Fringe projection with a sinusoidal phase grating · Fringe projection with a sinusoidal phase...

Fringe projection with a sinusoidal phase grating

Elena Stoykova,* Georgi Minchev, and Ventseslav SainovCentral Laboratory of Optical Storage and Processing of Information, Bulgarian Academy of Sciences,

Acad. G. Bonchev, Bl. 101, P.O. Box 95, 1113 Sofia, Bulgaria

*Corresponding author: [email protected]

Received 17 February 2009; revised 27 June 2009; accepted 3 August 2009;posted 4 August 2009 (Doc. ID 107634); published 18 August 2009

Phase-shifting profilometry requires projection of sinusoidal fringes on a 3D object. We analyze the vis-ibility and frequency content of fringes created by a sinusoidal phase grating at coherent illumination.Wederive an expression for the intensity of fringes in the Fresnel zone and measure their visibility and fre-quency content for a grating that has been interferometrically recorded on a holographic plate. Both eva-luation of the systematic errors due to the presence of higher harmonics by simulation of a profilometricmeasurement and measurement of 3D coordinates of test objects confirm the good performance of the si-nusoidal phase grating as a projective element. In addition,we prove theoretically that in comparisonwitha sinusoidal amplitude grating this grating produces better quality of fringes in the near-infrared region.Sinusoidal phase gratings are fabricated easily, and their implementation in fringeprojectionprofilometryfacilitates construction of portable, small size, and low-cost devices. © 2009 Optical Society of America

OCIS codes: 050.1950, 050.1960, 050.5080, 070.6760, 100.2650.

1. Introduction

Projection of a light pattern with a regular structureis a highly sensitive noncontact technique for mea-suring the three-dimensional (3D) shape of an object[1,2]. The shape information is encoded in the de-formed light pattern that is reflected by the objectand captured with a CCD camera. Decoding is per-formed by appropriate algorithms provided the posi-tions of the camera, the projector, and the objectare known. Nowadays, many areas of human activ-ity such as machine vision, computer-aided design,manufacturing, engineering, virtual reality, culturalheritage protection, and medical diagnostics benefitfrom pattern projection profilometry in its variousmodifications.An attractive approach among structured light

methods is phase-measuring profilometry [3,4], orfringe projection profilometry, in which the para-meter being measured is encoded in the phase of atwo-dimensional (2D) fringe pattern described by aperiodic function f ½…�, f ∈ ½−1; 1�. In general, mostof the existing algorithms for phase retrieval pre-

sume a unidirectional sinusoidal profile of fringes,which is equivalent to introduction of a spatial car-rier into the projected pattern. A popular methodfor phase demodulation due to its pointwise perfor-mance, large dynamic range, and high spatial reso-lution and accuracy is the phase-shifting techniquebased on sinusoidal fringe projection of at least threepatterns that have undergone equal (conventionalcase [5,6]) or arbitrary (generalized case [7,8]) phaseshifts with respect to one another. The phase-shiftingalgorithm is inherently free of errors only at perfectsinusoidal fringe projection. Violation of the assump-tion f ½…� ∝ cosð…Þ leads to systematic errors in theevaluated phase and in the reconstructed shape ofthe object [9–11].

Projection of purely sinusoidal fringes is not aneasy task. High-contrast sinusoidal fringes can beproduced by interference of two enlarged and colli-mated light beams, thus making possible large-depthand large-angle measurements [12–14]. Interfero-metric systems, however, are complex and prone toenvironmental influence, which restricts their out-door usage. Better stability is achieved in profilo-metric devices based on gratings or spatial lightmodulators but at the expense of higher harmonicsin the projected pattern [14–18]. Systems with one,

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4774 APPLIED OPTICS / Vol. 48, No. 24 / 20 August 2009

dual-frequency, and multifrequency amplitude orphase grating projection at incoherent orcoherent illumination have been reported [17,18].Different approaches have been tested to ensuregood sinusoidality of fringes in grating systems. Amethod that has quickly gained popularity is defo-cusing projection of a Ronchi grating. The analysisin [16] proves that a phase-shifting algorithm withan odd number of steps and a proper choice of thedegree of defocusing may diminish phase errors toless than 0:005 rad. The easy production of a binaryRonchi grating, e.g., using computer-controlled LCDprojectors or digital micromirror devices is an addi-tional advantage of this method. Recently, a specialarea modulation grating for sinusoidal structure illu-mination was proposed and tested in [19]. A pro-grammable spatial light modulator for digitalfringe projection offers flexibility in choosing a fringeprofile and spacing as well as an option for fast andaccurate phase shifting [20,21]. The drawbacks arethat these systems are comparatively expensiveand suffer from luminance nonlinearity and higherharmonic content of the fringes. Suppression ofhigher harmonics both in gratings and in computer-generated fringes can be done at the stage of digitalprocessing by specially designed error-compensatingalgorithms that have been developed over the yearsto overcome the systematic errors caused by nonli-nearity of phase shifters, projecting and recordingdevices, and the presence of higher harmonics. Theserather elaborate algorithms often show a controver-sial behavior: the more efficient they are in suppres-sing the systematic error, the more vulnerable theymay become to random error sources [9,22]. Somemore flexible techniques for suppression of higherharmonics have been recently developed such as, forexample, a signal subspace method [23] or gradient-based shift estimation method [24] at the expense,however, of additional computational burden. Ob-viously, the best way to benefit from the high ac-curacy of the phase-shifting fringe projection profilo-metry is to ensure high quality of the projectedfringes. In addition, as stated in [25], a good phase-measuring profilometric system should provide goodcontrast of fringes in a large depth range, and itshould be built from a small number of low-cost com-ponents to ensure its portability and reliability.The above-listed requirements are in line with the

goal of this paper, which is to check the quality offringes produced by a sinusoidal phase grating (SPG)at coherent illumination in the Fresnel zone from thepoint of view of phase-shifting fringe projection pro-filometry. As is well known, a monochromatic plane-wave illumination of a periodic structure (grating)creates its exact replicas at equally spaced Talbotplanes, which are located at 2nL2=λ from it, whereL is the grating spacing (pitch), λ is the light wave-length, and n is an integer, as well as many otherimages at distances 2nL2=mλ from the grating,where m is an integer. Divergent illumination pro-duces a magnified replica of the grating in Talbot

planes [26,27], which in this case are located atincreasing distance from it. Due to its potential tobe used for imaging, beammanipulation, and metrol-ogy, the Talbot effect evokes constant research in-terest [28–39]. Fourier optics and multiple-slitdiffraction have been applied to predict positions ofinteger and noninteger Talbot planes [33]; the Talboteffect in the deep Fresnel region was investigatedin [34]. The Talbot effect for beams of an arbitrarywavefront and a Gaussian beam in particular forRonchi and binary phase gratings was studied in [32]by using a Fresnel diffraction integral. Variation ofthe fringe contrast for some types of grating as a peri-odic function of the distance was studied in [34].A phase-only diffraction grating used for creation ofTalbot array illuminators was described in [35]. In[36] 3D Talbot-effect-based profilometry is realizedusing wavelength scanning to shift continuously theTalbot self-image of a grating in the longitudinal di-rection. Especially interesting for the study in thispaper is utilization of Talbot self-imaging under co-herent divergent illumination of a sinusoidal ampli-tude grating (SAG) for profilometry of 3D diffuseobjects, as described in [28].

The paper is organized as follows: Subsection 2.Apresents an expression for the intensity of fringescreated by a SPG in the Fresnel zone and considerstheir visibility and frequency content; Subsection 2.Bcompares the quality of fringes created both by aSPG and a SAG at different wavelengths. In Section 3a profilometric measurement is simulated with aSPG as a projective element, and the systematic er-rors due to the presence of higher harmonics areevaluated. Section 4 gives the experimental resultsof test measurements of relative 3D coordinates bymeans of a fringe projection system with a SPG thathas been interferometrically recorded on a holo-graphic plate.

2. Fresnel Diffraction Pattern

A. Solution of the Diffraction Integral

A thin SPG located in the (x,y) plane with gratinglines parallel to the y axis is characterized withtransmittance [40]

τðx; yÞ ¼ exp�jm × sin

�2π x

L

��

¼X∞q¼−∞

JqðmÞ exp�j2πq x

L

�; ð1Þ

where m is the modulation parameter, L is the grat-ing period along the x axis, and Jq is a Bessel func-tion of the first kind, order q. For convenience, we willconsider the problem as 2D; its generalization to the3D case is straightforward. The complex amplitudeUðx; zÞ of the light field created by the grating can befound using the transfer function approach [40]with the Fresnel diffraction transfer function orthe Fresnel diffraction integral:

20 August 2009 / Vol. 48, No. 24 / APPLIED OPTICS 4775

Uðx; zÞ ¼ expðjkzÞjλz exp

�jk2z

x2�Z∞

−∞

Uðx0; 0Þ

× exp�jk2z

x20

�exp

�−j

kzxx0

�dx0;

ð2Þ

where Uðx0; 0Þ is the complex amplitude of the lightfield in the plane immediately to the right of the grat-ing at z ¼ 0, x0 is the x coordinate in this plane, andk ¼ 2π=λ. If a grating is illuminated by a monochro-matic point source at distance d in front of it, the fieldUðx0; 0Þ in the paraxial approximation of the spheri-cal wave is given by

Uðx0; 0Þ ¼ A0τðx0Þ exp�jk2d

x20

�; ð3Þ

where A0 is a constant amplitude. Substituting Eqs.(2) and (3) into Eq.(1) leads to

Uðx; zÞ ¼ A0expðjkzÞjz

ffiffiffiffiffiλη

p exp�jk2z

x2� X∞

q¼−∞

JqðmÞ

×�ð1þ jÞ cos

�2ðπσqÞ2

kη

�þ ð1 − jÞ sin

�2ðπσqÞ2

kη

��; ð4Þ

where σq ¼ qL −

xzλ and η ¼ 1

d þ 1z. Hence, the intensity

of the diffraction pattern is given by

Iðx; zÞ ¼ 2A20d

zλðzþ dÞ ½S21ðx; zÞ þ S2

2ðx; zÞ�

¼ 2A20d

zλðzþ dÞ ISðx; zÞ; ð5Þ

with

S1 ¼X∞q¼−∞

JqðmÞ cos�2ðπσqÞ2

kη

�;

S2 ¼X∞q¼−∞

JqðmÞ sin�2ðπσqÞ2

kη

�:

ð6Þ

The term in the square brackets in Eq. (5) can bewritten as follows:

IS ¼X∞q¼−∞

J2q þ

X∞q¼−∞

×X∞

p ¼ −∞

p ≠ q

JqJp cos½ðq2 − p2Þα − 2ðq − pÞβ�; ð7Þ

where we have introduced the notations

α ¼ πλzdL2ðzþ dÞ ; β ¼ πxd

Lðzþ dÞ : ð8Þ

Keeping in mind that J−q ¼ ð−1ÞqJq, we obtain

Iðx; zÞ ¼ 2A20d

zλðzþ dÞ�I0 þ IVðzÞ sin

2πxdLðzþ dÞ þΘðx; zÞ

�;

ð9Þ

where

I0 ¼X∞q¼∞

J2qIVðzÞ

¼ 2

�X∞q¼0

J2qJ2qþ1 sinð2qþ 1Þα

þX∞q¼1

J2qJ2q−1 sinð2q − 1Þα�Θðx; zÞ

¼ 2X∞q¼0

X∞p¼1

½J2qJ2qþ2pþ1 sinð4pβÞ

× sinfαð2pþ 1Þð2pþ 4qþ 1Þg þ cosð4pβÞ× ðJ2qJ2qþ2p cosf4αðqþ 2pÞgþ J2qþ1J2qþ2pþ1 cosf4αpðqþ 2pþ 1ÞgÞ�:

In general, the intensity distribution Iðx; zÞ givenby Eq. (9) is not sinusoidal. We see that Iðx; zÞ con-sists of sinusoidal terms that are responsible forthe periodicity of the created diffraction patternalong the x and z axes. The contribution of the non-sinusoidal term Θðx; zÞ along the x and z axes de-pends on the grating parameters (spacing, L, and themodulation parameter, m), as well as on the illu-mination conditions (wavelength, λ, and source loca-tion, d).

At illumination with a unit-amplitude normally in-cident plane wave we have d → ∞. In this case theFresnel diffraction pattern is a periodic structurealong the x and z axes that consists of alternatingzones with a phase-reversed contrast. The spatialperiod of the first harmonic along the x axis dependsonly on L. The spatial period of the first harmonicalong z axis depends on all three parameters: L, λ,and m. The alternating zones with a phase-reversedcontrast along the z axis are located between theplanes that corresponds to

zn ¼ 2nL2

λ ; n ¼ 1; 2…;

z0n ¼ ð2nþ 1ÞL2

λ ; n ¼ 0; 1; 2…: ð10Þ

The planes located at zn and z0n are Talbot planesthat contain a “perfect” grating image with uniform


intensity distribution equal to unity. The length ofthe zone along the z axis with a phase-constant con-trast of fringes for the first harmonic is equal toz0n − zn ¼ L2=λ, for the second harmonic to L2=2λ aswell as for the qth harmonic to L2=qλ. This meansthat the second harmonic is eliminated in the planesparallel to the grating and located at

z00n ¼ ð2nþ 1=2ÞL2

λ ; n ¼ 1; 2…; ð11Þ

but the amplitude of the third harmonic is maximalin these planes. For illustration, Fig. 1(a) depicts theFresnel intensity distribution of the light trans-mitted by a SPG with L ¼ 0:025 cm and m ¼ 0:2 as afunction of the distance, z, for λ ¼ 660nm. Two zonesof a phase-reversed contrast are shown. The frequen-cy content of the fringe pattern as a function of z ispresented in Fig. 1(b).

As should be expected, the modulation parameterm has a strong influence on the harmonic contentof the diffraction pattern. The first-order harmonicdominates at all values of m. The degrading effectof higher-order harmonics becomes unacceptable atm greater than 0.3. The zones that are close to z ¼ z00n,n ¼ 1; 2;… appear to be optimal for pattern projec-tion because, at least theoretically, the energy in thefirst harmonic reaches maximum for m up to 0.6–0.8at z ¼ z00n, whereas the second harmonic is missing inthese planes. The location of these optimal planesalong the z axis varies with the wavelength.

The rapidly varying contrast of the fringes due tothe Talbot effect makes impossible profilometry oflarge-scale objects at plane-wave illumination. In-deed, the distance between the Talbot planes in thiscase is of the order of several centimeters [41]. To in-crease the spatial zone of equal contrast, the gratingshould be illuminated by a divergent beam. If a grat-ing is illuminated by a monochromatic point sourceat distance d from it, a magnified image of the grat-ing is formed at planes located at [28]

1zþ 1d¼ λ

nL2 ð12Þ

with a spatial period ML, where MðzÞ ¼ 1þ z=d isthe magnification factor and n is an integer. So, bya proper choice of the distance d, the number of Tal-bot planes can be reduced to one. With increasing λ,the location of this single plane moves closer to thegrating. This is clearly seen in Fig. 2, which depictsvariation of the fringe visibility with distance for d ¼12 cm and a SPG with L ¼ 0:025 cm and m ¼ 0:2 atdifferent wavelengths. The visibility is calculatedfrom [28]

VðzÞ ¼ Ið0; zÞ − I½L=2MðzÞ; z�Ið0; zÞ þ I½L=2MðzÞ; z� : ð13Þ

The property of the SPG to create high-qualitycontrast fringes in a large region along the z axis

Fig. 1. Intensity distribution (a) and spatial frequency spectrum(b) of the light transmitted by a SPG in the Fresnel zone as a func-tion of the distance z from the grating at plane-wave illumination.For convenience, the part of the spectrum at zero frequency isomitted. The grating parameters are spacing L ¼ 0:025 cm, mod-ulation parameter m ¼ 0:2; the wavelength is λ ¼ 660nm.

Fig. 2. Fringe contrast in the Fresnel zone as a function of thedistance from a SPG at divergent illumination: L ¼ 0:025 cm,m ¼ 0:2.


at divergent illumination is illustrated in Fig. 3,which shows the diffraction pattern and the spectralcontent of the fringes at z ¼ 1m from the gratingwhen the light source is positioned at d ¼ 12 cm infront of the grating. Since the spread of the patternalong the x axis is kept the same with z, the energycontent of the fringes in Fig. 3 is gradually decreas-ing with z. The wavelength is 830nm. At this wave-length and for the chosen grating parameters thefirst harmonic dominates everywhere for the consid-ered range of distances. More than that, the con-tribution of the second harmonic is in practice negli-gible. The height of the peak of the first harmonicalong z is a sum of a constant background and an os-cillating term whose period increases with the dis-tance from the grating.

B. Comparison with a Sinusoidal Amplitude Grating

It is interesting to compare the fringes created by theSPG in the Fresnel zone with the fringes of a SAGwith transmission [28],

τAðxÞ ¼ A0 þ 2A1 cosð2πx=LÞ; ð14Þ

where A0 and A1 are constant amplitudes. The inten-sity of the diffraction pattern in the Fresnel zone atillumination with a point source located at distance din front of the SAG is given by [42]

IAðx; zÞ ¼ A20 þ 2A2

1 þ 4A0A1 cos� πλzdL2ðzþ dÞ

�

× cos�

2πxLð1þ z=dÞ

�þ 2A2

1 cos�

4πxLð1þ z=dÞ

�¼ A2

0 þ 2A21 þ 4A0A1 cos α × cos 2β

þ 2A21 cos 4β: ð15Þ

As follows from the above expression, the ratioη between the energy concentrated in the secondand first harmonics varies strongly with z. At givenvalues of L, d, and λ there exist regions along the zaxis with a dominating second harmonic. From thecondition

zmfh ¼ nL2

2λd − nL2 ; n ¼ 2kþ 1; k ¼ 0; 1; 2…;

ð16Þ

we obtain the distance zmfh at which the first har-monic is missing. For the z values in the vicinity ofa given zmfh value the ratio η substantially increases.This means that the regions close to a given zmfhshould be strictly avoided. As seen from Fig. 4, whichgives zmfh as a function of the distance d for differentwavelengths at L ¼ 0:025 cm, the acceptable valuesof d in the near-infrared region, where powerfuldiode lasers are available, should be of the order of8–9 cm. Figure 5 shows the visibility of fringescreated by the SAG with L ¼ 0:025 cm at A0 ¼ 8A1for d ¼ 10 cm and A0 ¼ 4A1 for d ¼ 8 cm. IncreasingA1=A0 at d ¼ 10 cm leads to an unacceptable rise ofthe energy in the second harmonic. The case of A0 ¼4A1 and d ¼ 8 cm corresponds to fringes with thesame contrast as the fringes created by the SPG

Fig. 3. Diffraction pattern (top) and spectral content (bottom) ofthe fringes along the z axis at divergent illumination for d ¼ 12 cm,λ ¼ 830nm, L ¼ 0:025 cm,m ¼ 0:2. The zero-order term is omittedfor clarity.

Fig. 4. Distance zmfh at which the first harmonic is missing as afunction of the location of the illuminating point source; the mean-ing of the parameter n is clarified in Eq. (16).


with L ¼ 0:025 cm and m ¼ 0:25 cm (Fig. 2). To com-pare the spectral quality of fringes that have beencreated by the SPG and SAG with the same visibilityin the near-infrared region, we evaluate the ratio ηfor both gratings at distances varying from z ¼ 0:5mto z ¼ 1m. The spacing of both gratings is L ¼0:025 cm; for the SPG we have m ¼ 0:2, whereas forthe SAG the condition A0 ¼ 4A1 is fulfilled. The loca-tion of the point source in front of each grating is cho-sen to ensure the optimal fringes, i.e., fringes withthe least possible contribution of the second harmo-nic. The evaluation results shown in Fig. 6 for twowavelengths confirm the much better performanceof the SPG in the near-infrared region.

3. Simulation

To evaluate the systematic errors of a 3D surfacereconstruction due to the presence of higher harmo-nic content in the SPG created fringes we simulateda profilometric measurement of a dome in the case ofa conventional cross-axes optical arrangement of theprofilometric system [1]. In this arrangement, a lightprojector that consists of a point light source and aSPG creates a regular fringe pattern on the object,

Iðx; yÞ ¼ I0ðx; yÞ þ IVðx; yÞf ½φðx; yÞ þ ϕ�; ð17Þwhere φðx; yÞ is the phase term related to the mea-sured parameter and ϕ is an additional phase termintroduced during the formation of the waveform f .The object reflects the deformed light pattern whenobserved at another angle. Analysis of the deformedimage captured with a CCD camera at normal view-ing direction to a certain reference plane R connectsthe phase differenceΔφ ¼ φ − φR at the camera pixelwith coordinates (x ¼ iΔx, y ¼ jΔy, i ¼ 1;…Nx, j ¼1;…Ny) to the depth (or height) h of the current pointof the object surface by the expression [1]

hðx; yÞ ¼ L

�lpdþ 1

��sin θ þ ½lc − ðlp þ dÞ� cos θ

lcðlp þ dÞ�

−1

×�1þ x × sin θ

lp þ d

�2Δφðx; yÞ; ð18Þ

where Δx, Δy give the sampling steps along x and yin the recording plane, respectively; Nx, Ny are thenumber of samples along both axes; φRðx; yÞ is thephase distribution corresponding to the referenceplane; lc, lp are the distances from the centers of theexit pupils of the projector and the camera to the ori-gin of the coordinate system on R, respectively; and θis the angle between the axes of the projection andrecording optical systems that lie in the same plane.The simulation includes registration of two sets of Nfringe patterns with and without the object at the fol-lowing system parameters: L ¼ 0:025 cm, m ¼ 0:2,λ ¼ 830nm, d ¼ 12 cm, Nx ¼ Ny ¼ 256, and Δx ¼Δy ¼ 0:075mm at z ¼ 1m; 0:1mm at z ¼ 1:5m and0:13mm at z ¼ 2m. The dome parameters in all si-mulated experiments are as follows: the dome iscut from a sphere with a radius of 145 pixels, andthe dome height at the center with respect to the re-ference plane is 55 pixels. In this way the dome oc-cupies the same area within the “recorded” fringepattern at different distances of the object from thegrating. Since the goal of the simulation is to esti-mate the influence of higher harmonics on the accu-racy of reconstruction of a 3D surface, it is performedwithout any noise sources.

The reconstruction of the 3D surface included thefollowing steps:

i. Simulation of projection and recording of fringepatterns; the recorded intensities Iiðx; yÞ were simu-lated using Eq. (9), in which the terms up to q ¼ 4were kept; the variation of the fringe profile withz on the surface of the object was also taken intoaccount.

ii. Calculation of the wrapped phase maps, φðx; yÞand φRðx; yÞ, for the object and the reference plane byusing a four-step algorithm (N ¼ 4),

φðx; yÞ ¼ arctanI4ðx; yÞ − I2ðx; yÞI1ðx; yÞ − I3ðx; yÞ

; ð19Þ

or a five-step algorithm (N ¼ 5),Fig. 5. Fringe contrast in the Fresnel zone as a function of thedistance from a SAG at divergent illumination; L ¼ 0:025 cm.

Fig. 6. The ratio between the energy concentrated in the secondand first harmonics in the fringes created by a SAG with A0 ¼ 4A1

and a SPGwithm ¼ 0:2 as a function of the distance from the grat-ing; for both gratings L ¼ 0:025 cm.


φðx; yÞ ¼ arctan2½I4ðx; yÞ − I2ðx; yÞ�

I1ðx; yÞ − 2I3ðx; yÞ þ I5ðx; yÞ; ð20Þ

where Iiðx;yÞ ¼ I0ðx;yÞþ IVðx;yÞf ½φðx;yÞþ ði−1Þπ=2�are the recorded fringe patterns.iii. Unwrapping by a quality-guided algorithm

[43] of both phase maps.iv. Calculation of the phase difference Δφ ¼

φ − φR.v. Calculation of the 3D surface hðx; yÞ using Eq,

(18) and evaluation of the reconstruction errorhðx; yÞ − hðx; yÞ, where hðx; yÞ describes the realsurface.

Figure 7 illustrates the above-listed steps de-picting the gray scale 8 bit maps of bφðx; yÞ, φRðx; yÞ,hðx; yÞ, and hðx; yÞ − hðx; yÞ. As can be seen, the recon-struction error, which has a systematic nature, isregularly distributed within the dome area takingpositive and negative values. To estimate effectivelythe magnitude of this error, we found the maximumpositive value, εþi , i ¼ 1;…256, and the maximumnegative value, ε−i , i ¼ 1; 2;…256, of the differencehðx; yÞ − hðx; yÞ in each row of the simulated 256 ×256 fringe pattern and calculated the maximumspread of the reconstruction error in a row as εi ¼εþi − ε−i , i ¼ 1;…256. Figure 8 gives an example ofthe performed calculations. We see that the quanti-ties εþi , ε−i exhibit more or less constant behaviorwithin the dome area. This means that we coulduse the average value �ε of εi to characterize the ac-curacy of reconstruction; averaging is performed overall rows within the dome area. The parameter �ε giveson average the maximum possible difference be-tween the positive and negative deviations of thereconstructed 3D surface from the real one. The re-construction error outside the dome area is zero inthe absence of noise. Figure 9 presents the resultsfrom the simulation at three distances from the

Fig. 7. Grey scale 8 bit maps of (top) wrapped phase distributionsφðx; yÞ and φRðx; yÞ corresponding to the object (left) and the refer-ence plane (right); (bottom) reconstructed dome surface hðx; yÞ(left) and distribution of the reconstruction error hðx; yÞ − hðx; yÞ(right); L ¼ 0:025 cm, m ¼ 0:2, z ¼ 1m, λ ¼ 830nm, d ¼ 12 cm,θ ¼ 36°.

Fig. 8. Deviation of the reconstructed surface from the real sur-face averaged in a row as a function of the row number;L ¼ 0:025 cm, m ¼ 0:2, θ ¼ 36°.

Fig. 9. Mean value of the maximum spread of the deviation be-tween the reconstructed surface and the real surface as a functionof the angle θ; L ¼ 0:025 cm, m ¼ 0:2.

Fig. 10. (top) Optical arrangement for determination of fre-quency content of patterns projected by a sinusoidal phase grating:DL, diode laser; L1, L2, lenses; SPG, sinusoidal phase grating;GGS, ground glass screen. (bottom) Cross-axes optical arrange-ment for profilometric measurement; RP, reference plane.


SPG for the four-step algorithm (19). The systematicreconstruction errors obtained with the five-step al-gorithm are practically the same. The value of �ε inmicrometers is given as a function of the illumination

angle θ. As it should be expected, registration of thedeformed fringe pattern at a larger angle improvesthe measurement accuracy, but thus the shadowedarea of the object also increases. We see that by aproper choice of all parameters of the profilometricsystem, the systematic error in reconstruction of a3D surface can be made negligibly small (less than20 μm between the maximum elevation and depres-sion on the reconstructed surface with respect to thereal one).

4. Experimental Check

To evaluate the visibility and the frequency contentof the fringes created by the SPG, we performed testmeasurements at λ ¼ 660nm and 830nm using theoptical arrangement in Fig. 10 (top). The divergentlight beam (expanded by the lens L1) from the diodelaser DL illuminates the grating SPG. The distancebetween the light source and the grating remainedfixed. We recorded the fringe patterns projected ontothe ground glass screen, GGS, at varying distancez (as indicated in Fig. 10, top) for two different

Fig. 11. Visibility of the fringes created by a SPG with L ¼0:025 cm (experiment).

Fig. 12. Experimentally determined frequency content of fringes created by a SPG with L ¼ 0:025 cm: (a), (b) λ ¼ 660nm; (c),(d)λ ¼ 830nm; ðaÞ; ðcÞ − d� ¼ 9 cm, ðbÞ; ðdÞ − d� ¼ 10:5 cm; the zero-order term is omitted for clarity.


positions of the projector lens L2 with respect to thegrating. The phase grating was recorded on a high-resolution silver halide holographic plate HP-650,laboratory production of CLOSPI-BAS (Bulgaria)by using an interferometric optical arrangementwith a He–Ne laser (λ ¼ 632:8nm, 30mW). An inter-ference pattern was generated using adjustable Mi-chelson interferometer that provided equidistantsinusoidal fringes whose period could vary in broadlimits. The chemical processing of the holographicplates was realized with a fixing developer, which en-sured formation of colloidal silver grains, i.e., practi-cally phase recording, low level of noise, and highdiffraction efficiency up to 70% in the near-infraredregion. The required modulation of the recordedinterference patterns was achieved by proper selec-tion of exposures to fall into the dynamic range0:5–1:5mJ=cm2.Figure 11 depicts the visibility of fringes as a func-tion of the distance from the SPG for the two posi-tions of the lens L2 with respect to the grating,characterized by the distance d�. We see that starting

from some distance away from the Talbot plane, thefringes created by the SPG are characterized with agood constant visibility. Figure 12 presents the spec-tral content of the experimental fringes created by aSPG with L = 0.025 cm at illumination with 660 and830nm. The contribution of the second harmonic ismuch less in comparison to the first harmonic. Theresult of the profilometric measurement is given inFigs. 13 and 14. A five-step algorithm was used forphase retrieval. To remove the speckle noise fromthe recorded fringe patterns, we adopted a signal-dependent multiplicative noise model and applied ahomomorphic transformation combined with a locallinear minimum mean-square error estimator [44].Figure 13 depicts the reconstructed 3D surface of atest object (a dome). Figure 14 gives a cross sectionof this reconstruction through the dome apex alongthe diameter subtended by the dome circle. The re-constructed profile is compared to the profile mea-sured with a stylus. The standard deviation of thedifference hðx; yÞ − hðx; yÞ between the reconstructedand real profile was evaluated to be 81 μm, which cor-responds to less than 1% relative error at the domeapex. The standard deviation was estimated usingtwo measurements with the stylus, each of themyielding 27 points. The maximum absolute value ofhðx; yÞ − hðx; yÞ is about 330 μm.

5. Conclusion

In conclusion, the analysis made by solution ofthe Fresnel diffraction integral confirms that the si-nusoidal phase grating can serve as a projectionelement in a profilometric system. The obtained so-lution permits calculation of the grating diffractionpattern at plane-wave and divergent illumination.A proper choice of grating parameters at divergentillumination ensures practically sinusoidal fringesof constant contrast in a large spatial region. The si-mulation of the profilometric measurement madewith the chosen parameters yields systematic errorsof the 3D reconstruction that can be neglected. Theusage of the sinusoidal phase grating is especiallyprofitable in the near-infrared region where it out-performs the sinusoidal amplitude grating by the de-gree of suppression of higher harmonics. In the nearinfrared, the sinusoidal phase grating ensures prac-tically constant spectral content and visibility offringes at large distances from the grating, thusmak-ing measurement of deep scenes possible in a largedynamic range. Using near-infrared wavelengths formeasurement of the 3D coordinates is advantageousfor several reasons [45]. First, high-power low-costdiode lasers are available in practically the wholenear-infrared region. Second, especially for a gratingrecorded on silver halide holographic plates and de-veloped to form nano-sized colloidal silver grains, ab-sorption strongly decreases in the near infrared, andhence diffraction efficiency becomes almost twice ashigh in comparison to the visible region. Third, mea-surement of coordinates or displacement could becombined with simultaneous acquisition of the color

Fig. 13. Reconstructed 3D surface of a dome (the five-step algo-rithm has been used for phase retrieval).

Fig. 14. Comparison of the reconstructed profile through thedome apex with the real one measured with a stylus.


coordinates of the object in the visible spectral range.The experimental results obtained for the fringecontrast are consistent with the theoretical conclu-sions. The profilometric measurement performedwith a dome as a test object shows the high accuracyof the 3D reconstruction. The sinusoidal phase grat-ing is easy to fabricate, and its implementation infringe projection profilometry makes possible con-struction of portable low-cost devices.

This work is supported by the European Commis-sion within the Sixth Framework Programme (FP6)under grant 511568. Data processing was partiallysupported by VSSoft Co.

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