Digital Moiré Fringe-Scanning Method for Centering a Circular Fringe Image

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igital moire fringe-scanning method forentering a circular fringe image

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A digital moire fringe-scanning method for centering a circular fringe image is proposed. The image ofa nondiffracting beam, whose cross section is a circular fringe, is first downloaded onto a computer. Theimage is then superposed with a digital circular grating, whose center is close to the center of the image,to generate circular moire fringes. Changing the phase of a digital grating can cause moire fringescanning. The global center of the image can be calculated by use of sets of the scanned picture.Because all the image data are used for the calculation, the effect of random noise on centering is greatlyreduced and the center position resolution can reach the order of a subelement of a CCD. The mea-surement of spatial straightness is discussed. © 2004 Optical Society of America

OCIS codes: 120.0120, 120.4120, 100.0100, 100.2650, 120.5050, 120.3930.

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. Introduction

traightness is a basic term in precision engineeringeasurement. Because of the common knowledge

hat light travels along a straight line, a narrow lighteam such as a Gaussian beam is generally used as aatural straightness datum although there are manyroblems with this method. There are two basicethods for use of light datum. One is the intensityethod, which uses a four-quadrant photodetector orposition-sensitive detector to determine the center

f light intensity of the beam spot.1 This method cane used to measure straightness in two directions andas a long measurement range. The other is the

nterference method that is used to determine theoint of equal optical path length.2 This method hasigh accuracy but can measure straightness only inne direction. Because the two interference beamsave a divergent angle and the distance between thewo reflecting mirrors is fixed, to avoid the two lighteam spots outside the mirrors, the measurementange of the interference method is much shorterhan that of the intensity method. Therefore, except

The author �[email protected]� is with the Department ofnstrumentation, School of Mechanical Science and Engineering,uazhong University of Science and Technology, Wuhan, Hubei30074, China.Received 30 July 2003; revised manuscript received 9 February

004; accepted 18 February 2004.0003-6935�04�142833-07$15.00�0© 2004 Optical Society of America

or low accuracy, the intensity method has moreerit in practice.The basic restriction on the accuracy of the inten-

ity method is that the spot diameter and the focalepth of the light beam conflict because of diffraction.or the intensity method, the sensitivity of the beampot position detection is determined directly by theize of the sensor, which means that high accuracynd the long range of the straightness measurementannot be satisfied simultaneously for a conventionalight beam.

A nondiffracting beam3,4 �also called a Besseleam� is a new form of light beam and was proposednd demonstrated experimentally by Durnin in 1987.he cross section of the nondiffracting beam is a cir-ular fringe pattern �Fig. 1�. The intensity distribu-ion along the radial direction coincides with a Besselunction �Fig. 2� and remains unchanged along theropagation axis. This is called a nondiffractingeam because its cross section remains unchangeduring propagation regardless of how small its cen-ral spot �Fig. 1� is. This is an advantage that otheright beams such as a Gaussian beam do not have.o the nondiffracting beam can be used as a new formf natural straightness datum having both a longocal depth and a small central beam spot.

To use a nondiffracting beam for straightness mea-urement, it is critical to pinpoint the circular fringeattern center for measurement accuracy. In the-ry, the track of maximum intensity in the centralpot area is a straight line, and the departure fromhis line is a straightness error of the measurand.hus, the simplest method is to use the central spot

10 May 2004 � Vol. 43, No. 14 � APPLIED OPTICS 2833

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f a Bessel beam directly and use the center of theentral spot as the straightness datum.

There are several problems with the central spotethod. One is that, because of the influence of

ackground light, scattered light, and wave-front ab-rration, the shape and light intensity distribution ofhe central spot is not perfect and often changes sig-ificantly and unpredictably even to the point of split-ing into several spots. There is some unavoidablencertainty in the determination of the maximum

ntensity point within the central spot area. Thised to the measurement uncertainty of straightness.

theoretical study shows that the smaller the cen-ral spot size, the greater its influence on measure-ent uncertainty.5 Therefore, under conventional

ackground and noise there is a minimal acceptableize of the central spot. When the central spot ismall, its influence is high and the center uncertaintys great. When the central spot is large, the size ofhe photodetector is also large and the absolute posi-ion resolution of the detector is correspondingly low.herefore, by use of the aiming method with only theentral spot it is difficult to obtain high accuracy.

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834 APPLIED OPTICS � Vol. 43, No. 14 � 10 May 2004

Another problem is caused by the CCD image de-ector. Because the cross section of the nondiffract-ng beam is a circular fringe pattern and the diameterf the central spot is generally smaller than 0.1 mm,t is not practical to use a four-quadrant detector or aosition-sensitive detector sensor to measure the po-ition of the central spot directly. Therefore, thenly suitable detector is an image sensor such as aCD. Because the size of a CCD element pixel ispproximately 5–14 �m, the position resolution of aCD and the centering accuracy of the central spot is,f course, greater than that of the CCD element,hich does not satisfy the accuracy for almost all the

ases.The key to solution of the above problem is to avoid

he use of only the central spot to determine the center.new datum of straightness must be found. In the-

ry, not only the track �Fig. 3, track0� of the centeroint of the central spot is a straight line, but theracks of the centers of the first ring and all the otherings �Fig. 3, track1 and track2� are also straight lines.ach track can be used as the straightness datum.ut it is obvious that each ring has its own center andecause of background noise and other influences,uch as the wave-front error, these centers do not co-ncide with each other and the tracks of each ring alsoo not coincide with each other. One can find manyenters in the cross section of a nondiffracting beamnd several straightness data. Which datum shoulde selected? Experimentally it has been determinedhat, although there are serious disturbances in manyocal areas, the global center of the ring fringe patternppears to remain unchanged. This phenomenon isasy to understand since, because there are more datan the outer rings, the influence of noise on the centers less than that of the inner rings. If a common cen-er or global center of all rings is defined, it will be theost stable point because it uses all the data of the

ntire image.I present a mathematical method to define and

alculate the common center of all the rings of aondiffracting beam. The technique used in thisethod is moire fringe scanning, which is similar to

hat used in conventional scanning interferometry.ecause of the amount of data in the calculation of

he common center point, subelement resolution cane obtained and the accuracy of straightness mea-urement can be significantly improved.

ig. 1. Experimental photograph of a nondiffracting beam. CSepresents the central spot.

Fig. 2. Nondiffracting beam. CS represents the central spot.

ig. 3. Illustration of the tracks at the center of each ring for theondiffracting beam.

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. Principleigure 4�a� is an experimental photograph of a non-iffracting beam. It is the starting point of our

ethod. The figure appears to be a set of rings hav-ng the same center. I aim to determine the correctefinition of a common center or a global center of

ig. 4. Calculation procedure of the common center of ring patterns by digital moire fringe scanning: �a� original photograph ofondiffracting beam f �x, y�, �b� f �x, y� times g�x, y�, �c� M �x, y�, �d� �0�x, y�, �e� three-dimensional image of phase cone surface ��x, y�, �f �ommon center, where the cross represents �x , y �.


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hese rings and to find a suitable method to calculatehe defined common center.

If the theoretical center of the nondiffracting beams x0, y0, the spatial frequency along the radial direc-ion is �f, and the ring spacing is df � 2��f, themage of the measured nondiffracting beam can bepproximately expressed as

f � x, y� � A� x, y�sin��f R � � � N� x, y�, (1)

here R is the distance to the center �x0, y0�,

R � �� x � x0�2 � � y � y0�

2�1�2; (2)

is a constant; A�x, y� represents the intensity dis-ribution �for a Bessel beam, it is easy to observe that�x, y� � 1��1 � bR�, where b is a constant, is a goodpproximation, which is a slowly changing function;nd N�x, y� represents background and light inten-ity.A computer was used to generate a digital fringe

attern having many rings with a common center atx0, y0�:

g� x, y� � sin��g R � �, (3)

here �g is the spatial frequency along the radialirection and is a phase parameter. One can thenultiply f �x, y� with g�x, y�, point by point, to obtain

f � g �12

A� x, y�cos��g � �f� R � � �

�12

A� x, y�cos��g � �f� R � � �

� N� x, y�sin��g R � �, (4)

hich is shown in �Fig. 4�b��. If we let �g � ��f ��, where �� is a small quantity compared with �f,

he first term of Eq. �4� is a low-frequency term andhe other two terms are high-frequency terms.herefore, after low-pass filtering of Eq. �4� we obtaindigital moire fringe pattern �Fig. 4�c��:

M� x, y� �12

A� x, y�cos��g � �f� R � � �, (5)

n which the spatial frequency is ��g � �f � and thepacing of the moire ring is dm � 2��g � �f �.From Eq. �5� it is easy to determine that each point

n M�x, y� varies sinusoidally with . So if wehange from 0 to 2, we can calculate the phase ofach point in the pattern6:

�0� x, y� � ��g � �f� R � � tan�1��0

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M cos d � ,

(6)

hich is shown in Fig. 4�d�, where the gray changerom black to white when the phase change is greater


han 2. To avoid a greater than 2 phase jump ando make the phase change a continuous operation ofnwrapping yield

�� x, y� � �0� x, y� � 2k, (7)

here k is selected to unwrap the phase pattern data.After the unwrapping operation we obtain a new

hase pattern ��x, y�. From Eq. �6� it is obvious that�x, y� increases linearly with R, the distance be-ween �x0, y0� and �x, y�. So the figure of ��x, y� is aonelike surface �Fig. 4�e��.

Then ��x, y� is fitted with a standard cone surface,nd we obtain a point �xm, ym�, which is the peakoint of the standard cone surface. If the nondif-racting beam is ideal and there is no noise, �xm, ym�ill certainly coincide with �x0, y0�.Because the cone ��x, y� is determined by the data

f all the rings and its peak �xm, ym� coincides withx0, y0� under ideal conditions, �xm, ym� can be used askind of common center of the ring patterns �Fig.

�f �� and Eqs. �2�–�7� provide the method for deter-ination of the center.It can be seen that, when increases, the moire

ringe emerges at the center one by one and thennlarges at the outer side. This is a process of fringecanning that is realized by software other than byhanging the optical path difference in conventionalnterferometry,6 which means it is digital fringe scan-ing. With conventional optical fringe-scanningechniques, because of the use of figure data obtainedy digital moire fringe scanning, the fringe phasenformation can be effectively isolated from back-round and noise. There is, however, one problem.t the beginning of fringe scanning, we assume that

he real center �x0, y0� of the nondiffracting beam isnown and use it as the center of the digital ringrating g�x, y�. If the center is not known precisely,hat should be done?At first we estimate the initial center of the non-

iffracting beam and use it as the starting point ofenter �xg, yg�, which differs from but is close to theeal center of the nondiffracting beam, and measurepatial frequency �f along the radial direction.hen, we multiply �f by a factor near 1.0 �for exam-le, 1.1� to obtain �g that is nearly equal to �f andenerate a digital ring grating:

G� x, y� � sin��g R� � �, (8)

here R� � ��x � xg�2 � �y � yg�2�1�2 and is a phaseonstant. Similar to Eq. �4�, we multiply G�x, y�ith f �x, y� and obtain

f � G �12

A� x, y�cos��g R� � �f R � � �

�12

A� x, y�cos��g R� � �f R � � �

� N� x, y�sin�� R� � �, (9)
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here the first term is also the low-frequency termhat represents the moire fringe. The other twoerms are high-frequency terms that can be filteredway. Let us consider the phase of the moire term:

�� x, y� � �g R� � �f R

� �g�� x � xg�2 � � y � yg�

2�1�2

� �f�� x � x0�2 � � y � y0�

2�1�2. (10)

ecause the direction of the coordinates is arbitrary,g � y0 will not lose generality. We then obtain

�� x, y� � �g�� x� � �x�2 � � y��2�1�2

� �f�� x��2 � � y��2�1�2, (11)

here x� � x � x0, y� � y � y0, and �x � xg � x0.ecause the moire fringe is on the track where ��x, y�

s equal to a constant C, for the polar coordinateystem we obtain

�g2 � �f

2��2 � 2��x�g2 cos � � C�f��

� �g2�x2 � C2 � 0, (12)

here x� � � cos � and y� � � sin � are used. Con-idering that � � 0, the solution of Eq. �12� is

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oire fringe curves corresponding to C � 2, 4, 6,tc. are shown in Fig. 5, where �g � 2�10, �f ��11.Analysis of Eq. �13� and Fig. 5 shows that, if there

s a deviation between the centers of the digital ringrating and the nondiffracting beam, the center of theoire rings will also deviate and the shape of theoire fringe would not be a perfect circle. The curve

eformation is serious in the area near the center �x0,0�. But when �x is not large compared with theringe spacing of the nondiffracting beam, the shapef the moire rings at the outer area still appears to beircular. Therefore, we can still use the digitaloire fringe-scanning method described above to

enerate a phase surface, fit it with a standard coneurface, and find its peak point to be the commonenter �xm, ym� of the moire rings. Even though theurface shape difference in the center area is rela-ively great, the outer part of the phase surface ispproximately a cone. Because there are more dataoints in the outer rings than in the inner rings, theeak point of the fitted cone represents the center ofhe outer moire rings.

The relation between the centers of the moire ringsnd the digital ring grating can be obtained from Eq.13�:

xm � x0 � ��0 � ��2 ��g�x

��g � �f�,

y � y � 0, (14)
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hich means that displacement of the digital gratings amplified by a factor of �g��g � �f �. We can nowonstruct a formula for the calculation of �x0, y0� fromq. �14�:

x0 � xm � � xm � xg��g

�f,

y0 � ym � � ym � yg��g

�f, (15)

here the direction of the coordinate is selected ar-itrarily.If x0 � xg and y0 � yg, the center of the beam can

till not be determined. We then let xg � x0 and yg �

0 and recalculate the new �x0, y0�. After some iter-tion, we finally obtain

� x0 � xg�2 � � y0 � yg�

2 � 0,

hich means that the center of the nondiffractingeam has been determined.

. Algorithm

ccording to the above principle, an iteration algo-ithm is designed as follows:

�1� For a given nondiffracting beam pattern f �x, y�ith an assumed initial center �x , y � and spatial

ig. 5. Figure of moire rings with center deviations of �a� �x � 3nd �b� �x � 6. The black cross represents the moire fringe centerx , y �.

� ��C�f � �x�g

2 cos �� C�f � �x�g2 cos ��2 � �C2 � �g

2�x2��g2 � �f

2��1�2

��g2 � �f

2�. (13)

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requency along radial direction �f, the generatingigital ring grating is

g� x, y� � sin��g R � �,

here �g � 1.1�f.�2� Multiplying f �x, y� with g�x, y� point by point

ields

p� x, y� � f � x, y� g� x, y�.

�3� Low-pass filtering p�x, y� to obtain moire pat-ern

M� x, y� � filter� p� x, y��,

here filter is a function of MATLAB or some otheromputer language.

�4� Increasing by � � 2�n, where n is deter-ined according to the accuracy requirement, n � 20

s generally sufficient to obtain a new g�x, y�.�5� Repeating �2�–�4� to obtain a series of Mj�x, y�.�6� Calculating phase �0�x, y�:

�0� x, y� � ��g � �f� R �

� tan�1��j�1

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Mj� x, y�sin�2j�n�

�j�1

n

Mj� x, y�cos�2j�n�� .

�7� Unwrapping �0�x, y� to eliminate phase changesreater than 2:

�� x, y� � unwrap��0� x, y�� 0� x, y� � 2k.

�8� Fitting ��x, y� with an ideal cone c�x, y� withenter at �xc, yc� and parameter A and c0,

c� x, y� � A�� x � xc�2 � � y � yc�

2�1�2 � c0

y the least-squares method,

εj � �x,y

�c� x, y� � �� x, y��23min,

o obtain parameter �xc, yc�, which is the peak point ofhase cone ��x, y�, and used it as the center of theoire ring:

xm � xc, ym � yc.

�9� According to Eqs. �15�, constructing a new esti-ation of �x0 y0� from �xg yg� and �xm ym�, and using

t as the new �xg�, yg�� yield

xg� � x0 � xm � � xm � xg��g

�f,

yg� � y0 � ym � � ym � yg��g

�f.

�10� Repeating �7�–�9�. When �xm � xg� � � andy � y � � � �e.g., ε � 10�4�, the center of the digital
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ing grating is considered to coincide with the centerf the nondiffracting beam and the iteration stop.

The results at each stage of the above iteration arehown in Fig. 4.

. Numerical Analyses of Noise Sensitivity of thelgorithm

n a real experiment, there are basically two kinds ofnfluence on an optical fringe pattern. One is ran-om light intensity that is caused by laser speckle,nwanted background illumination, and electronicoise. The other is macroscopic fringe deformationhat is usually caused by asymmetric wave-front ab-rration such as coma. Because the proposed digitaloire fringe-scanning method depends on the shape

f fringe rings, the macroscopic fringe deformationould certainly cause movement of the resulting

ommon center and therefore should be avoided.In principle, the effect of random noise on the pro-

osed centering algorithm can be analyzed by a statis-ical method,6 but it is too complicated. Instead wesed a digital random noise experiment to simulatehe effect. First, we generated a series of noise im-ges with variance �n by use of the random noise func-ion of any computer language �e.g., the Imnoiseunction of MATLAB� and added it to the measure-ent image. We then calculated the changed centers

nd their variance. Table 1 lists the simulation resultf the relation among �n, �m, and �0.Note that, because the light intensity of the center

oint of a nondiffracting beam is much greater thanhat of the outer fringes, when the center intensity isniform to 1, the noise with � � 0.01 is already seri-usly a gray image with intensity between 0 and 1.ven though the number of digital experiments isnite, it can still be concluded qualitatively that theroposed center algorithm is not sensitive to randomoise. Because the order of variance of the commonenter is far less than 1, which means the error is far

ig. 6. Experimental arrangement for the straightness measure-ent by use of a nondiffracting beam. BEP, beam expansion

rocess.

Table 1. Numerical Simulation of Noise Effectsa

Noise Level�n

Variance of the MoireFringe Center �m

Variance of theCommon Center �0

0.010 0.497 0.1130.005 0.299 0.0340.0025 0.153 0.0130.001 0.073 0.009

a�m and �0 are measured in pixels.

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ess than the size of one sensor element of the CCDmage detector, the resolution of this digital moireringe-scanning method can reach the subelement or-er of a CCD.

. Application

he digital moire fringe-scanning method has beenpplied to the spatial straightness measurementFig. 6�. The laser beam is expanded and collimatedy the beam expander and then passed through anxicon lens to generate a nondiffracting beam. Theeam is projected directly to the sensor surface of a

ig. 7. Experimental results of the spatial straightness measure-ent: �a� *, x-direction data; o, y-direction data; and �b� space

traightness curve of the slide.

CD detector. The CCD detector is attached to theeasurand on a slide. When the slide moves, theCD moves simultaneously. As the CCD moves, the

ight spots of the nondiffracting beam on the surfacef the CCD change. At a series of slide positionsith uniform spacing the light spot pictures are

aken and saved. These pictures are then processedo determine the coordinates of the center of eachicture by the digital moire fringe-scanning methodroposed in this paper. Finally these data representhe straightness of the measured slide. Figure 7�a�hows the experimental data in two directions andig. 7�b� shows it in three dimensions.

. Conclusion

y digital moire fringe scanning, the common centerf a nondiffracting beam that is a ring fringe patternan be determined accurately from a noisy back-round. Because of the use of all the data in thentire pattern, one can reach an accuracy of subele-ent resolution. The proposed method is a key

echnique for the application of a nondiffracting beamor the measurement of spatial straightness.

This research was supported by the National Nat-ral Science Foundation of China, project 59805006.

eferences. K. C. Fan and Y. Zhao, “A laser straightness measurement

system using optical fiber and modulation techniques,” Int. J.Mach. Tools Manuf. 40, 2073–2081 �2000�.

. S.-T. Lin, “A laser interferometer for measuring straightness,”Opt. Laser Technol. 33, 195–199 �2001�.

. J. Durnin, J. Miceli, Jr., and J. H. Eberly, “Diffraction-freebeams,” Phys. Rev. Lett. 58, 1499–1501 �1987�.

. J. Durnin, “Exact solutions for nondiffracting beams. I. Thescalar theory,” J. Opt. Soc. Am. A 4, 651–654 �1987�.

. Z. Bin and L. Zhu, “Diffraction property of an axicon in obliqueillumination,” Appl. Opt. 37, 2563–2568 �1998�.

. D. Malacara, Optical Shop Testing �Wiley, New York, 1992�,Chap. 13, p. 409.


Digital Moiré Fringe-Scanning Method for Centering a Circular Fringe Image

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