposed as OP3 � OH � HP3, where H isthe normal projection of P2 on the same plane. Itfollows that

OH � NgOP2, (4)

where the second-order tensor Ng � I � hm � hm isthe normal projector on a plane normal to hm, whichis assumed to be collinear to ng. R is the dyadic prod-uct or tensor product. Furthermore, the fact that tri-angles �P3HP2� and �P3OS� are homothetic impliesthat |HP3|�|OP3| � |HP2|��lg � Lg� or, more gen-erally,

HP3 � �(OP2 · hm)

lg � LgOP3, (5)

where (·) is the scalar product. Substitution of Eqs. (4)and (5) into the decomposition expression results inthe following:

NgOP2 � �1 �(OP2 · hm)

lg � Lg�OP3. (6)

In Appendix A, we show that g · Mr � gM · r, whereM is any second-order tensor that represents anoblique or normal projector, and g and r are anyfirst-order tensors. Using this relation in conjunctionwith Eq. (6), the extension of Eq. (3) in the spacerelative to the projection center S leads to

Dp(P) � Dp(P2)

�m�p

[1 � (OP2 · hm)�(lg � Lg)]ggNg · OP2

� constant, (7)

where m � lg��lg � Lg� is a factor that is inverselyproportional to the optical magnification of the pro-jection optical system.

The difference between the observed and projectedfringe-order functions at point P can be expressed asfollows:

�D(P) � Do(P) � Dp(P) � Dp(P1) � Dp(P2). (8)

Considering that the plane of reference and the objectare both located far from the illuminating source andthe array of sensors in the camera, it suggests that P1is located in the close vicinity of P2. In light of thisfact, Eq. (8) is seen as the variation of the projectedfringe-order function on the reference plane at pointP2. According to Eq. (B9) in Appendix B, this varia-tion can be written as

�D �m�p

[1 � (OP2 · hm)�(lg � Lg)]ggNg · P2P1

�mp ggNg · P2P1, (9)

given that �OP2 · hm� �� lg � Lg�. Knowing that thetriangles �PSR� and �PP2P1� are homothetic, we canderive the following connection:

P2P1 �e(�r · km)

lc � Lc � (�r · km) q, (10)

assuming that the reference plane is parallel to thegrid plane of the camera and thereby normal to vectorq. The combination of Eqs. (9) and (10) induces

�D �mp

e(�r · km)lc � Lc � (�r · km) ggNg · q. (11)

If the reference plane is furthermore located far fromthe observation optical system, which can be trans-lated as ��r · km� �� lc � Lc�, then we have

�D � S · �r, S �mp

elc � Lc

(gg · q)km, (12)

as ggNg · q � gg · q (see Appendix C for more details).Using the height z of point P with respect to thereference plane, it also takes the following form:

�D(P) �mp

elc � Lc

(gg · q)z(P) � Sz(P). (13)

S is the sensitivity vector of the optical setup. It liesparallel to the axis of the observation optical system.�r denotes a vector variable P0P giving the positionof the arbitrary chosen point P on the object’s surfacewith respect to a given reference point P0 on thereference plane. In this way, the profile of the objectcan be derived from the intensity distribution on the


camera grid that can be expressed as

I(P) �A(P)

2 {1 � cos[2�Dp(P) � 2��D(P)]}. (14)

It is worth recalling that this study aims at exploringthe displacements generated by the contractions ofhighly absorbing and scattering muscular systems.These out-of-plane displacements are deduced fromtheir three-dimensional profiles before and afterloading.

B. Expression of the Resolution and Measurement Range

Based on this analysis, it is important to bring tolight some key features of the measurement system,namely, its resolution and the measurement range.In doing so, the variation of the difference of theprojected and observed orders is written as

d(�D) � � (�D) · d�r � S · d�r � Sdz. (15)

This suggests that the smallest variation of heightthat can be measured by the optical measurementsystem is achieved when the variation of �D is min-imized and the sensitivity is maximized. As it ap-pears from the expression of the sensitivity vector,the setup has its maximal sensitivity along the direc-tion km of the observation system. In addition to theprojected fringe spacing m�p, the maximal sensitivityvalue depends on the angle between the optical axesof the projection and observation systems as inferredfrom expressions e�lc � Lc and �gg · q�. If the illumi-nating intensity, the reflectance, and the projectedfringe-order functions are slowly varying compared to�D, the differentiation of the intensity distribution onthe camera grid is such that

dI � �A sin{2�[Dp � �D]}d(�D). (16)

In practice, after the spatial and temporal samplings,the recorded intensity is coded into 256 gray levelswhose minimum variation is 1. Therefore, the mini-mum variation of �D is written as

min|�(�D)| �1

� max(A). (17)

This expression implies that a good visibility of theprojected fringe system is a prerequisite for theachievement of a good resolution. To this end, as ittranspires from Eq. (2), the reflectance R of the objecthas to be maximized. The illuminating intensity Is

can also be maximized by focusing a beam with asymmetric profile, such as that of a Gaussian beam,near the object and, thus providing a quasi-uniform,clear illumination.

Besides, the projected order can also be expressedas

Dp �mp

1mc

ggNg · rc � constant, (18)

where rc denotes the vector OcPc taken on the cam-era. mc � lc��lc � Lc� is a factor that is inverselyproportional to the optical magnification of the obser-vation optical system. Assuming that the referenceplane is parallel to the plane of the camera grid, Eqs.(13) and (18) are combined to derive the local fre-quency, which is defined as the gradient of the re-corded intensity’s phase18 � � Dp � �D,

� � � �mp

1mc

gg �1

mcS�� z. (19)

The gradient operator � is considered with respect toa coordinate system related to the camera grid while�� is associated to a reference-plane coordinate sys-tem. As stipulated by the Nyquist sampling theorem,the fringe pattern should be spatially sampled atleast at twice the highest frequency in the interfero-gram. In the case of a parallel projection, this theo-rem can be expressed as

max| · gc| 1

2pc, (20)

where pc is the length of rectangular effective pictureelements (pixels) that compose the camera grid. gc

denotes the characteristic unit vector located in thecamera grid plane and is perpendicular to the widthof the rectangular pixels. Since S � gc, the inequality(20) takes the following forms:

S max(|�� z · gc|) mc

2pc�

mp , (21)

provided that �m�p� �mc�2pc�. This implies that thespacing of the observed fringe pattern should be big-ger than the distance between two successive col-umns of pixels. The inequality (21) sets a limit to theprocess of maximization of the sensitivity vector, andconsequently to the resolution of the measurementsystem. This limit depends on the height variationson the object’s surface in addition to the properties ofthe optics, the grating, and the array of sensors in thesetup. Therefore, some strong surface slopes, such asthose that characterize human faces, are susceptibleto induce an undersampling of some regions of theobserved fringe pattern.

3. Illustration

An illustration of the design of a profilometric systemthat performs precise 3D shape measurements with-out suffering from undersampling and its conse-quences is presented in this section. Using Eq. (13)and inequality (21), the modulus of the sensitivityvector and its limits are computed for two differentobservation lenses, which are thought to be relativelyinexpensive, with two focal lengths (f � 25 mm and


f � 50 mm). For simplicity, we assume that the op-tical axes of the projection and observation systemsintersect at P0 on the reference plane. Considering aprojection lens of 50 mm focal length, the image of agrating is formed at a distance of 580 mm from thefirst principal point of the lens. This distance is suf-ficiently large compared to the height of differentmuscular groups in the body so that the assumptionsmade in the previous section are fulfilled. The fringepattern is imaged by the observation lens whose firstprincipal point is located at 550 mm. Different sizesof the grating periods are taken into account startingwith an initial pitch of 5 lines�mm, while the lengthof the pixel is 10 �m. Figure 4 shows which of thegrating pitches satisfy the condition �2pc�mc� �p�m�. For the 25 mm focal-length lens, the cutoffspacing value is of 26.36 lines�mm, while it is of52.72 lines�mm in the case of the 50 mm focal-lengthlens. The modulus of the sensitivity vector of thesetup is plotted in Figs. 5(a) and 5(b) versus thegrating pitch. The limits of the sensitivity are alsoplotted considering four different angles (10°, 20°,30°, 40°) that represent the variation of the heighton the surface of different muscular groups such asthe forearm muscles and the human back. Accord-ingly, the maximal sensitivity is likely to be ob-tained for grating pitches that range from20 lines�mm up to 25 lines�mm in the first case andfrom 40 lines�mm up to 50 lines�mm in the secondcase. Overall, the projection lens of 50 mm focallength of the system provides the highest values ofthe sensitivity of the system. Yet, the 25 mm focal-length lens allows the exploration of contractions in abigger surface. We then evaluated the resolution ofthe system and demonstrated its ability not only toretrieve the shape of different muscular systems butalso to measure the displacements induced by theircontractions. Accordingly, a nontelecentric, cross-optical axes measurement system that is fully de-scribed in Ref. 19 is built up and characterized. ARonchi grating �10 lines�mm� is projected onto thelumbar region using a projection optical system com-posed of a 50 mm �f�1.4� objective lens and a 200 Whalogen lamp. The deformed fringe pattern is imaged

through a 25 mm �f�1.4� objective lens onto a CCDcamera. A reference plane is placed perpendicularlyto the axis of the camera at a distance of 535 mm andat 575 mm from the projection lens. The observedfringe pattern is distributed such that one fringe cov-ers five pixels, which fulfills the sampling condition(21). The variation of the phase is extracted from theintensity distribution using the Fourier-transformmethod18,20,21 and converted into height distribu-tion. The use of Fourier-transform technique stemsfrom the fact that it relies on the acquisition of onlyone image. It allows us to verify that the measuredentity is not varying in time while maintaining theloading parameters constant. Indeed, a variation ofthe temperature, for example, can cause changes inthe profile of the muscles. Thanks to their focallengths and f-numbers, the lenses produce depth offields that are large enough to generate a good vis-ibility at this distance. This allows the calibration ofa volume of 106.6 mm � 73.1 mm � 50 mm. Thecalibration procedure enables not only the phase-to-height conversion but also the evaluation of the sen-sitivity of the system as shown in Table 1. Theexperimental value of the sensitivity, which is Sexp

� 0.32 mm�1, turns out to be close to the computedone, Stheo � 0.35 mm�1.

To estimate the resolution of the system, a whiteplane with no curvature is then moved from an initialposition, which corresponds to the position of the ref-

Fig. 4. (a) Determination of the grating pitches that satisfy thesampling condition p�m 2pc�mc for two projection lenses of25 mm and 50 mm focal lengths, respectively; pc � 10 �m.

Fig. 5. Computed sensitivity of the setup and its limits are plottedas function of the grating pitch for different variations of the height(10°, 20°, 30°, 40°): (a) 25 mm focal-length projection lens, (b)50 mm focal-length projection lens.


erence plane, with a 10 �m incremental motion alonga distance of 100 �m. Ten equidistant images of theobserved fringe pattern of the plane are then grabbedand processed. Given the accuracy of the system, theresolution is estimated to be of 10 �m, as shown inFig. 6. The evaluation of the accuracy of the systemdepends on the accuracy of the inline translationstage on which the plane is attached. Overall, despitethe fact that the 10 lines�mm grating does not yieldthe maximal value of the sensitivity, the obtainedresolution is thought to be satisfying. Indeed, it al-lows the exploration of the muscles’ contraction with-out including other entities such as the vibration ofthe muscles or the variation in the profile of the skin.

In this way, the 3D profile of a human hand that ischaracterized by its height variations is successfullyreconstructed as illustrated in Fig. 7. Furthermore,Fig. 8(a) shows the observed fringe pattern of a hu-man lower back with dark skin. The reconstructedshape is displayed in Fig. 8(b). More importantly, onehas to ensure that the system is able to measuredisplacements on living objects. In particular, it hasto quantify the changes induced by contractions inthe muscle groups. Consequently, the measurementsystem is used to assess the forearm muscular dis-placements during isometric contractions generatedby grasping performances at different levels. Basi-cally, an apparatus that is fully described in Refs. 22and 23 allows an individual to grasp a cylindricalsensor while provided with a visual feedback of theexerted force. The setup and the protocol are de-signed so that no variation in the length of the mus-cles can take place during the exertion. The forearmis then placed in the calibrated volume perpendicularto the axis of the camera. The subject is asked to

successively exert force at 30% and 40% of his�hermaximum voluntary contraction (MVC), which is de-fined as the maximum force that an individual canexert during a very short period of time. Indeed, themechanical setup, protocols and the conditions of ex-ertions are designed to generate variation in the crosssection of the forearm muscles without causing anylateral displacement or shaking that can affect the

Table 1. Characteristics of Measurement System

Measurement volume �mm3� 106.6 � 73.1 � 50Resolution ��m� 10Sensitivity �mm�1� 0.32Accuracy [%] 4%

Fig. 6. Evaluation of the resolution of the system.

Fig. 7. Three-dimensional reconstruction of a human hand.

Fig. 8. Reconstruction of the lower back profile with dark skin.


measurement. Two images that correspond to theobserved fringe patterns of the forearm muscles inthe two mentioned states are grabbed and processed(see Fig. 9). The difference between the two recon-structed profiles provides a displacement that is acombination of pure deformation induced by the vari-ation of the muscles’ cross section, and the rigid bodymotion of the whole forearm. The region located nearthe elbow does not include any muscular structure.Therefore, the points in this region are mainly un-dergoing rigid body displacement. Its subtraction re-sults in a pure deformation that translates thevariation of the muscles’ cross section, as demon-strated in Fig. 10.

4. Conclusions and Discussion

The present work explores the potential use of theFTP technique in the in vivo studies of muscle con-tractions. Their values are derived from the recon-

structed profiles of the muscle groups that arecovered by a highly absorbing and scattering skin.In this context, the implementation of the FTP tech-nique requires the maximization of its sensitivity.In doing so, we propose a new tensorial analysis ofthe technique that brings out a clear formulation ofits sensitivity vector. It follows that the setup hasits maximal sensitivity along the direction of theobservation system. Its value depends not only onthe projected fringe spacing but also on the anglebetween the optical axes of the projection and ob-servation optical systems. This maximization pro-cess also plays an important role in providing anappropriate contrast of the projected fringe pat-terns and reaching a good resolution. Nevertheless,the maximal sensitivity value has a limit that is setby the sampling theorem. When the spacing of theobserved fringe pattern is greater than twice thesize of a pixel of the camera grid, this limit is ex-pressed in terms of their difference and the heightvariations on the surface.

The resulting simulation based on this analysisproved to be efficient in determining key parametersfor the design of an inexpensive setup with a maxi-mum sensitivity to the amount and rate of changes inthe surface of a living and absorbing object. Accord-ingly, an experimental setup that enables the mea-surement of the out-of-plane displacements in avolume of 106.6 mm � 73.1 mm � 50 mm with a10 �m resolution is built up and characterized. Thedisplacements of a plane are accurately measured.The obtained sensitivity is in agreement with thecomputed one. In addition to the reconstruction of theshape of living objects, the resulting measurementsystem is also used to evaluate the deformation of theforearm muscles that are induced by grasping exer-tions. Yet, unlike the evaluation of the plane’s dis-placements, these measured deformations can bequantitatively validated only if the results are re-peatable under the same conditions. To that end, themeasurement system has to be involved in biome-chanical studies that use rigorous protocols as is thecase in the study conducted in Ref. 19.

Appendix A

We establish the relation gM · r � g · Mr, where gand r are any first-order tensors, while M is anyoblique projector (second-order tensor) that can beexpressed as M � I � �k � n��k · n� · k and n are,respectively, the direction of projection and the vectornormal to the plane of projection. Accordingly, wehave on the one hand

gM · r � gI �k � nk · n · r

� gI �k · gk · n n · r

� (g · r) �k · gk · n (n · r). (A1)

Fig. 9. Variation of the cross-section of forearm muscle groupduring grasping exertions: (a) intensity distribution during agrasping exertion at 30% of the maximum voluntary contraction(MVC); (b) intensity distribution during a grasping exertion at 40%of the MVC. (c) Profiles reconstruction of the forearm musclesbefore and after the contraction.

Fig. 10. Distribution of the out-of-plane displacements generatedby the contraction of the forearm muscles.


Similarly, on the other hand, the same developmentleads to

g · Mr � g · I �k � nk · n r

� g · Ir �n · rk · n k

� g · r �n · rk · n (g · k). (A2)

Thus, Eq. (A1) is equal to Eq. (A2).

Appendix B

The expression of the projected fringe-order functionon the reference plane can be straightforwardly writ-ten as

Dp(r) � aggNg · u(r) with Ng � I � hm � hm.(B1)

Indexes have been omitted for simplicity. Here, a is aconstant factor, gg is a first-order tensor, and Ng is asecond-order tensor. u is a vectorial function of thevector coordinate r of an arbitrary point P on theobject’s surface. In our case, this function takes theform u�r� � r�b�r�, where b is a scalar function of r.Functions u and b are similar in terms of the order ofcontinuity and derivation. Therefore, the differentia-tion of function Dp can be written as

dDp(r) � � Dp(r) · dr. (B2)

� stands for the derivative operator. It is recom-mended to check Schumann’s book11 for more detailson the differentiation of tensors. Using the Eq. (B1),it follows that

�Dp(r) � a{[� � (ggNg)]u(r) � [� � u(r)]ggNg}� aggNg[� � u(r)] (B3)

as � � �ggNg� � 0. Thus,

dDp(r) � aggNg[� � u(r)] · dr. (B4)

The central derivative term of u in Eq. (B4) can bedecomposed in the form

� � u(r) �1

b(r) (� � r) �1

b2(r){[�b(r)] � r}

�1

b(r) I �1

b2(r){[�b(r)] � r}. (B5)

In the present case, function b can be merely writtenas b�r� � cr · hm, where c is a constant factor. Thegradient of b is then given by

�b(r) � c{(� � r)hm � (� � hm)r}� c(� � r)hm � cIhm � chm. (B6)

It follows that

� � u(r) �1

b(r) I �c

b2(r)(hm � r). (B7)

Consequently, the differentiating term becomes

dDp(r) � aggNg � 1b(r) I �

c

b2(r)(hm � r)� · dr. (B8)

Since �hm � r� · dr � �r · dr�hm, it implies that thevector �hm � r� · dr is collinear to vector hm. As Ng isthe normal projector along the direction hm onto aplane normal hm, the second term of the Eq. (B8)vanishes. From which, one can conclude that

dDp(r) � aggNg ·dr

b(r). (B9)

Appendix C

In order to show that ggNg · q � gg · q, let us considerthe development of the first term of this equality,which can be rewritten as ggNg · q � gg · Ngq, follow-ing the previous development. As Ng � I� hm � hm, it follows that

Ngq � q � (hm � hm)q � q � (q · hm)hm. (C1)

Therefore,

ggNg · q � gg · q � (q · hm)(gg · hm) � gg · q, (C2)

since gg � hm.

This work was supported by the French MyopathicDisease Association and the French National Re-search Council (CNRS).

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