Correction for rigid body motion in moire topography
A. T. Andonian
The effect of different types of rigid body motion in moire topography is discussed. Procedures to eliminatesuch movements are described. A technique to assist measurement of changes in human bodies by meansof moire contouring is presented from the standpoint of practical applications.
1. IntroductionA moire pattern results whenever two similar but not
quite identical arrays of lines are arranged so that onearray can be viewed through the other. The geometricinterpretation of the moire patterns was first publishedby Tollenaar.1 Since then several authors have de-scribed the application of moire in strain analysis,2-6vibration analysis,7 and in contour mapping. 8' 9
An easy way to describe a 3-D shape is to draw thecontour lines. A shadow moire technique for contourmapping involves positioning a grating close to an objectand observing its shadow on the object through thegrating.1 0-'2 The size of the object to .be mapped,however, is restricted to the grating size.
A method proposed by Oster and Nishijima13 to findthe depth difference of two surfaces was applied onliving bodies by Takasaki.9 Moire contours producedby this method show the difference between the twocontour line systems, and they represent the total out-of-plane displacement of the object including local de-formations as well as rigid body translations and rota-tions. To eliminate the latter two the body has to berigidly fixed. Hence the method of subtractively en-gaging two contour line systems 9 is useful if the objectis fixed, but it is almost irrelevant if deformations in aliving body resulting from muscle activities aresought.
This paper will describe a method for separating localdeformations from resultant displacements if small-scale rigid body movements are allowed in thesystem.
II. Generation of Moire Patterns with PointIllumination and Point Observation at Finite Distance
Referring to Fig. 1, an object to be mapped is placednext to an equispaced grating with line spacing so. Thex axis is taken along lines of the grating which lies in thex -y plane, and the z axis is taken perpendicular to thegrating surface. The moire pattern in the small areaaround P on the object can be represented by the fol-lowing light intensity equation9:
Ip = (IoIK)t1 + cos27r(yQ - YR)/SO1, (1)
where Io is the intensity of illumination and K is theattenuation constant. Considering similar trianglesPRQ and PCL, we get
(YQ -YR)/d = h/(1 + h). (2)
Equation (1) can then be rewritten as
Ip = (Io/K)J1 + cos27r[dh/so(I + h)J, (3)
which is a nonlinear periodic function of h for given Io,d, 1, and so. Light intensity around P reaches itsmaximum value as the argument of the cosine term ofEq. (3) approaches an even multiple of r or
2rdhso(+h = 2Nir,
The author is with University of Illinois at Chicago Circle, De-partment of Materials Engineering, P.O. Box 4348, Chicago, Illinois60680.
where N is an integer and it is known as fringe order inmoire analysis. In a moire photograph it is possible todetermine the order of any fringe experimentally.1 0
The depth of the Nth bright line can be obtained byrearranging Eq. (4) as follows:
hN = Nl/(d/so - N). (5)
The moire pattern obtained by point illumination andpoint observation does not have a constant fringe in-terval as a result of the nonlinearity of hN with respect
Fig. 2. A 3-D surface with a rectangular x-y projection.
Fig. 1. Shadow moire setup.-
to N. Moreover, the moire fringes are affected bycentral perspective geometry.
In the following sections the effect of different typesof rigid body motion will be discussed, procedures toeliminate such movements will be described, and atechnique to assist measurement of changes in humanbodies by means of moire contouring will be presentedfrom the standpoint of practical applications.
111. Effect of Rigid Body Motion In Moire Topography
A. Parallel TranslationIf an object is displaced on a plane parallel to the
grating, the moire pattern observed (or photographed)at a fixed point changes slightly due to central per-spectivity. Surface profiles generated from thesecontour lines, however, remain unaffected as long ascentral perspectivity is properly accounted for (seeAppendix A).
B. Translation of Object Perpendicular to GratingEquation (5) can be rearranged to give the Nth fringe
order as
N= hNdsO(I + hN)
(6)
When hN is increased by an amount Ah, N increasesproportionately by an amount AN such that
N + AN = (hN + Ah)d/so(l + hN + Ah). (7)
Although every point on the object experiences the samedisplacement due to the nonlinear nature of Eq. (7), thecorresponding change in fringe order is not uniform.This nonuniformity effects the 3-D surface generationprocess if the absolute orders of moire fringes are notdetermined experimentally.' 0
C. Rotation of Object About an Axis Perpendicular toGrating
The amount of rotation can be found by measuring,in the moire photograph the rotation of a straight linejoining two target points. Correction of moire patternis needed, however, to compensate for the change incentral perspectivity if the rotation is about an axisother than the axis of perspectivity (see Appendix A forfurther explanation).
D. Translation and Rotation About x and y AxesFigure 2 shows a 3-D surface bounded by four curves
intersecting at points A, B, C, and D. If the projectionsof AD and BC on the grating surface are parallel to thex axis, the rotation of the surface about the x axis willnot effect projected lengths of AD and BC. Similarly,there will be no change in the projected length of AB orCD when the surface is rotated about the y axis, pro-vided that the projections of AB and CD are parallel tothe y axis.
The amount of translation along any direction orrotation about any axis can be determined by measuringthe displacements of a set of target points marked onthe object. Consider Fig. 2. Let a, b, c, and d be thedisplacements of points A, B, C, and D along the z axis,respectively. From Eq. (5)
a = NAf i NAild/so - NAf d/so-NAi
NBl NBidiso - NBf dso - N&i
C = Nc/I Ncild/so-Ncf d/so-Nci
d = NDfI NDild/so NDf d/so - ND
(8)
(9)
(10)
(11)
where N. = fringe order at point . before displace-ment, and N..f = fringe order at point . after displace-
ment. If d is much larger than so, Eqs. (8)-(11) can besimplified as
(NAf - NAi)Ian d
b (NBf-N i)lId/so
(Nc1 - Nci)Id/so
(12)
(NDf -NDO 31diso
Consider an arbitrary point on the 3-D surface de-fined by coordinates xp, yp, and zp. The displacementof point P can be written in terms of a, b, c, and d as
P'= a+ (d-a) BC
PXA YP-YAI[\cub da, BC AB
+ (b - a) YPYA (13)
In deriving Eq. (13) only rigid body movements wereconsidered. If the surface deforms locally, the resultantdisplacement-of point P will have two components:
P = P10C1 + PP', (14)
where Plocal = local deformation, and PP' = rigid bodymovement.
The left-hand side of Eq. (14) can be evaluated in-dependently as
P (Npf - Np )lso/d, (15)
where Npi and Npf are initial and final fringe orders atpoint P. Substituting Eqs. (12), (13), and (15) into Eq.(14) and solving for local displacement,
This approximation is satisfactory if d/so is much largerthan N. Equation (16) should be modified if this as-sumption is inappropriate.
IV. ExperimentsA shadow moire setup large enough to view the back
of a full-size living body has been constructed. Theplane grating was made by stretching nylon thread of1-mm thickness on a steel frame using two long screwsas pitch guides. The grating was painted flat black toprovide high contrast. The subject was illuminated bytwo 1000-W quartz light bulbs. The bulb filamentswere aligned along an axis parallel to the lines of thescreen. The moire fringes were recorded with a 35-mmSLR camera with a 50-mm focal length. The pupil ofthe camera and the line of illumination were aligned ona vertical plane parallel to the grating. Grating lineswere horizontal.
Figures 3 and 4 show the contour line system of acylindrical container with the longitudinal axis parallel
Fig. 3. Moire map of a container with the cylindrical axis parallel Fig. 4. Moire map of a container after the cylindrical axis has beento the grating. rotated with respect to the grating.
Fig. 5. Comparison of profiles derived from the moire fringe patternsof Figs. 3 and 4.
to the grating and with known rotations about the x andy axes. Moire patterns were used to produce cross-sectional profiles along MM and NN (see Fig. 5). Inderiving these profiles corrections were applied to ac-count for central perspectivity. The screen frame wasequipped with a horizontal reference line near the planeof the screen which intersected the optical axis of thecamera. A reference contour was selected below thehorizontal line. The absolute fringe order of this ref-erence contour line was determined by measuring in themoire photograph the vertical distance between thehorizontal line and the intersection of its shadow withthe reference fringe.
From the geometry hN the distance between gratingand the reference fringe was determined as
hN = y*. (S.F.), (17)
where y = vertical distance, in the moire photograph,between the horizontal line and the intersection of itsshadow with the reference fringe, and S.F. = scale factorrelating distances on the image plane to the ones on thescreen. The absolute fringe order was then related tohN as follows:
N = hNd/so(hN + 1). (18)
To prove the power and accuracy of the proposedtechnique the profile along NN was derotated andtranslated analytically using Eq. (16). The optical in-formation at target points A, B, C, and D was used toeliminate rigid body movements of the second config-uration relative to the first one. Figure 6 shows theresultant profiles with an excellent match.
Calculations revealed that the error involved in theprocedure of derotation was <2% for rigid body rota-tions below 10°. For larger rotations it is hard to justifythe comparability of two profiles along a given directioneven if the errors produced during the derotation pro-cedure are small. A probabilistic error analysis is pre-sented in Appendix B.
Figures 7 and 8 show moire contour lines of the backof a living body without and with surface muscle con-tractions, respectively. Certain muscle groups wereexternally stimulated by two carbon electrodes. Thesubject was belted against a rigid support around thebelly. Target point A was placed on the skin close to
-- - profile dong MM
- profile along NN
IlOmm
0
Fig. 6. Comparison of moire profiles after analytical derotation andtranslation.
the spine. Points B and C were marked behind theshoulders where the local muscle activities were negli-gible.
Fringe orders along horizontal lines were read fromthese moire photographs by placing them in a NikonC6-2 optical comparator. Optical data were then fedinto a computer program for generating cross-sectionalprofiles on the back at comparable heights. The opticalinformation at the target points was used to derotate thecontour line system in Fig. 8 relative to the one in Fig.7.
Subtractively engaging the two contour line systemsas described in Ref. 9 would have been irrelevant sincethe net difference represents local muscle contractionsas well as rigid body movements.
A set of back profiles is shown in Fig. 9. Muscle ac-tivities are more pronounced along upper levels whichare closer to the electrodes. When differences amongthe back profiles are analyzed in conjunction with anatlas of back muscles, it can be concluded that trapezius,deltoid, and serratus anterior muscles were activatedduring electrical stimulation.
V. Discussion and ConclusionThe accuracy of the technique is calculated to within
0.2% in measuring depth (see Appendix B) and <2% forthe procedure of derotation. The accuracy in the pic-ture plane depends solely on the distortion of the cam-era lens which was negligible.
The proposed technique has been successfully usedto measure surface muscle contractions, and the utilityof this method in medical research has been shown.
The author wishes to express his appreciation andsincere thanks to A. B. Schultz and the Research Boardof University of Illinois at Chicago Circle for their helpand support in this work. The assistance of R. Haugenand M. Ghattineh is also appreciated.
Appendix A. Perspective CorrectionIn Fig. 10 an object is viewed through the grating. If
the scale factor relates the distances on the image planeto the ones on the grating, point P on the object will beseen at P on the grating as viewed through the cameralens. Thus, due to central perspectivity there will bea reduction in the x and y coordinates of point P.However, the moire photograph gives us the fringe order
Equation (B5) yields the following maximum allowableerrors:
RN 0.25 fringe order, Rd 9.63 mm,RI < 15.12 mm, R,, C 0.02 mm.
Large values of RI and Rd suggest that a reasonableerror in measuring I or d will have a negligible effect.
Moreover, this error will be systematic since and d areconstant for the entire data set. The probable error inN can be eliminated by determining the absolute fringeorders experimentally. Thus Ro is the major source oferror and it is due to the nonunformities of the grating.This conclusion necessitates an extreme care and pre-cision in constructing the screen. Random errors in s0would cause jagged fringes which would ordinarily besmoothed in the data interpretation process. Sys-tematic errors in so would be very small in a well-con-structed apparatus. Accordingly, the probable errorfor the prescribed conditions could be even smaller thanpredicted.
References1. D. Tollenaar, "Moire: Interferentieverschijnselen bij Raster-
druk," Amsterdam Institut voor Grafische Techniek (1945).2. R. Weller and B. M. Shepard, Proc. Soc. Exp. Stress Anal. 6, 35
(1948).3. F. K. Lightenberg, Proc. Soc. Exp. Stress Anal. 12, 83 (1954).4. S. Morse, A. J. Durelli, and C. A. Sciammarella, ASCE J. EM 86,
4, 105 (1960).5. P. S. Theocaris, Exp. Mech. 4, 153 (1964).6. D. Post, Exp. Mech. 5, 368 (1965).7. J. Der Hovanesian and Y. Y. Hung, Appl. Opt. 10, 2734 (1971).8. D. M. Meadows, W. 0. Johnson, and J. B. Allen, Appl. Opt. 9,942
(1970).9. H. Takasaki, Appl. Opt. 9, 1467 (1970).
10. H. Takasaki, Appl. Opt. 12, 845 (1973).11. W. Jaerisch and G. Makosch, Appl. Opt. 12, 1552 (1973).12. C. Chiang, Appl. Opt. 14, 177 (1975).13. G. Oster and Y. Nishijima, Sci. Am. 208, 54 (1963).
Meetings Calendar continued from page 1221
1982June
20-25 Light Microscopy in Biological Research course, WoodsHole M. Maser, Marine Biological Lab., Woods Hole,Mass. 02543
21-25 11th Int. Laser Radar Conf., Madison J. Edwards, 11thILRC, Space Science & Eng. Ctr., 1225 W. Dayton St.,Madison, Wisc. 53706
21-25 Advanced Infrared Technology course, Ann Arbor Eng.Summer Conf., 200 Chrysler Center, N. Campus, U.Mich., Ann Arbor, Mich. 48109
21-2 July NATO Advanced Study Inst. on Image Sequence Pro-cessing & Dynamic Scene Analysis, Braunlage/Hanz,W. Germany T. Huang, Coordinated Sci. Lab., U. Ill.,1101 W. Springfield Ave., Urbana, Ill. 61801
22-25 12th Int. Quantum Electronics Conf., Munich OSA,Mtgs. Dept., 1816 Jefferson Place, N. W., Wash., D.C.20036
26-11 July NATO-ASI Laser Applications to Chemistry, SanMiniato F. Arecchi, Istituto Nazionale Di Ottica,50125 Arcetri-Firenze, Italy
5-7 Applications of Laser-Doppler Anemometry to FluidMechanics Int. Symp., Lisbon F. Durst, Sonderfor-schungsbereich 80 An Der Universitat Karlsruhe, 7500Karlsruhe 1, Postfach 6380, F.R.G.
7-9 8th Int. Symp. on Machine Processing of RemotelySensed Data, W. Lafayette D. Morrison, PurdueU./LARS, 1220 Potter Dr., W. Lafayette, Ind.47906-1399
12-16 Optical System Design course, Rochester K. Teegarden,Inst. of Optics, U. Rochester, Rochester, N.Y. 14627
