application of generalized grating imaging to pattern projection in three-dimensional profilometry

Application of generalized grating imaging to patternprojection in three-dimensional profilometry

Koichi Iwata,1,* Yusuke Sando,2 Kazuo Satoh,2 and Kousuke Moriwaki2

1Lumino, 18-1, Daishi, Kawachi-Nagano, Osaka, 586-0041, Japan2Technology Research Institute of Osaka Prefecture, 2-7-1 Ayumino, Izumi, Osaka, 594-1157, Japan

*Corresponding author: k‐[email protected]

Received 5 January 2011; revised 1 July 2011; accepted 26 July 2011;posted 2 August 2011 (Doc. ID 140634); published 8 September 2011

The theory of generalized grating imaging for a one-dimensional grating is applied to a pattern projectionsystem in pattern projection profilometry. Contrast of the projected grating image is calculated undervarious conditions. The results help to determine the conditions suitable for obtaining high contrastgrating images in a large space. Although the gratings required for the profilometry are hexagonal,the theory for two-dimensional gratings is prohibitively complex. Therefore, the projection systemwas designed using the one-dimensional theory. The projection system using two-dimensional hexagonalgratings was constructed and experiments were done with it. The result agrees approximately with thetheoretical calculations for one-dimensional gratings. This suggests that the one-dimensional theorymaybe used for estimating the approximated behavior for hexagonal gratings for use in pattern projectionprofilometry. Some discussions are given for the application of the projection system for profiling themannequin or human body. © 2011 Optical Society of AmericaOCIS codes: 110.6880, 120.2830, 050.2770, 110.6760.

1. Introduction

Pattern projection profilometry using the principlesof triangulation has been used to measure three-dimensional profiles of many kinds of objects includ-ing the human body, teeth, and industrial parts [1].The schematic diagram of the system is shown inFig. 1. In this system, the grating pattern is projectedby a projector on the object to be measured. The pro-jected pattern on the object surface is viewed from acamera located in the direction different from thatof the projector. The image of the grating detected bythe camera is deformed, corresponding to the objectheight from the reference plane. With the data of thedeformed grating, the object profile is calculated.

There are many methods for measurement [1,2].Among them, the Fourier transform method has amerit that the measurement can be made with aone-shot image [3]. Recently, new Fourier transform

profilometry was proposed which expands the rangeof the measurement height [4]. The method con-tained two new ideas. One was to use a hexagonalgrating for projection and to process its image accord-ing to a new algorithm. The other was to project apattern through a new projection system. In [4], thenew processing algorithm of the projected hexagonalgrating image was explained in detail, but its projec-tion principle and systemwere explained only briefly.We shall explain them in detail in this paper.

In conventional pattern projection systems, anenlarged grating pattern is projected through a lenswhich enlarges a grating pattern formed by, forexample, a liquid crystal display or a grating [5,6].The new projection system consists of two gratingswith slightly different pitches and a light source. Noprojection lens is used and the system is simple. Inaddition, a LED can be used as the light source.

The principle of the new projection system is basedon the generalized grating imaging [7–12]. Manypapers have been published on the principle of thegeneralized grating imaging, but they did not aim

0003-6935/11/265115-07$15.00/0© 2011 Optical Society of America

10 September 2011 / Vol. 50, No. 26 / APPLIED OPTICS 5115

at the application to profilemetry. In the presentpaper, the theory is applied to the projection systemfor profilometry. We select the profile measurementof a mannequin body described in [4] for an example.Contrast of the projected grating in space is calcu-lated as a function of the distance from the projectorto the object. The calculation shows that high con-trast fringes are obtained at a definite distance deter-mined by the distance between the two gratings.High contrast in a large space is found to be obtainedwith a light source of small size and large wavelengthwidth. This result shows that the projection systemis suitable to pattern projection profilometry.

The system in [4] employs a two-dimensionalhexagonal grating in order to obtain multiple datafrom a projection. Thus, in this paper, experi-ments are made with a two-dimensional hexagonalgrating although the calculation is made for a one-dimensional grating. The experimental result agreesapproximately with the analytic results for a one-dimensional analysis. This confirms that the theorycan be used for designing a projection system withtwo-dimensional hexagonal gratings.

2. Generalized Grating Imaging

Figure 2 is a schematic diagram for the generalizedgrating imaging. We use two gratings G1 and G2 ofdifferent pitches. A kind of moiré fringes are ob-served on a plane different from the gratings [10,11].

In Fig. 2, Ls is an incoherent quasi-monochromaticlight source of wavelength λ. Pitches of the two grat-ings G1 and G2 are p1 and p2. Distance from thesource to the first grating is expressed by L0, distancefrom the first grating to the second grating by L1,distance from the second grating to the observationplane by L2, and the total distance by LT ¼ L0þL1 þ L2. The coordinate on the source plane isdenoted by x and that on the observation plane isdenoted by X.

For simplicity, discussion in this paper is limited tothe case where both the gratings are amplitude grat-ings of a rectangular shape, although other type of

gratings can be used for the same purpose. In thiscase the diffraction coefficients an, bm of the firstand second gratings are expressed as

a0 ¼ 1; an ¼ sinðπnγaÞ=ðπnγaÞ n ¼ �1;�2;…b0 ¼ 1; bm ¼ sinðπmγbÞ=ðπmγbÞ m ¼ �1;�2;…

;

ð1Þ

where γa and γb are the opening ratios of the rectan-gular gratings.

When the size 2S of the light source is much largerthan the pitches of the gratings p1, p2, intensityon the observation plane B is calculated in a simpleprocedure. It consists of summation of many sinusoi-dal intensity variations with the fundamental fre-quency μ1 and higher order frequencies jμ1. The jthorder is formed by the two waves, as shown in Fig. 2.One is diffracted in the nth order in the first gratingand in the mth order in the second grating and thesecond is diffracted in the ðnþ jÞth and ðm − jÞth,respectively. The total intensity is expressed by thesummation of the sinusoidal intensity with variousfrequencies as

IðXÞ ¼X∞j¼−∞

ampðjÞ cos½2πXjμ1 þ jΨ1�; ð2Þ

where

ampðjÞ ¼ Wðjμ2ÞAðjÞBðjÞ;

AðjÞ ¼X∞n¼−∞

ananþj cos½ð2nþ jÞjE�;

BðjÞ ¼X∞m¼−∞

bmbm−j cos½ð2m − jÞjF�; ð3Þ

Wðjμ2Þ ¼Z

S

−Sexp½i2πjμ2x�dx ¼ sinð2πjSμ2Þ=ð2πjSμ2Þ;

ð4Þ

μ1 ¼ −½L0=p1 − ðL0 þ L1Þ=p2�=LT ; ð5Þ

μ2 ¼ −½ðL1 þ L2Þ=p1 − L2=p2�=LT ; ð6Þ

Fig. 1. (Color online) Schematic diagram of a pattern projectionprofilometer.

L0 L1 L2

p1

p2

Ls G1 G2 B

L20

B0

R R0

x

X

2S

LT0

LT

n

n+j

m

m-j

Fig. 2. Optical system for generalized grating imaging.

5116 APPLIED OPTICS / Vol. 50, No. 26 / 10 September 2011

E ¼ −ðπλL0=p1Þμ2; F ¼ ðπλL2=p2Þμ1; ð7Þ

Ψ1 ¼ 2πfε1=p1 − ε2=p2g: ð8ÞIn Eq. (8), ε1 and ε2 are the lateral displacements of

the first and second gratings.The pitch P of the fundamental frequency is

given by

P ¼ 1=μ1: ð9ÞThe contrast CðjÞ of the grating image with fre-

quency j=P on plane B is defined by the equation

CðjÞ ¼ 2Wðjμ2ÞCAðjÞCBðjÞ;where CAðjÞ ¼ AðjÞ=Að0Þ;

CBðjÞ ¼ BðjÞ=Bð0Þ: ð10Þ

A negative sign in these contrasts means the reversalof dark and bright parts.

Amplitude and pitch of the grating image changeaccording to the location L1 and L2 of the gratingsand the diffraction coefficients an and bm. The dif-fraction coefficients in Eq. (1) is large and positivefor small n andm. Therefore, to obtain large absolutevalues for Að1Þ and Bð1Þ in Eq. (3), the absolutevalues of the cosine factors should be large.

According to Eq. (3), the amplitude of the gratingimage is largest when μ2 ¼ 0 or E ¼ 0. Thus the posi-tion L20 of the observation plane B0 where the ampli-tude is largest is expressed by the equation

L20 ¼ L1p2=ðp1 − p2Þ; ð11Þwhich is derived from Eq. (6). The pitch P0 of thefundamental fringe on the plane B0 is given withEqs. (5) and (11) as

1=P0 ¼ −1=p1 þ 1=p2: ð12ÞOn this plane, B0, F in Eq. (7) becomes

F0 ¼ πλL1=ðp1p2Þ: ð13ÞHigh contrast fringe for j ¼ 1 on the plane B0 isobtained when F0 ¼ νπðν: integerÞ, because all thecosine factors in Eq. (1) are minus one with the lar-gest absolute value. Thus for obtaining high contrastfringes on the plane B0,

L10 ¼ νp1p2=λ: ð14ÞThe fundamental pitch P on the plane other than

B0 is given by the equation

P ¼ 1=μ1 ¼ P0ðL0 þ L1 þ L2Þ=ðL0 þ L1 þ L20Þ; ð15Þwhich means that the fringes on the plane L2 aredetermined as the projection of the fringes of pitch

P0 on the plane L2 ¼ L20 with the projection centeras the light source.

The effect of the spectrum width is calculated asthe sum of the intensity for different wavelengths.If the light has the broad rectangular spectrum ofwidth 2δ with the central wavelength λ0, the resul-tant intensity Iδ can be obtained from Eq. (2) as

IδðXÞ ¼Z λ0þδ

λ0−δIðXÞdλ: ð16Þ

3. Calculation of Contrasts for Projection Systems

In [4] the grating pattern was projected using twogratings based on the generalized grating imaging.For the profilometry, measurable height range isimportant. Thus the project fringes should havehigh contrast in a large range of L2. In this sectioncontrast of fringes are calculated based on the abovetheoretical results for a grating suitable for the pro-filometry aimed for measuring a mannequin orhuman body [4].

The system in [4] adopted amplitude gratings ofγa ¼ 1=2 and γb ¼ 1=2. The pitches p1 and p2 were100 and 96:8 μm, and the wavelength λ was 617nm.Contrast Cð1Þ for the fundamental pitch P of thegrating image is calculated in Fig. 3, because inthe profilometer only the fundamental pitch is usedfor calculating the height. High contrast is obtainedfor L1 ¼ 15:8νmm and L2 ¼ 474νmm, as calculatedby Eqs. (11) and (14). These dimensions are suitablefor the profilometer for ν ¼ 1. The fundamental pitchP0 on the high contrast plane is calculated to be3:0mm by Eq. (15). It corresponds to the spatialresolution of the image and is suitable to the bodymeasurement.

Figures 3–6 show the theoretical results for one-dimensional grating based on the equations inSection 2. Contrast Cð1Þ in Fig. 3 is calculated as afunction of L1 and L2 with Eq. (10) for the conditionsL0 ¼ 5mm and S ¼ 0:5mm. This source size can beattained with a high power LED. In Figure 3, the

Fig. 3. Contrast of fringes as a function ofL1 andL2. p1 ¼ 100 μm,p2 ¼ 96:8 μm, λ ¼ 617nm, L0 ¼ 5mm, S ¼ 0:5mm.


position of maximum contrast given in Eq. (11) isshown in a broken line. The contrast is higher than−0:8 along the solid line L1 ¼ 15:8mm in the widerange from L2 ¼ 300mm to 1000mm. This is suffi-cient for body measurement. The minus sign of thecontrast shows that the intensity variation is re-versed compared with the intensity of the wave justbehind the grating. In addition, Fig. 3 shows that thedistance setting between the two gratings G1 and G2is not so stringent for the purpose of measuring thedimension of the mannequin body.

The effect of source size S is shown in Fig. 4. InFig. 4, L1 ¼ 15:8mm and L0 ¼ 5mm, but the sourcesize S and L2 are changed. Contrast for L2 ¼ 474mmstays high even if the source size changes, but thehigh contrast range for L2 decreases as S becomeslarge. The appropriate source size should be selected.

The effect of L0 is shown in Fig. 5. In Fig. 5, L1 ¼15:8mm and S ¼ 0:5mm, but L0 and L2 are changed.Contrast for L2 ¼ 474mm stays high even if L0changes, but the high contrast range decreases asL0 becomes large.

The effect of the spectrum width δ is calculatedaccording to Eq. (16) and shown in Fig. 6. In Fig. 6,contrast for L1 ¼ 15:8mm is calculated as a functionof L2 and the spectrum half width δ for λ ¼ 617nm.

We see that the effect of the spectrum width is notso severe and a conventional high power LED canbe used.

4. Design of a Projection System with Two HexagonalGratings

The preceding sections show the theory of general-ized grating imaging. Although the analysis is madefor one-dimensional gratings, hexagonal gratingswere used in [4]. To design the grating projection sys-tem, we shall apply the one-dimensional theory tothe hexagonal case.

Two-dimensional gratings have many differentfrequencies. The plural phase difference for the dif-ferent frequencies helps us to unwrap the measuredphase [4]. The hexagonal grating has three majorfrequencies with the same pitch. Thus three phasedata can be obtained at a point of the object. In ad-dition, hexagonal gratings in the pattern projectionprofilometry have an advantage that high spatialresolution can be obtained. This is because it has theclose-packed structure, and the distances betweenthe main frequencies in the spectrum plane arelargest among two-dimensional gratings [4] for thesame pitch. This is the reason for adopting the hex-agonal grating in the profiling system.

Usually a two-dimensional grating has differentpitches for different directions. Different pitchesresult in different projected grating pitches and dif-ferent high contrast positions, as seen from Eqs. (11),(12), and (14). Thus general two-dimensional grat-ings behave in a complex manner and theoreti-cal analysis for the two-dimensional grating iscomplicated.

However, a hexagonal grating has the same pitchfor three directions separated by 60°. Thus the one-dimensional theory may be used to predict its beha-vior. Thus we will construct a projection system withhexagonal gratings to see if the analytic result forone-dimension can be used for hexagonal gratings.

One problem to be determined is the two-dimensional shape of a unit lattice. Two examplesare shown in Fig. 7. These figures are obtained asfollows: at first we superpose three two-dimensional

Fig. 4. Contrast of fringes for L1 ¼ 15:8mmas a function of S andL2. p1 ¼ 100 μm, p2 ¼ 96:8 μm, λ ¼ 617nm, L0 ¼ 5mm.

Fig. 5. Contrast of fringes for L1 ¼ 15:8mm as a function of L0

and L2. p1 ¼ 100 μm, p2 ¼ 96:8 μm, λ ¼ 617nm, S ¼ 0:5mm.

Fig. 6. Contrast of fringes as a function of δ and L2. L1 ¼15:8mm, p1 ¼ 100 μm, p2 ¼ 96:8 μm, λ ¼ 617nm, L0 ¼ 5mm,S ¼ 0:5mm.


sinusoidal functions with different directions sepa-rated by 60° as

f ðx; yÞ ¼ cos½2πx=pþ ϕ� þ cos½2πðx=2þffiffiffi3

py=2Þ=p�

þ cos½2πð−x=2þffiffiffi3

py=2Þ=p�: ð17Þ

This function is binarized with a threshold of zero.Two examples in Fig. 7 are obtained for (a) ϕ ¼ 0 and(b) ϕ ¼ π=2. The parallelograms in the figures showunit lattices. The pitches are as shown as p1 and p2.The white area corresponds to the transparentregion and the black area corresponds to the opaqueregion. We adopted the pattern of Fig. 7(a) for theexperiment but adjusted it to have equal area forblack and white by enlarging the circle because thecalculations in the preceding section are made forγa ¼ γb ¼ 1=2. We decided to make two binary hexa-gonal amplitude gratings of ϕ ¼ 0.

The available light source was a LED (LumiledsLXHL-LH3C, λ ¼ 617nm) and its spectrum widthδ is about 10nm. The light emitting area is esti-mated as 2:2mm × 2:2mm, which corresponds toS ¼ 1:1mm.

In consideration of the spatial resolution, the pro-jection pitch P0, suitable for body profilometry, maybe about 3mm. In consideration of the fabrication fa-cility, preferable grating pitches p1 and p2 are about100 μm. Using Eq. (12), we decided p1 ¼ 100 μm andp2 ¼ 96:8 μm. These pitch values and the illumina-tion wavelength determined L1 ¼ 15mm by Eq. (14).The distance L20 of the highest contrast is calculatedas 453mm by Eq. (11). According to Fig. 4, the highcontrast region for L2 decreases to about 250mm dueto the large light source area of S ¼ 1:1mm, but thisrange is enough large for body measurement.

According to Fig. 6 the spectrum width does notdegrade the contrast of the projected pattern appre-ciably. In consideration of Fig. 5, distance L0 does notaffect the contrast when L0 is smaller than 10mm.We determined L0 ¼ 5mm. The two gratings areso arranged that their diffraction spectra are inthe same direction.

The half angle of constant intensity radiating fromthe employed LED is also about 30°. The bright areaon the plane, perpendicular to the optical axis, isdetermined by this angle. This illuminates a circle of

radius 200mm on the plane at L2 ¼ 400mm, whichis wide enough for a human body. Because the dis-tance from the LED to the second grating is 20mm,the half angle 30° corresponds to the second gratingof the area of 20mm × 20mm. This is a reasonablegrating size.

5. Experiments with the Projection System

With this system, we measured the contrast of theprojected grating by changing the distance L2. Inorder to explain the procedure of contrast calcula-tion, an example of a projected grating image isshown in Fig. 8(a). This image is obtained by directlytaking a photograph with a complementary metaloxide semiconductor image sensor located at a dis-tance of L2 ¼ 280mm. Figure 8(b) shows intensityvariations along the solid line in Fig. 8(a). With thiskind of intensity variations we calculated the con-trast by the expression

C ¼ Imax − Imin

Imax þ Imin; ð18Þ

where Imax corresponds to the points A and Imin to Bin Fig. 8(a).

The experimental result is shown in Fig. 9 in filledcircles. They are obtained by changing the position ofthe image sensor. The contrast corresponding toFig. 8 is indicated as E in Fig. 9. Figure 9 also showsthe contrast Cð1Þ as a function of LT calculated usingthe generalized grating imaging theory for L0 ¼5mm and L1 ¼ 15mm and S ¼ 1:1mm. We see

Fig. 7. Hexagonal binary gratings obtained from Eq. (15):(a) ϕ ¼ 0 (b) ϕ ¼ π=2.

Fig. 8. Projected grating image. (a) Intensity image. (b) Intensityvariation along the solid line in (a).


in Fig. 9 that the variation of contrast C for theexperiment is similar to contrast Cð1Þ of the one-dimensional theory. Figure 3 shows the contrast isnegative for the system with L1 ¼ 15mm. In accor-dance with this, the grating intensity has a largeintensity in the interior of the circle, although theoriginal gratings are transparent in the interior ofthe circle. The pitch of the grating image agreeswith the value predicted by Eq. (12). It should benoticed that C and Cð1Þ show similar variationalthough their definition is different.

The theoretical curve shows rapid contrast varia-tion in the left side of the figure. The variation isnot shown in the experimental results. The smallvariation is not important for designing the projec-tion system because we use only the good contrastregion. In addition, the contrast changed rapidlyaccording to L0, L1, and S as seen from Figs. 3–5.The rapid contrast variation may be blurred due tothe variation of these parameters.

The pitch of the projected grating changes accord-ing to Eq. (15). The pitches obtained by the imagesensor are plotted in open circles in Fig. 9. Thepitches calculated by Eq. (15) are shown in a brokenline.We should note that the pitches are measured asshown in Fig. 8.

Thus we see that one-dimensional theory may beused to predict the approximate behavior of two-dimensional hexagonal grating and can be used todesign a projection system with hexagonal gratings.

6. Discussion and Conclusion

This projection system consists of a high power LEDand two amplitude gratings which are not expensivecompared with other projection systems with lenses.

The fringe pitch on the plane determined by L2 isexpressed by Eq. (15). These equations express thatthe fringe pitch is constant on the plane L2, parallelto the grating plane. No distortion and curvature oc-curs on this plane as long as paraxial approximation

is valid. This conforms to the algorithm of profilome-try expressed in [4] which assumes constant pitch ona reference plane.

The distance between G1 and G2 is restricted byEq. (14). In order to have more flexibility in design-ing, its relaxation is desirable [13]. We have to inves-tigate a condition of relaxation for the presentprojection system.

An example of the hexagonal grating image pro-jected on a mannequin body is shown in Fig. 10,which is the same figure shown in [4]. The contrastis good for the whole space of the body. The heightrange of the body is about 130mm, and its widthrange is more than 300mm.

In consideration of the experimental results, wecan say that the one-dimensional grating imagingtheory can be used to predict the approximated beha-vior of two-dimensional grating as long as it is hex-agonal and of Fig. 7(a). Thus, the equations inSection 2 can be used to design a suitable systemwhen we construct other projection systems. How-ever, a theory of two-dimensional general gratingimaging has to be constructed if we want a moreexact design or two-dimensional gratings other thanhexagonal grating of the present type.

References1. N. D’Apuzzo, “Overview of 3D surface digitization technolo-

gies in Europe,” Proc. SPIE 6056, 605605 (2006).2. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-

measuring profilometry: a phase mapping approach,” Appl.Opt. 24, 185–188 (1985).

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

4. K. Iwata, F. Kusunoki, K. Moriwaki, H. Fukuda, and T. Tomii,“Three-dimensional profiling using the Fourier transformmethod with a hexagonal grating projection,” Appl. Opt. 47,2103–2107 (2008).

5. S. Kakunai, T. Sakamoto, and K. Iwata, “Profile measurementtaken with liquid crystal grating,” Appl. Opt. 38, 2824–2828(1999).

A

LT (mm)

cont

rast

,

CC

(1)

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

E

4

3

2

1

0

P(m

m)

Fig. 9. (Color online) Variation of contrast and pitch in relation toLT . Filled circles are the experimental results of the contrast. Solidcurve is calculated by the theory for a one-dimensional gratingwith S ¼ 1:1mm and L1 ¼ 15mm, L0 ¼ 5mm. The point E corre-sponds to Fig. 8. Broken line is calculated by Eq. (14), and the opencircles are the experimental result of the pitch.

Fig. 10. Hexagonal grating projected on a mannequin body.


6. E. Stoykova, G. Minchev, and V. Sainov, “Fringe projectionwith a sinusoidal phase grating,” Appl. Opt. 48, 4774–4784(2009).

7. K. Patorski, “The self-imaging phenomenon and its applica-tions,” in Progress in Optics, E. Wolf, ed. (North Holland,1989) Vol. 27, pp. 3–108.

8. D. Crespo, J. Alonso, and E. Bernabeu, “Generalized gratingimaging using an extended monochromatic light source,”J. Opt. Soc. Am. A 17, 1231–1240 (2000).

9. D. Crespo, J. Alonso, and E. Bernabeu, “Experimental mea-surements of generalized grating images,” Appl. Opt. 41,1223–1228 (2002).

10. K. Iwata, “Interpretation of generalized grating imaging,” J.Opt. Soc. Am. A 25, 2244–2250 (2008).

11. K. Iwata, “Interpretation of generalized grating imaging(farther analysis and numerical calculation),” J. Opt. Soc.Am. A 25, 2939–2944 (2008).

12. K. Iwata, “X-ray shearing interferometer and gener-alized grating imaging,” Appl. Opt. 48, 886–892(2009).

13. L. M. Sanchez-Brea, J. Aloso, and E. Bernabeu, “Quasicontin-uous pseudoimages in sinusodial grating imaging usingan extended light source,” Opt. Commun. 236, 53–58(2004).


application of generalized grating imaging to pattern projection in three-dimensional profilometry

Documents