in vivo measurement of lower back deformations with fourier-transform profilometry

In vivo measurement of lower back deformationswith Fourier-transform profilometry

Abdelmalek Hanafi, Tijani Gharbi, and Jean-Yves Cornu

Through the variation of their cross sections, the in vivo response of lower back muscles to low loadingin an upright seated posture is explored by the Fourier-transform profilometry technique. The maximi-zation of its sensitivity allows us to reach an adequate resolution for the evaluation of low-back displace-ments. Refinements of the fringe pattern analysis permit the minimization of errors. The experimentsshow an asymmetric distribution of the displacement during head rotation movements. Significantcontribution of the lower back to grasping exertions is also observed. These results are thought to beuseful for early defect detection in the lower back. © 2005 Optical Society of America

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

1. Introduction

The lower back plays an important role in humanposture as a force and movement transferring de-vice.1 As such, it undergoes the highest constraintdistributions in the body during the achievement ofdaily life tasks, namely, in seated postures. Thereforeit is subjected to disorders that cause biomechanicalchanges in postural, muscular, and mobility charac-teristics of the spine. Low back pain frequently occursamong the working population and induces consider-able cost in all western countries.2 In general, assess-ment of the changes in the lower back is of interest toseveral medical disciplines such as orthopedics, neu-rology, sports medicine, and occupational medicine.In addition to their electrical3 and mecanomyo-graphic4 activities, the behavior of low back musclesis described by the changes in their cross sections.5 Inthis way, several studies have been devoted to theexploration of the behavior of the lower back underheavy load.6,7 Conversely, only few studies have fo-cused on the effect of low load8 induced in this region

by small mechanical perturbations that resemble theconditions of work in seated postures.

In this context, optical profilometric techniques,namely, those that are based on incoherent illumina-tion, allow for accurate, noninvasive, nontraumatic,and low-cost measurement of the changes and thecontractions of different muscle groups through thevariation of their cross sections.5 For example, themoiré projection has already been used for the recon-struction of the human back profile.9 In this way, wechoose to adapt and implement the Fourier-transform profilometry (FTP) technique10 for the invivo quantification of the out-of-plane displacementsof lower back muscles that are induced by low loads inthe seated posture.

In this paper we first review the principle of theFTP technique and the influence of the highly absorb-ing and scattering properties of the skin on the fringepattern. The measurement setup and its character-istics are described. Finally, we show the results ofour biomechanical experiments.

2. Instrumentation

A. Optical Profilometry

The FTP technique has been widely explored andused for different applications since its introductionby Takeda and Mutoh.10 As depicted in Fig. 1, it isbased on a triangular configuration that consists of aprojection arm, the object under investigation, and anobservation arm. The curved surface of the objectcauses the image of the projected fringe pattern to bespatially modulated. It manifests itself by a variationof the spatial frequency of the fringe pattern around

A. Hanafi ([email protected]) and T. Gharbi are with the Labo-ratoire d’Optique P.M. Duffieux, Institute of Microtechniques ofFranche-Comté, Université de Franche-Comté, Route de Gray, Be-sançon F-25030, France. J.-Y. Cornu is with the Laboratoired’Explorations Fonctionnelles et Biomécanique Appliquée, CentreHospitalier Universitaire, Université de Franche-Comté, Be-sançon F-25000, France.

Received 1 July 2004; revised manuscript received 13 November2004; accepted 13 November 2004.

0003-6935/05/122266-08$15.00/0© 2005 Optical Society of America

2266 APPLIED OPTICS � Vol. 44, No. 12 � 20 April 2005

a central frequency, which corresponds to the projec-tion of the same fringe pattern on a flat surface po-sitioned parallel to the grating and located in thevicinity of the object (see Fig. 1). In this way, theprofile of the object is structurally coded. The inten-sity I of the observed fringe pattern can be expressedas follows:

I(Pc) � I(P) � Is(P)Tg(P)R(P)

�V(P)

2 �1 � cos�2� Dp(P) � 2� �D(P)��, (1)

in an arbitrary point P taken on the surface. Is is theilluminating intensity and Tg is the grating’s trans-mittance function; R denotes the reflectance functionof the muscle groups; V represents the visibility of theobserved fringe pattern; Dp is the projected orderfunction; and �D represents the variation of the pro-jected fringe order function on the reference plane.When both the projection and the observation opticalsystems are located far enough from the referenceplane, the object’s height z, which is considered withrespect to this plane, is related to �D through thefollowing relation10:

�D(P) � �mp

elc

z(P). (2)

Here m is a factor that is inversely proportional to themagnification of the projection optical system and pdenotes the grating’s spacing. The sensitivity of thesystem has to be maximized for the FTP technique tobe appropriately used for the measurement of theamount and rate of changes on the surface of lowerback muscles. These changes are such that the heavyloading induced by weightlifting tasks is found toyield displacements ranging from 1 up to 3 cm in thelumbar region.11 Consequently, we assume that

small loads produce displacements smaller than1 cm. On the other hand, the resolution of the mea-surement system should be higher than �1 �m,which corresponds to the amplitude of the muscularvibrations.4 The maximization of the sensitivity isrequired to offset the variation of the reflectance toreach such a good resolution over an area of120 mm � 100 mm. This area corresponds to a regionthat lies between the upper of the first lumbar ver-tebrae L1 and the lower part of L5. It also includes allthe surrounding muscular groups that support thestresses and tensions.

B. Interaction between the Projected Fringe Pattern andthe Skin

The skin is a multilayered, randomly nonhomoge-neous medium with strong scattering and absorbingproperties.12 It is composed of four main layers: stra-tum corneum, epidermis, derma, and subcutaneousfat. As shown in Fig. 2, the surface skin specularlyand diffusely reflects a portion of an incident illumi-nation and the other portion is transmitted into thelayers.13 After a series of scattering events in thelayers, part of the transmitted light would probablyemerge from the surface to contribute to the overalldiffused light. The scattering properties of the skinare determined by the difference of the refractiveindices of its structural components and their ar-rangement. The collagen fibers in the dermal layers,for example, are reported to be a source of light scat-tering.14 By contrast, the other part of the transmit-ted light is absorbed, namely, in the epidermis andderma. The absorption properties of the skin dependon the distribution of blood vessels, the contents ofchromophores, and water. The optical properties ofthe skin vary significantly with the wavelength. In-deed, the proteins and nucleic acids in the skin tis-sues are highly absorbing of ultraviolet radiationswhereas longer wavelengths in the infrared regionare absorbed by water.13 As for the visible and near-infrared spectral regions, low absorption takes placein the skin.13 The stratum corneum and epidermisare strongly absorbent within this spectral window incomparison with the other layers.12 For all the layers,

Fig. 1. Description of the FTP-based measurement setup.

Fig. 2. Some aspects of light–skin interaction.

20 April 2005 � Vol. 44, No. 12 � APPLIED OPTICS 2267

the absorption is low for radiations ranging between600 and 800 nm.12 The reflectance that is related tothe optical scattering and absorption properties of theskin through an approximative form of the radiativetransport equation varies slowly in this spectralrange. A variation of the reflectance can be generatedby a change in the preferential orientation of thecollagen fibers in the dermal layers.15 Indeed, thesefibers are lying parallel to the skin and are orientedalong the tension lines. Therefore a stretching of theskin caused by body posture can change the directionof minimal scattering. Furthermore, changes in theconcentration of the absorbers driven by differentprocesses such as the oxygenation of tissues duringheavy muscular contractions, exposure to UV light,or hot water are also susceptible to cause a variationof the reflectance.16 Along with the appropriate choiceof the illumination, the low loading conditions of thelower back muscles in the upright posture take intoaccount the minimization of the effects of blood flow,skin stretching, and low temperature.

In addition, the specular reflection depends on therefractive index of the stratum corneum, the angle ofincidence, and the roughness of the skin.13 If thesurface of the skin is smooth enough, the reflectedintensity is maximal in the direction of the mirrorlikereflection and attenuates gradually in all other direc-tions. The specular reflection is susceptible to deformthe fringe pattern as illustrated in Fig. 3(a). Therough aspect of the skin decreases the effect of thespecular reflection on the fringe pattern because ofthe light diffusion in all directions. This quasi-Lambertian aspect is attributed to the structuralchanges caused by pores, follicles, wrinkles, and hair(see Fig. 2). However, their distribution can locallydeform the projected fringes as depicted in Figs. 3(b)and 5.

C. Description of the Measurement Setup

Following the requirements established in Subsec-tion 2.A, a nontelecentric, cross-optical axes setup isbuilt up. Its schematic diagram is shown in Fig. 1. Itis based on the projection of a 10-lines�mm Ronchigrating onto the lumbar region by use of a projectionoptical system composed of an f�1.4, 50-mm focal-length objective lens, and a 200-W halogen lamp. Useof a source that emits light mainly in the visible rangeaims to minimize the absorption in different skinlayers as highlighted in Subsection 2.B. The de-formed fringe pattern is imaged through an f�1.4,25-mm objective lens onto a 752 � 582 array of pixelsin a Sony XC75CE CCD camera operating in CCIR

(Consultative Committee for International Radio)mode. The size of the rectangular active part of thepixel is 8.6 �m � 8.3 �m. The grating is fixed on anadjustable rotation stage that allows the observedfringes to lie parallel to the camera grid lines. Thiscombination is carried by a linear stage that enablesthe monitoring of the spacing of the projected fringesalong with the focal length of the lens. Similarly, theCCD camera is mounted on a rotation stage thatallows us to control the angle between the axes of theprojection and the observation optical systems. Allthese degrees of freedom are used to maximize thesensitivity of the setup and to reach a high resolutionas pointed out in Subsection 2.A. A reference plane isplaced perpendicular to the axis of the camera at adistance of �535 mm. The distance between the pro-jection and the observation optical systems is�220 mm. Because of their f-number, the lenses pro-duce depth of fields that are large enough to generatea good visibility factor at this distance. The CCDcamera is attached to a Matrox Pulsar 45-MHz PCI(peripheral component interface) frame grabber.

D. Phase Retrieval and Calibration Procedures

To retrieve the phase, we successively bandwidthlimited the intensity using a Hamming window andFourier transformed employing the discrete Fourier-transform method.17,18 The Fourier transform of theintensity becomes

�(�) � �(�) �12 �(�) � �(� � �0)

�12 �(�) � �*(�� 0), (3)

where � is the spatial-frequency vector, � is theFourier transform of the complex function1�2 exp�2�j�Dp � �D� and �* is its conjugate, andthe � denotes the convolution product. The intensitydistribution is such that one fringe covers five pixels,which fulfills the sampling theorem.18 Thereby,peaks centered, respectively, at the origin and thecentral frequencies �0 appear separately in the Fou-rier domain as indicated in Fig. 4(b). The centralfrequency �0 corresponds to the projection of the samefringe pattern on a flat surface positioned parallel tothe grating and located in the vicinity of the referenceplane. Spectral contributions of optical imperfections,random noise, and structural noise (e.g., hair, pores)are also contained in this domain. Some of these spu-rious contributions are isolated whereas others canbe located inside the baseband of interest as shown inFig. 5. By isolating, filtering, translating to the originthe baseband that is centered at �0, and finally in-verse Fourier transforming it, we obtain the termIp�P � 1�4 V�Pexp�2�j�D�P� [see Fig. 4(c)]. Thenthe phase variation and the order variation can berecovered to an accuracy that depends on the mini-mization of errors as follows:

Fig. 3. Fringe pattern under the influence of (a) strong specularreflection and (b) structural elements such as pores.


2��D(P) � arctanIm[Ip(P)]Re[Ip(P)]�, (4)

where Im and Re denote, respectively, the imaginaryand real parts of a complex. Accordingly, an asym-metrical rectangular filter window with an adaptiveband limit is used to select the baseband and elimi-nate spurious terms. Its center is first evaluated byan automated procedure that searches frequencieswith the second maximum amplitude in the fre-quency domain of the reference plane’s fringe pat-tern. Then the adequate rectangular band limit isdetermined by searching local minima in the vicinityof the limits of a rectangle predetermined by the ex-perimenter. This refinement is carried forward bysmoothing down data around the localized minima.In this way, the optimum filter window is defined. Itreduces the loss of data caused by inappropriate trun-cation or the inclusion of unwanted noisy compo-nents, which can result in errors that account for halfof the total errors in the retrieved phase.18 The re-maining noise is spread from the center of the re-trieved phase variation window to its edges as aresult of the convolution of the noise and the inversetransform of the optimum filter window.17,18 Thusonly two columns of pixels at the edge of the retrieveddiscontinuous phase variation are affected as illus-

trated in Fig. 4(d). However, structural noise effects,such as the effect of hair distribution (see Fig. 5), canbe removed from the baseband of interest by slightlychanging the projected fringe spacing with the risk ofeffecting the sensitivity of the setup. The discontin-uous phase variation that results from use of the

Fig. 5. Effect of hair distribution on the spectrum of the fringepattern.

Fig. 4. Three-dimensional reconstruction of a lower back profile from its modulated fringe pattern by use of a discrete Fourier-transformmethod for phase retrieval. rd, radians.


numerical inverse tangent of the selected peak func-tion is then unwrapped with respect to a referencepoint10,17 to obtain the continuous profile of the hu-man lower back as shown in Figs. 4(e) and 4(f). Thereference point is arbitrarily chosen in the center ofthe window to avoid the affected edges. The resultingphase profile is then converted into height by thecalibration procedure.

Indeed, a 200 mm � 150 mm white-painted mirroris mounted on a motorized linear stage. It is movedperpendicular to the camera grid along a distance of50 mm in the depth of field with an incremental mo-tion of 100 �m from the initial position of the refer-ence plane. A series of 500 equidistant images aregrabbed and processed to retrieve the distribution oftheir phase variations. Note that, for all the plane’spositions, the retrieved phase variation is unwrappedaccording to the same pixel as a reference point. Fig-ure 6(a) shows the phase jumps that are generatedevery �1.7 mm by the fringe pattern sliding as theplane is moving. Therefore an unwrapping correctionwith respect to the reference point at the initial po-sition is required. The resulting phase variation var-ies linearly over the height in the depth of field for allthe pixels as highlighted in Fig. 6(b). The data curves

for each pixel are then fit by a second-order polyno-mial function that provides three matrices of coeffi-cients. This enables us not only to confirm thelinearity of the curves but also to evaluate the sensi-bility of the system, which is 2� for each 3.079 mm. Aresolution of 10 �m is reached over a volume of106.6 mm � 73.1 mm � 50 mm. The displacementsof a plane were measured with a precision of 4%. Thesystem was characterized by reconstructing thethree-dimensional shapes of both nonliving and liv-ing objects.

3. Biomechanical Experiments and Results

This optical system is then involved in protocols thataim to qualitatively or quantitatively identify anddiscern different classes of the human lower backbehavior under low load. A subject is seated on anorthopedic chair fixed to a platform plane that isinclined by an angle of 10° to the horizontal plane asshown in Fig. 7. With the knees bent under the seat,the inclination causes the upper body to naturallyadopt the erect upright posture by swivelling thelumbar region around the pelvis.19 This is carried outwithout constraining the human central nervous sys-tem. To avoid lateral body movements, the subject isthen attached to the chair at the level of the hips bya belt. The physical implications of such a posture arethat tensions are developed within the back liga-ments making the muscles ready for loading. Headmovements and grasping exertions are thought togenerate both voluntary and synergistic contractionsin the lumbar region.19,20 Accordingly, two main pro-tocols based on these activities are defined to load thelower back muscles. Unlike heavy loading, this lowloading does not require a considerable amount ofblood flow. Thus the reflectance of the lower back isnot affected by the increase in the amount of absorb-ers in the blood. In addition, the temperature is main-tained constant during the tests to avoid any effect onthe reflectance as pointed out in Subsection 2.B. Theoptical system, that is carried by a tripod, is posi-tioned so that the lower back is parallel to the planeof the camera. This relies on a rapid procedure thatconsists in recovering the phase profile of three lineschosen from the top, middle, and low parts of theimage. Before loading, the three displayed profilesare closely identical and symmetric. Any tilt of thecamera grid plane with respect to the lower backwould result in a significant difference between thephase profiles or an asymmetry in the profile of eachline. Despite the fact that respiration generates nosignificant deformation in the lower back structure,21

the images of the initial, the loading, and the unload-ing states are acquired at the end of the exhalationphase. After being processed, the images of the mod-ulated fringe pattern deliver two profiles that areconverted into height [see Fig. 8(a)]. Note that each ofthe resulting profiles is expressed relative to an ar-bitrarily chosen point on the camera grid. Its corre-sponding point on the lower back in the initial stateis subjected to a displacement. This implies that thedetermination of the deformation field induced by theFig. 7. Biomechanical setup.

Fig. 6. (a) Phase jumps generated during the calibration proce-dure and (b) the corrected phase.


loading requires the knowledge of the displacementof at least one point on the object’s surface. The regionof the minima in the profiles represents the outer ofthe lower back vertebra, which is submitted to rigidbody motion only. Given the conditioning of the pro-tocols and the generated small loads, the measureddisplacements are thought to be sufficiently small.This enables us to assume that a point in the regionwithin a pixel before loading does not move laterallyout of this area after loading. The minima are thenlocalized and the unwanted rigid displacement is sub-tracted as illustrated in Fig. 8(b). In Fig. 8(c), theresulting deformation for a line in the deformationfield is displayed. Practically, four successive imagesof the lower back are grabbed at each state (initial,loading, and unloading). These images are processedand compared with each other to determine the sta-

bility of the lower back when subjected to such asuddenly applied loading and held constant thereaf-ter.

A group of 11 subjects without any apparent ab-normalities and with different skin colors took part ina series of five successive tests with a 10-min restperiod. Each test deals with the exploration of thecontractions induced by head rotation movements,head flexion motions, and grasping exercises. Themeasurements that are based on the full head rota-tion movements reveal an asymmetrical distributionof the out-of-plane displacements depending on thedirection of rotation as it appears in Figs. 9(a) and9(b). Indeed, these tests show that the deformation ofthe muscle groups located on the right side of thelower back are higher than those of the left side whenthe head is rotated toward the right side, and viceversa. For all subjects, this distribution pattern wasrepeatedly observed during the tests as illustrated inFig. 10. Those results suggest that a noticeable neu-roplanned strategy is associated with head rotationmovements to control the lower back muscles in theseated upright posture as pointed out by Gracovetsky

Fig. 9. Asymmetrical displacement distribution generated byhead rotation toward (a) the left side and (b) the right side. (c)Displacement values associated with head rotation toward theright side.

Fig. 8. Deduction of the lower back deformation after subtractionof the unwanted rigid motion.

Fig. 10. Repeated asymmetrical distribution induced by head ro-tation movements.


and Farfan.22 The root-mean-square values of theresulting displacements are computed for the areasdelimited by rectangles on each side of the deforma-tion field [see Figs. 9(a) and 9(b)]. In the case of therotation of the head toward the right, typical rmsvalues corresponding to this side of the deformationfield range approximately from 1 up to 3 mm,whereas the left side undergoes small deformations ofless than 1 mm as shown in the example in Fig. 9(c).A similar range of values is obtained in the left rota-tion case. Nevertheless, these values seem to varywith the subject’s body structure. Indeed, one subjectwith large muscular groups generates the smallestdeformation while carrying out these tests. However,the head flexion movements appear not to generatesignificant deformations in the lower back muscularstructures. For almost all subjects, the rms displace-ment values are less than 100 �m. Some of the re-sulting displacements in the deformation field areeven found to be negative. This can be attributed tothe stretches of the skin and not to any muscularcontraction.

The third type of test resembles a monotonic, in-cremental loading. It is based on the grasping proto-col described in Ref. 23. Two angles (180° and 90°) ofthe forearm with respect to the arm are considered.During these tests, subjects are instructed to applyforces successively at 60%, 70%, 80%, 90%, and 100%of their maximum voluntary contraction (MVC) dur-ing 5 s each by grasping a cylindrical sensor whilethey are provided with a visual feedback of the ex-

erted force. As it appears from the induced distribu-tion of the displacement in Figs. 11(a) and 11(b), bothsides of the lower back contract during the graspingexertions at a high percentage of the MVC for all thesubjects. When computed, typical values of the rmsdisplacement values range from 3 up to 6 mm. Fur-thermore, as shown in Figs. 12(a) and 12(b), the lowerback response to the grasping exertions starts around80% of the MVC for the majority of the subjects. Forthe subject with large muscle groups, this thresholdvalue is higher.

4. Conclusions and Discussion

In the present study we explored the contribution ofthe optical Fourier-transform profilometry (FTP)technique to the in vivo study of the seated humanposture. The profiles are retrieved from the modu-lated fringe pattern of the lower back by use of adiscrete Fourier transform associated with asymmet-ric rectangular filters. With these optimum filter win-dows, the errors that affect the edge of the phasewindow are minimized to two columns from five col-umns. An analysis of the light–skin interaction al-lows the determination of the spectrum of theilluminating source and some conditions of the pro-tocols that allow the minimization of the effect of theabsorption. The resulting experimental setup allowsthe accurate exploration of a volume of 106.6 mm� 73.1 mm � 50 mm with a 10�m resolution.

This system is used in a series of biomechanicalprotocols that are based on head movements andgrasping exertions to explore isometric contractionsof the lower back region in the seated upright pos-ture. The explored area in the lumbar region coversthe L1–L5 vertebra, the outer of which is used as arigid body reference to deduce the out-of-plane dis-placements generated by the loading. The full headrotation movements are found to repeatedly produceasymmetrical distributions of the displacement for 11subjects. Significant displacements are located in theright side of the lower back when the head is rotatedtoward the right direction, and vice versa. This sug-gests that there is a correlation between the headrotation angle and the induced displacements in thisregion. The repeatability of the results demonstratesthe reliability of the measurements. The head flexionmovements seem to repeatedly generate no signifi-cant displacements. This can be related to skinstretches and not to the muscular contractions. Onthe other hand, the incremental loading tests arebased on successive grasping exertions at differenthigh percentages of an individual’s maximum volun-tary contraction. Two postures of the forearm withrespect to the upper arm are considered. These testsreveal the contribution of both sides of the lower backto the grasping starting at 80% of the MVC. Overall,this application highlights some strategies developedin the lower back region by the central nervous sys-tem during head movements and grasping exertionsin the seated upright posture. These strategies can beused for the early detection of defects and dysfunc-tions in the muscular structure of this part of the

Fig. 11. Displacement distribution generated by grasping exer-tions at 80% of the MVC when the angle of the forearm to the upperarm is of (a) 180° and (b) 90°.

Fig. 12. Deformation induced by incremental loading based ongrasping exertions when the angle of the forearm to the upper armis of (a) 180° and (b) 90°.


human body. To this end, tests on a wide variety ofsubjects are required to confirm our deformation val-ues and findings. Note that the obtained resolution ishigh enough to detect skin stretches whereas the rmsvalues of the out-of-plane displacements generatedby the contraction of the muscles are higher than1 mm. Reducing the resolution to 200 �m, for exam-ple, would certainly help in exploring bigger areas.By doing so, some markers can be used outside thelumbar region to evaluate the eventual lateral rigidbody displacement and, incidentally, to verify the as-sumption on which a point in the region within a pixelbefore loading does not move laterally out of this areaafter loading.

This research was supported by the French Myo-pathic Disease Association and the French NationalResearch Council.

References1. G. Jayaraman, A. A. Nazre, V. McCann, and J. B. Redford, “A

computerized technique for analyzing lateral bending behaviorof subjects with normal and impaired lumbar spine,” Spine 19,824–832 (1994).

2. J. W. Frymoyer, “Magnitude of the problem,” in The LumbarSpine, J. N. Weinstein and S. W. Weisel, eds. (Saunders, Phil-adelphia, Pa., 1990), pp. 32–38.

3. A. F. Mannion, G. A. Dumas, J. M. Stevenson, and R. G.Cooper, “The influence of muscle fiber size and type distribu-tion on electromyographic measures of back muscle fatigabil-ity,” Spine 23, 576–584 (1996).

4. A. Courteville, T. Gharbi, and J. Y. Cornu, “Noncontact MMGsensor based on the optical feedback effect in a laser diode,”J. Biomed. Opt. 3, 281–285 (1998).

5. T. R. Dangaria and O. Naesh, “Changes in cross-sectional areaof Psoas major muscle in unilateral Sciateca caused by discherniation,” Spine 23, 928–931 (1992).

6. G. B. J. Andersson, R. Örtengren, and A. Schultz, “Analysisand measurements of the loads on the lumbar spine duringwork at a table,” Biomechanics 13, 513–520 (1979).

7. A. Rohlmann, G. Bergmann, F. Graichen, and H. Mayer, “In-fluence of muscle forces on loads in internal spinal fixationdevices,” Spine 23, 537–542 (1996).

8. T. P. Hedman and G. R. Fernie, “In vivo measurement of

lumbar spinal creep in two seated postures using magneticresonance imaging,” Spine 20, 178–183 (1995).

9. T. E. Denton, F. M. Randall, and D. A. Deinlein, “The use ofinstant moiré photographs to reduce exposure from scoliosisradiographs,” Spine 17, 509–512 (1992).

10. M. Takeda and K. Mutoh, “Fourier transform profilometry forthe automatic measurement of 3-D object shapes,” Appl. Opt.22, 3977–3982 (1983).

11. S. Gracovetsky, H. F. Farfan, and C. Lamy, “The mechanism ofthe lumbar spine,” Spine 6, 249–262 (1981).

12. I. V. Meglinski and S. J. Matcher, “Computer simulation of theskin reflectance spectra,” Comput. Methods Programs Biomed.70, 179–186 (2003).

13. B. Wilson and S. L. Jacques, “Optical reflectance and trans-mission of tissues: principles and applications,” IEEE J. Quan-tum Electron. 26, 2186–2199 (1990).

14. I. S. Saidi, S. L. Jacques, and F. K. Tittel, “Mie and Rayleighmodeling of visible-light scattering in neonatal skin,” Appl.Opt. 34, 7410–7418 (1995).

15. S. Nickell, M. Hermann, M. Essenpreis, T. J. Farrell, U.Krämer, and M. S. Patterson, “Anisotropy of light propagationin human skin,” Phys. Med. Biol. 45, 2873–2886 (2000).

16. M. Shimada, Y. Yamada, M. Itoh, and T. Yatagai, “Melaninand blood concentration in human skin studied by multipleregression analysis: experiments,” Phy. Med. Biol. 46, 2385–2395 (2001).

17. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transformmethod of fringe-pattern analysis for computer-based topogra-phy and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).

18. D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-patternanalysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).

19. K. Black, P. McClure, P. Ocs, and M. Polansky, “The influenceof different sitting positions on cervical and lumbar posture,”Spine 21, 65–70 (1996).

20. C. Y. Su, J. H. Lin, T. H. Chien, K. F. Cheng, and Y. T. Sung,“Grip strength in different positions of elbow and shoulder,”Arch. Phys. Med. Rehabil. 75, 812–815 (1994).

21. G. Ferrigno, P. Carnevali, A. Aliverti, F. Molteni, G. Beulcke,and A. Pedotti, “Three-dimensional optical analysis of chestwall motion,” J. Appl. Physiol. 77, 1224–1231 (1994).

22. S. Gracovetsky and H. Farfan, “The optimum spine,” Spine 11,543–573 (1986).

23. A. Hanafi, J. Y. Cornu, T. Gharbi, and A. Courteville, “Anapproach to the evaluation of hand strength and the ability tocontrol force given visual biofeedback,” J. Eng. Mech. 123,1215–1218 (1997).


in vivo measurement of lower back deformations with fourier-transform profilometry

Documents