Optics and Lasers in Engineering 50 (2012) 256–261

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Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

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

Transformation of phase to (x,y,z)-coordinates for the calibration of a fringeprojection profilometer

Jesus Villa a,n, Marıa Araiza a, Daniel Alaniz a, Rumen Ivanov b, Marvin Ortiz a

a Laboratorio de Procesamiento Digital de Senales, Facultad de Ingenierıa Electrica, Universidad Autonoma de Zacatecas, Av. Ramon Lopez Velarde #801,

C.P. 98000, Zacatecas, Mexicob Facultad de Fısica, Universidad Autonoma de Zacatecas, Calz. Solidaridad Esquina Paseo de la Bufa s/n, C.P. 98060, Zacatecas, Mexico

a r t i c l e i n f o

Article history:

Received 6 May 2011

Received in revised form

3 August 2011

Accepted 23 August 2011Available online 6 September 2011

Keywords:

Fringe projection profilometry

Calibration

Phase measuring

66/$ - see front matter & 2011 Elsevier Ltd. A

016/j.optlaseng.2011.08.005

esponding author.

ail address: jvillah@uaz.edu.mx (J. Villa).

a b s t r a c t

In this work we present a phase to (x,y,z)-coordinates transformation method for the calibration of a

fringe projection profilometer. Our technique is divided in two parts: (1) phase to z transformation (for

axial calibration) based on the typical polynomial fitting which uses a flat plane placed at several z

positions to measure the phase of the projected fringes. (2) Phase to x and y transformation (for

transverse calibration) based on the use of a crossed gratings pattern and a Fourier phase measurement

method to determine x and y coordinates at several z positions. As will be shown the use of the crossed

gratings pattern and the Fourier phase measurement method for transverse calibration is advantageous

in several aspects: an unique image can give us x and y information at once. It provides x and y

coordinates at each pixel in the image avoiding the use of interpolation methods. We present some

experimental results and explain the viability of the proposed technique.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The 3D dimensioning is a very important task for severalapplications such as manufacturing, design and entertainment.One of the most common ways to do this is by means ofcoordinate measuring machines (CMMs) which use mechanicalparts and sensors. However, the use of CMMs has severaldisadvantages, among them the high cost and low speed ofmeasurement. On the other hand, non-contact optical methods,such as the structured-light-based (fringe projection) techniques[1–9] are rapidly increasing its acceptation in industrial applica-tions, mainly because of its high speed and better cost-benefit.

A key aspect for accurate measurements using fringe projec-tion techniques is the calibration procedure for obtaining real(x,y,z)-coordinates. Several calibration methods have been pro-posed, among them the neural network based [10,11], polynomialapproach [12,13] and model-based [14–19]. In general, the neuralnetwork and polynomial techniques use a z stage to processseveral planes along z-axis to find a rule for phase to depth

conversion, while the model-based techniques consist of usingcontrol points to model the camera-projector system.

In many fringe projection systems only axial calibration (phaseto depth conversion) is considered [8,10,11], however in mostapplications transverse coordinates are also important. Sitnik

ll rights reserved.

et al. [13] proposed the transverse calibration by the use of adotted plane mounted on a z stage, and the conversion betweenpixels and real coordinates were realized using polynomial inter-polation for a set of planes along z-axis. This methodology,however, is cumbersome owing that requires accurate methodsto detect markers’ centers and complicated surface interpolationalgorithms. An interesting work reported by Liu et al. [12] usesthe principle of phase to transverse coordinates by using sinusoi-dal gratings along horizontal and vertical directions sequentiallyplaced in planes along z-axis. This methodology is advantageousowing that phase detection techniques have high degree ofaccuracy and spatial sampling is realized over every pixel in theobserved image.

In this work we present a transformation method of phaseto (x,y,z)-coordinates for the calibration of a fringe projectionsystem. Our proposal uses a temporal phase measurementtechnique [20,21] and polynomial fitting to process depth infor-mation. To process transverse coordinates we propose aFourier based method for phase measuring to process a patternof crossed gratings positioned at several planes along z-axis.As will be shown the use of this kind of pattern and theFourier method has the advantage that phase measurement forhorizontal and vertical calibration can be fast and easily per-formed at once.

This paper is organized in the following way: in Section 2 weexplain the proposed method for phase to (x,y,z)-coordinatestransformation. In Section 3 we present some experimentalresults and discussions. Finally conclusions are given in Section 4.

J. Villa et al. / Optics and Lasers in Engineering 50 (2012) 256–261 257

2. The principle of fringe projection profilometry

A basic fringe projection system consists of a digital videoprojector and a digital camera (a CCD camera). The projector isused to project the fringe pattern onto the measured object whilethe CCD camera captures the distorted fringe image (Fig. 1). Thebasic principle of this system consist of processing the observedimage captured by the CCD camera. This image contains distortedfringes owing to the object’s dept variation. The key task in theprocessing of the fringe pattern is the so-called phase detection.Phase information is important because it is directly related withobject’s depth information. Several phase detection techniqueshave been proposed for this purpose, among them the classicalFourier technique [5] and the temporal technique [20,21].Although phase detection is the key of fringe projection profilo-metry, phase to dept transformation is essential for obtaining realdimensions, even more, transformation of pixel image coordi-nates to real transverse coordinates is also necessary to comple-tely measure object’s real (x,y,z)-coordinates. The method todetermine the rules for these transformations is the so-calledcalibration.

3. Phase to (x,y,z)-coordinates transformation

To analyze the problem of phase to (x,y,z)-coordinates trans-formation, consider the real coordinates (x,y,z) of a point in theobject to be measured. The phase f measured at the pixel (i,j)which correspond to (x,y,z) are related by means of the followinggeneric transformation functions:

f¼ T1ðx,y,zÞ, i¼ T2ðx,y,zÞ, j¼ T3ðx,y,zÞ: ð1Þ

These functions are generally non-linear and dependent on thesystem’s parameters. On the other hand, the relation betweencoordinates (x,y,z) and ði,j,fÞ can be modeled by means of [12]

z¼ TðfÞ, x¼mxzþ lx, y¼myzþ ly: ð2Þ

In this case T is a monotonic non-linear function (see reference[12] for more details). A key aspect of these equations is that thedetermination of z depends only on f, and (x,y) coordinatesdepend on z at the given pixel, this means that to determine x

and y coordinates it is necessary to first determine z. Function T

Fig. 1. Schematic diagram of a basic profilometer. The projected fringes and y-axis

are perpendicular to the plane of the figure.

and coefficients mx, lx, my and ly are also dependent on thesystem’s parameters which are different for each pixel and maybe determined by means of a calibration procedure. Note that inEqs. (2) the spatial dependence (i,j) is omitted for simplicity.

3.1. Phase to dept transformation (axial calibration)

In this section we describe the proposed technique to findparameters for modeling function TðfÞ. Before this, we shoulddiscuss the phase detection technique used for this purpose.Although several techniques may be used, the main drawbackof them is that spatial phase unwrapping must be carried out tofind the continuous phase. In our case we used the temporalphase measurement method by Huntley et al. [20,21] which cangive us the corresponding spatial phase f already unwrapped. Inparticular, the reversed exponential fringe sequence were imple-mented in our system. In this method the fringe frequency ischanged over time for t¼ s,s�1,s�2,s�4, . . . ,s=2, where s¼ 2n�1,being n the number of phase values to be estimated. The fringepattern is projected over the field of view such that the fringephase ranges in the interval ð�tp,tpÞ. For each t value four phase-steeped fringe patterns are projected and acquired to compute aset of n wrapped-phase values at each pixel in the image. In thisway a one-dimensional phase unwrapping process is carried outover t. In Fig. 2, we show a set of n¼7 unwrapped-phase valuesfor three different pixels. The graphs represent lines with slope owhich is proportional to the object’s dept.

For axial calibration we use the value of o to determine z foreach pixel. Polynomial fitting can be performed by measuring aset of o values for several z values. This may be done by movingalong the z-axis a flat plate using a linear stage (Fig. 3). Themeasurement of o is repeated as the flat plate is successivelytranslated to different z positions. Fig. 4 shows o�z graphs fortwo different pixels in a real experiment, note that the relation isalmost linear, so a low order polynomial may be used to fit thecurve. With our experience, we determined that a fifth orderpolynomial fitting was enough for our purpose.

3.2. Image-coordinates to real transverse-coordinates

transformation (transverse calibration)

The transverse calibration consist of finding the proper set ofparameters mx, lx, my and ly of Eq. (2) for each pixel (i,j) in theimage. One way to do this is by using a flat plate, perpendicular toz-axis, marked with a set of equally spaced circles [13]. Detectingthe center of circles in the image we can know the coordinate (x,y)at the given (i,j) and z. The process is repeated as the flat platewith markers is successively translated to different z positions. In

0 5 10 15 20 25 30 35 40 45 50 55 60 64−100

−50

0

50

100

150

200

t

Pixel 1Pixel 2Pixel 3

φ(t)

Fig. 2. Unwrapped phase at three different pixels in a real experiment.

y

x

z

Motorizedlinear stage

Flat plate or crossedgratings pattern

Fig. 3. Motorized linear stage for axial and transverse calibration.

−2 −1.5 −1 −0.5 0 0.5 1 1.50

5

10

15

20

z

Pixel (400,400) Pixel (100,400)

ω (rad/t)

Fig. 4. o�z graphs for two different pixels in a real experiment (z is in

centimeters).

Fig. 5. Images of the crossed gratings pattern at (a) z¼0 cm and (b) z¼20 cm.

Note the mark at the center to identify the origin.

J. Villa et al. / Optics and Lasers in Engineering 50 (2012) 256–261258

this way the parameters may be estimated by simple linearfitting. The drawbacks of this technique is that effective circledetection may be complicated, even more, for every z position, acomplicated surface interpolation algorithm is required to deter-mine (x,y) in pixels without markers.

Our technique to determine mx, lx, my and ly consists of using apattern of crossed gratings along x and y directions instead ofcircular markers (Fig. 5). In this way, by processing the acquiredimage to determine the fringe phases along both x and y direc-tions (Fx and Fy), and knowing that fringe pitch P of the grating iswell defined, we can easily deduce that x¼ ðP=2pÞFx andy¼ ðP=2pÞFy. These formulas, however, can be applied just if weuse a simple marker on the pattern to identify the origin of thex�y plane.

Consider the pattern of crossed gratings image can be modeledusing the Fourier series expansion as

gðx,yÞ ¼X1

n ¼ �1

anei2pn½uxxþfðx,yÞ� �X1

m ¼ �1

bmei2pm½vyyþfðx,yÞ�

¼ a0b0þa0b1ei2p½vyyþfðx,yÞ� þa1b0ei2p½uxxþfðx,yÞ�

þa0b�1e�i2p½vyyþfðx,yÞ� þa�1b0e�i2p½uxxþfðx,yÞ� þ � � � , ð3Þ

where fðx,yÞ represents the phase modulation owing to systemdistortions, such as misalignment and camera aberrations. Now, ifwe define

cðx,yÞ ¼ ei2pfðx,yÞ, ð4Þ

then we can rewrite (3) as

gðx,yÞ ¼ a0b0þa0b1cðx,yÞei2pvyyþa1b0cðx,yÞei2puxx

þa0b�1cnðx,yÞe�i2pvyyþa�1b0cnðx,yÞe�i2puxxþ � � � , ð5Þ

where n denotes complex conjugate. Then, the Fourier transformGðu,vÞ ¼F fgðx,yÞg is expressed as

Gðu,vÞ ¼ a0b0dðx,yÞþa0b1Cðu,v�vyÞþa1b0Cðu�ux,vÞ

þa0b�1Cnðu,�v�vyÞþa�1b0Cnð�u�ux,vÞþ � � � : ð6Þ

Isolating second and third terms from (6) by means of filterscentered at ð0,vyÞ and ðvx,0Þ respectively, we obtain

Gyðu,vÞ ¼ a0b1Cðu,v�vyÞ and Gxðu,vÞ ¼ a1b0Cðu�ux,vÞ: ð7Þ

Taking their inverse Fourier transforms we obtain the complexsignals

gyðx,yÞ ¼F�1fGyðu,vÞg ¼ a0b1ei2p½vyyþfðx,yÞ�,

gxðx,yÞ ¼F�1fGxðu,vÞg ¼ a1b0ei2p½uxxþfðx,yÞ�: ð8Þ

Now, defining Fyðx,yÞ ¼ vyyþfðx,yÞ and Fxðx,yÞ ¼ uxxþfðx,yÞ,then

WfFyðx,yÞg ¼ Argfgyðx,yÞg and WfFxðx,yÞg ¼ Argfgxðx,yÞg, ð9Þ

J. Villa et al. / Optics and Lasers in Engineering 50 (2012) 256–261 259

where W is the so-called wrapping operator, so that to obtain Fx

and Fy we must apply a phase unwrapping process.The use of this kind pattern and the phase detection described

above for transverse calibration have great advantages: phases Fx

and Fy can be fast and automatically determined at once usingfast Fourier transformations, simple filtering and phase unwrap-ping processes; experimental lighting conditions do not severelyaffect phase measurements; it works as a surface interpolatingtechnique which results in a high spatial sampling density overthe entire image. On the other hand, using a dotted plane [13],circle center detection and surface interpolation are cumbersomeand slow processes, and experimental lighting conditions mayseverely affect the circle center detection task.

4. Results and discussions

In this section we present some experimental results andmake a discussion about the effectiveness of the proposed phaseto (x,y,z)-coordinates transformation. First, we describe theexperimental setup for this purpose. We used a commercial SVGALCD projector and a Firewire CCD camera. Both the projector andcamera were controlled through a 2.5 GHz Pentium 4-based

Fig. 6. Fourier transforms of (a) Fig. 5a and (b) Fig. 5b. The circles indicate the

lobes to be isolated.

Fig. 7. Profiles along horizontal lines of (a) Fig. 6a and (b) Fig. 6b.

0 2 4 6 8 10 12 14 16 18 20−7.5

−7.25

−7

−6.75

−6.5

−6.25

−6

z

0 2 4 6 8 10 12 14 16 18 20z

x

Measured XLeast−squares fitting

7.2

7.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

y

Measured YLeast−squares fitting

Fig. 8. (a) Graphs of experimentally obtained z�x and its corresponding least-

squares fitting (at pixel (250,200)). (b) Graphs of experimentally obtained z�y and

its corresponding least-squares fitting (at pixel (100,300)).

J. Villa et al. / Optics and Lasers in Engineering 50 (2012) 256–261260

computer using Matlab for fringe projection, image acquisitionand the fringe image processing methods above described. Weused a motorized linear z-stage (Fig. 3) to mount the flat plate andthe pattern of crossed gratings for axial and transverse calibrationrespectively. The image size for the experiments was 640 �480.

4.1. Phase to dept conversion

We implemented the phase to dept conversion (in fact, o to z

conversion, where o is phase/t [20,21]) measuring o in a set ofL¼41 planes positioned at z¼ 0,0:5,1,1:5,2, . . . ,20 cms. The func-tion z¼ TðoÞ was approximated with a fifth order polynomialTðoÞ �

P5k akok by means of least-squared fitting. Once com-

puted oði,j,zÞ the mean absolute error

E¼

P9ze�z9N

ð10Þ

Fig. 9. (a) Intensity of a noisy simulated pattern. (b) Its Fourier transform

(magnitude). (c) Error of the recovered phase using the Fourier method.

was E¼0.025 cm, where ze represents the set of z values esti-mated over all pixels for all measured planes, and N is the totalnumber of estimated sites.

4.2. Phase to transverse-coordinates conversion

The transformation to real coordinates (x,y) by means of Eq. (2)(once determined mx, lx, my and ly by means of least-squared fitting)depends on the z value at the given pixel (i,j) (note that all mx, lx, my,ly, z, x and y are dependent on (i,j)). The x, y and z values, used for thefitting, were determined applying the phase measuring Fouriermethod above described, capturing and processing a set of crossed

Fig. 10. Measurement of a real object. (a) Face of a manikin, (b) a mesh of the

measured (x,y,z)-coordinates, and (c) a profile along y direction. All dimensions are

in centimeters.

J. Villa et al. / Optics and Lasers in Engineering 50 (2012) 256–261 261

gratings images positioned at z¼ 0;1,2;3, . . . ,20 cm. Fig. 5 showsthe images of the crossed gratings pattern at z¼0 and z¼20 cms(fringe pitch P¼1 cm). The magnitude of their respective Fouriertransforms are shown in Fig. 6. The circles indicate the lobes to beisolated (Eq. (7)) to determine Fx and Fy. Horizontal profiles alongcentral line of Fig. 6a and b are shown in Fig. 7. Note the narrowspectrum owing to low phase-modulation fðx,yÞ, which is advanta-geous because it can be easily filtered. In our experiments we used asimple Gaussian filter. Fig. 8 shows the z�x and z�y graphs fordifferent pixels. Each graph shows the experimentally determined x

and y, and its corresponding linear least-squared fitting.In order to have a close idea of the effectiveness of the method to

compute real coordinates (x,y) from phase, we performed a simu-lated experiment using a one-dimensional synthetic pattern of 640data, generated with

IðxÞ ¼ aðxÞþ0:8 cos2p16

xþfðxÞ� �

þnðxÞ, ð11Þ

where aðxÞ ¼ �0:2ððx�320Þ2=maxððx�320Þ2ÞÞþ1:4, and fðxÞ ¼1:2pððx�320Þ3=maxððx�320Þ3ÞÞ, being n(x) uniformly distributednoise ranging from �0.125 to 0.125. Observing fðxÞ we can seethat maximum phase variation is 1:2p rad which represents a fringeshifting of 0.6P. Fig. 9 shows the intensity of the fringe pattern, itscorresponding Fourier spectrum and the phase-error applying asimple Fourier method for phase recovery with a Gaussian filter(mean¼40 and s2 ¼ 40). A drawback using the Fourier method isthe phase error in image borders, which must be consideredapplying the technique proposed here. Rejecting the borders, in thisexperiment the maximum phase error is about 0.06 rad whichrepresents a fringe shifting of about 0.0095P (0.95% of P). It isrelevant to remark that, even with larger x and y error measureswith this method, the least-squares fitting to determine parametersmx, lx, my and ly reduces the x and y errors in real experiments.

Fig. 10a shows the image of an object to be measured. A meshof the measured (x,y,z)-coordinates is shown in Fig. 10b. Fig. 10cshows a profile along y direction.

5. Conclusions

We have proposed a phase to (x,y,z)-coordinates transformationtechnique which can be used to calibrate a fringe projection systemfor 3D measurement. In particular, the phase to transverse coordi-nates transformation technique proposed here is advantageous,owing that just from one image the Fourier method can easy andefficiently provide both horizontal and vertical coordinates at once.Even if the Fourier method provides larger x and y errors, theirinfluence in measurements is minimized by the linear least-squaresfitting used to determine mx, lx, my, ly parameters.

Acknowledgments

We acknowledge the partial support for the realization of thiswork to the Secretarıa de Educacion Publica of Mexico throughthe project PIFI-2010.

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