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Optics and Lasers in Engineering 48 (2010) 877–881

Contents lists available at ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Iterative estimation of the topography measurement by fringe-projectionmethod with divergent illumination by considering the pitch variation alongthe x and z directions

Amalia Martınez a,n, J.A. Rayas a, H.J. Puga b, Katia Genovese c

a Centro de Investigaciones en Optica, A.C. Apartado Postal 1-948, C.P. 37000, Leon, Gto., Mexicob Instituto Tecnologico de Leon, Leon, Gto., Mexicoc Dipartimento di Ingegneria e Fisica dell’Ambiente, Universit �a degli Studi della Basilicata, Viale dell’Ateneo Lucano, 10-85100 Potenza, Italy

a r t i c l e i n f o

Article history:

Received 22 September 2009

Received in revised form

23 March 2010

Accepted 9 April 2010Available online 28 April 2010

Keywords:

Projected fringe

Non-collimated projections

Structured light techniques

66/$ - see front matter & 2010 Elsevier Ltd. A

016/j.optlaseng.2010.04.002

esponding author. Tel.: +52 477 4414200; fa

ail address: amalia@cio.mx (A. Martınez).

a b s t r a c t

This paper deals with the application of the fringe-projection method for the measurement of large

object surfaces. In this case, the generated fringe pattern is equivalent to that obtained by using

divergent beams, i.e. the relation between phase and height is not linear. In this work, both the pitch

variation in x-direction of the projected grating and the perspective problem of CMOS camera are taken

into account for the formulation of a novel algorithm that iteratively retrieves the correct surface point

positions. The methodology is applied to a large car component and the obtained results are compared

with those obtained with a commercial scanner (Z-Scanner 700).

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The retrieval of the three-dimensional shape of an object is anissue of great interest for a wide range of industrial [1], technical[2], and medical [3] applications. Coordinates measurementmachines (CMMs) are well established and widely accepted inindustry for mechanical part inspection. However, CMMs possesssome limitations such as the high cost, the low measurementspeed and the sparseness of the measurement points.

Alternatively, non-contact optical systems that use structuredlight are available for contouring 3D surfaces [4–6]. The rationalebehind structured light techniques consists in the projection of alight pattern of known geometry onto the surface to be measured.The desired 3D information can be obtained by analyzing thedeformation of the pattern onto the object with respect to thereference one [7].

Recently, there has been an ever more growing interest forfringe-projection techniques due to the availability of a fasterimage-projection and image-acquisition technology that makes itpossible 3-D shape measurement in real-time [8–10]. With thesetechniques, a regular fringe pattern is projected onto anobject surface and then viewed from another direction. Theprojected fringes are modulated by the topography of the object.The intensity distribution of the deformed fringe pattern as

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x: +52 477 4414209.

imaged in the plane of a CCD camera sensor, is then sampled andprocessed to retrieve the resulting phase distribution. Amongphase-measuring techniques, the phase shifting methods [11,12]are commonly used to increase measurement resolution byacquiring a set of n images corresponding to n known phase shiftsof the projected grating. Alternatively, the Fourier-transformanalysis [13] can be used to retrieve the phase distribution fromthe analysis of one single image. When implementing thesetechniques, a phase-to-height conversion algorithm related to thespecific optical set-up is required to correctly reconstruct the 3-Dcoordinates of the object surface. The system parameters involvedin the formulation of the algorithm include the angle between theaxes of the projector and the camera, the distance between thecamera lens and the CCD sensor, the distance between the cameraand reference plane, and the fringe frequency.

In this paper we describe some problems entailed with theimplementation of the fringe-projection method. Camera calibra-tion is a crucial issue for computer vision where many tasks requirethe computation of accurate metric information from images.The calibration of the camera consists in the definition of thetransformation which maps 3D points of a certain scene or objectinto their corresponding 2D projections onto the image plane ofthe camera sensor. The accuracy of the 3D reconstruction willbe affected by how reliably the camera and the projector will bemodeled into the system. Hence, the lens distortion of the cameraand the projector should be considered during calibration [14–16].

Barrel distortion occurs when the magnification at the centreof the lens is greater than at the edges. A higher quality lens can

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A. Martınez et al. / Optics and Lasers in Engineering 48 (2010) 877–881878

be used to limit this effect but this would results in an increasedcost of the image capture system. Barrel distortion is primarilyradial in nature, thus a relatively simple one-parameter model canaccount for the most of the distortion. A cost effective alternativeto an expensive lens is the use of algorithms based on fieldprogrammable gate arrays.

When contouring large surfaces, the use of telecentric systemswould restrict the size of the measured field to the diameter ofprojecting lens [17]. On the other hand, if the projection system isnot telecentric the lines of the projected grating are not moreequispaced in the reference plane. Contouring surfaces are thusno longer flat. There are several ways for taking into account thiseffect. For example, it is possible to use a specially designedgrating whose projection in the reference plane will give rise toequispaced lines [18]. Alternatively, other authors proposed aformulation which takes into account the relation of the fringepitch with the direction perpendicular to the viewing direction[7,19–21].

Finally, since finite distances are entailed in the imageformation process, the distortion due to viewing perspectiveshould be considered. Due to this effect, in fact, a point P on thesurface will be apparently at P0 when viewed through the grating(Fig. 1). By applying simple trigonometric relations the actualcoordinates (xr, yr) can be obtained from the measuredcoordinates (xa, ya): this permits the measured surface to bemapped into the actual surface thus correcting the viewingperspective [22].

Another parameter that could affect the topography measure-ment is the deviation of observed fringes from the sinusoidalwaveform. This is often due to nonlinearity of the detector and/orthe projector and it results in phase error and thereforemeasurement error [23,24]. In some works, a gamma correctionof digital projection to produce the ideal sinusoidal intensitydistribution have been proposed [25] or sinusoidal fringe patternsare generated by defocusing binary patterns [26]. From simulatedresults, it was observed that the ridge-based methods areless sensitive to the higher harmonics than methods based onforward and inverse transforms [27]. In the forward and inversetransform methods, higher harmonics extend into the funda-mental frequency component, thus affecting phase retrieval. Inthe ridge-based methods, the ridge will not be severely affectedby the higher harmonics, since the ridge has the highestamplitude of the spectrum and is usually quite far away fromthe harmonics. Therefore, these ridge-based methods are lesssensitive to the higher harmonic components.

x

CMOS

z

xa xr

z

lk

object lk

xa

z

xa − xr =

PP´

Fig. 1. Perspective transformation.

System calibration techniques have been developed to obtain themapping relationship between the phase distribution and the 3-Dobject-surface coordinates, without explicitly determining thesystem-geometry parameters. Instead, calibration parameters,which implicitly account for the system geometry, are determined[28,29]. In the calibration method, a plane is positioned successivelyat different positions from the camera. Usually, a marked point onthe first calibration plane is used as the origin of the world referencesystem, then the following calibration planes are chosen parallel tothe first one and their displacements with respect to the first planehave to be known with high accuracy. Therefore, to obtain goodcalibration results, a precise linear z stage has to be used. The maindrawback of the system comes from practical limitations, such as itsplane position restriction or the difficulty of calibrating bigmeasurement volumes.

In this work, we describe the rationale and experimentalarrangement of a 3-D measurement system that uses structuredlight illumination (fringe projection) to obtain the topography oflarge objects. An algorithm that considers both the perspectiveproblem and the pitch variation due to divergent illumination ishere developed and tested. With respect to the commercialscanner, the present approach allows to obtain a larger number ofmeasurement points and a faster data acquisition and processing.

2. Basic principle

Fig. 2 reports the basic setup for 3-D measurement by patternprojection. Fringe projection is widely used for measuring theshape and out-of plane deformation of small objects, however fewworks report its application large surfaces (above 1 m2). When thetopography measurement of large sized objects is entailed, it isunpractical to use parallel illumination. The primary issue to dealwith is the need to illuminate the whole object surface at once(along its width and depth) with a regular fringe pattern. In thiscase, due to the large size of the object, the illumination cannot beparallel. Since the illuminating beam needs to be divergent [2] theprojected fringes are no more parallel and equispaced. Indeed, ifthe illumination is assumed to be generated by two point sources,the surfaces of equal phase difference are hyperboloids andconsequently, the fringes created are curved and no moreequidistant [7,19,20,21]. The fringe-projection scheme withdivergent beam is showed in Fig. 3. A lens is used to project thegrating onto the object surface. By considering two planes 1 and 2it is possible to notice as the projected gratings shows a differentpitch G1 and G2 along the direction of the illumination beam.Moreover, if the object under study possesses a large depth, eventhe variation of the pitch along the z-axis should be included inthe analysis. In this case, the perspective problem should be takeninto consideration. In view of the arguments listed above, theheight results to be given by [7]

z¼DfUS ð1Þ

lp

lk

Projector

CMOS

x

z�0

Fig. 2. Fringe-projection geometry.

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Plane 2

x

CMOS

lp

lk

z

Grating

Plane1

G1

G2

Lens

�0

px0

Fig. 3. Schematic representation of a projected fringes system which shows the

variation of period.

Input Data

0xp , x, and y

Compute zwith Eq. (1)

Compute rx , ry and xrp with Eqs. (2) and

the calculated value of z

Compute a new value of z which is renamed rz with Eq. (1) and the new

values of

rx , ry and xrp instead of 0xp , x, and y

czz r ≤−

Stop

rzz =No

Yes

zzr =

Fig. 4. Flow chart of the developed correction algorithm to calculate the object

topography.

A. Martınez et al. / Optics and Lasers in Engineering 48 (2010) 877–881 879

where

S¼px0

2p cosy0 siny0þðlk�lpcosy0Þ

lplkx

� ��1

1þxsiny0

lp

� �2

ð2Þ

and

Df¼fo�fr ð3Þ

In Eq. (2), S [mm/rad] is the sensitivity of the system and itdepends on the set-up geometry, namely the pitch pxo ofthe fringes at x¼0, the angley0 between the projection andthe observation directions, the distance lp between projectorand reference plane for x¼0, and distance lk between the CCDand the reference plane at x¼0. This formulation of the sensitivitytakes into account that the fringes are neither rectilinearnor equispaced along the x-axis. In Eq. (3), f0 is the measuredphase map of the object and fr is the measured phase mapof a reference planar surface at z¼0. Hence, once the phasemap of the object and the reference plane and the sensitivityS have been determined, the object height can be easilycalculated.

2.1. Perspective transformations

Fig. 3 sketches a generic object as viewed from a camera. Dueto the perspective effect, a given point with coordinates (xr, yr)

will appear with coordinates (xa, ,ya). From similarity betweentriangles it is possible to obtain the following relations:

xr ¼ xa 1�z

lk

� �, yr ¼ ya 1�

z

lk

� �, pxr ¼ px0 1�

z

lk

� �ð4Þ

Eq. (4) represents the perspective transformation and must betaken into account e.g. when comparing a real object with thecomputer-generated corresponding geometry.

On the basis of the Eqs. (1–4), a routine was coded intoMathcad environment to retrieve the object topography accordingto the flow chart reported in Fig. 4. The developed routine allowsto progressively approach the correct value of the object pointlocation by iteratively updating the parameters of the system onthe basis of the current estimated point position.

3. Experimental setup and results

A typical setup fringe-projection profilometry is illustrated inFig. 5. In this work, the experimental system mainly consists in aDLP digital projector (Texas Instruments, size 0.7 inchs, resolution

1024�768 pix) and a CMOS camera (PixeLink, resolution

1280�1024) A computer generated continuously shiftingpattern with straight and equally spaced sinusoidal fringes isprojected onto the specimen surface and the reference plane andthen captured by the CMOS camera in the normal view direction.The proposed technique requires the geometric parameters of theexperimental setup to be precisely determined.

The object chosen as test case is a car part (fender with width:1.20, height: 0.41, and deep: 0.55 m approximately) which is fixedvertically on a tripod, as shown in Fig. 5. The camera is placed

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Fig. 5. Picture of optical system used in the fringe-projection technique: (a) object, (b) camera, and (c) multimedia projector.

Fig. 6. (a) Fringes projected on a car part, (b) Measured wrapped phase map, and

(c) Unwrapped phase map.

2593-253

602

409

216

23

-531 -237 57 352 646

y (mm

)

z (m

m)

x (mm)

Fig. 7. Shape of the car part obtained by fringe-projection technique: (a) Dark dots

plot is the corrected shape and (b) Gray dots plot is the non-corrected shape.

-531 -237 57 352 646

x (mm)y

(mm

)

259

3

-253

131

-125

Fig. 8. xy coordinates associated before and after of the perspective correction.

A. Martınez et al. / Optics and Lasers in Engineering 48 (2010) 877–881880

parallel to the object, at a distance lk of about 4.08 m. Theprojection head is positioned 0.81 m away on the left side(x¼� .81, y¼0, z¼4.08 m and lp¼4.16 m), resulting in a projec-tion angle of about 111, with respect to the viewing direction. Onehundred nine retro-reflective targets are fixed on the surface anddefine the reference points. Their position has been previouslymeasured with the commercial scanner (Z-Scanner 700: xy

accuracy up to 50 mm, resolution 0.1 mm in z), and they are usedto be compared with the obtained by projected fringes technique.Then a fringe-projection measurement is performed in orderto capture the car part phase-map. The method used in this work

to get the phase map is the phase-stepping technique of 4 steps.Fig. 6 shows the experimental data obtained in this study. Inparticular, Fig. 6a reports the image of the object with theprojected fringes onto it, Fig. 6b is the wrapped phase, and Fig. 6cis the unwrapped phase. Two 3-D height maps are shown in Fig. 7.The height map with dark dots is obtained by using the Eqs. (1)and (4) and the procedure implemented into the algorithmsketched in Fig. 4. A value of 0.1 mm for the stop criterionparameter c in the algorithm was chosen since this valuecorresponds to the z-resolution of the commercial scanner usedfor getting comparative results. With this choice, three iterationsare demonstrated to be sufficient to reach the value of c and quitthe loop. The topography obtained in this case is corrected.The height map with gray dots was calculated only using theEq. (1). A maximum difference of 7.74 cm is found between

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Fig. 9. Comparison between the shape obtained with projected fringes (gray solid

surface) and the Z-Scanner 700 (black solid surface): (a) without correction,

(standard deviation of 10.6 mm) and (b) with correction, (the standard deviation is

1.9 mm).

A. Martınez et al. / Optics and Lasers in Engineering 48 (2010) 877–881 881

the corresponding z coordinates of two 3D maps. Fig. 8 shows thex–y coordinates position before and after of the perspectivecorrection.

The so retrieved object topography was compared to thatmeasured with the commercial scanner. Fig. 9 presents the objecttopography obtained with the fringes projected technique withand without correction compared with the obtained with thescanner.

4. Conclusions

A fringe-projection method has been applied to retrieve theshape of a large object. The obtained results have been comparedwith those obtained with a commercial scanner.

A great discrepancy between topography data has beenobserved when a proper correction for perspective error andpitch variation due to divergent illumination has not beenconsidered.

The main contribution of this work is to demonstrate as, withthe due corrections, fringe projection represents a fast and costeffective method for the topography measurement of largeobjects.

Acknowledgments

The authors wish to acknowledge the financial support for thisresearch to the Consejo Nacional de Ciencia y Tecnologıa deMexico, CONACYT, under grant 48286-F.

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