Fourier transform profilometry for the automaticmeasurement of 3-D object shapes

Mitsuo Takeda and Kazuhiro Mutoh

A new computer-based technique for automatic 3-D shape measurement is proposed and verified by experi-ments. In contrast to the moire contouring technique, a grating pattern projected onto the object surfaceis Fourier-transformed and processed in its spatial frequency domain as well as in its space-signal domain.This technique has a much higher sensitivity than the conventional moire technique and is capable of fullyautomatic distinction between a depression and an elevation on the object surface. There is no requirementfor assigning fringe orders and interpolating data in the regions between contour fringes. The techniqueis free from errors caused by spurious moire fringes generated by the higher harmonic components of thegrating pattern.

I. IntroductionThe method of moire contouring is a well-known

technique for 3-D shape measurement.1 Recent in-terest in the technique has been an automatic mea-surement based on data processing by computer.2 -4

For this purpose it is essential to have means to (1) makeautomatic distinction between a depression and an el-evation from a contour map of the object, (2) assignfringe orders automatically including those separatedby discontinuities, (3) locate the center lines of broadfringes by correcting unwanted irradiance variationscaused by nonuniform light reflection on the objectsurface, and (4) interpolate the regions lying betweenthe contour lines. To satisfy these requirements, Ide-sawa et al.

2 proposed the scanning moire method, andMoore and Truax4 proposed the phase-locked moiremethod. Further, we proposed another method calledFourier-transform profilometry (FTP) which is betterfor automatic measurement by computer processing.5

The idea of the FTP stemmed from the observation thatall these cumbersome requirements mentioned abovearise merely from attempting computer-based auto-matic measurement by means of a moire contouringtechnique that was originally developed for fringeanalysis by human observation rather than by computerprocessing. Since FTP does not use moire fringes, it isfree from all the difficulties associated with the moire

The authors are with University of Electrocommunications, 1-5-1,Chofugaoka, Chofu, Tokyo 182, Japan.

Received 9 August 1983.0003-6935/83/243977-06$01.00/0.(© 1983 Optical Society of America.

contouring technique. Another great advantage of FTPis that it has a much higher sensitivity than the con-ventional moire technique. It can detect a shape vari-ation much less than one contour fringe in moire to-pography. Our previous paper 5 described a generalprinciple applicable to both profilometry and interfer-ometry, but experiments were given only for interfer-ometry. The purpose of this paper is to give a morespecific description of the principle of FTP and topresent experimental results of-3-D shape measurementby FTP.

11. Optical GeometryOptical geometry is similar to that of projection moire

topography,6 7 but in FTP the grating image projectedon an object surface is put directly into the computerand processed without using the second grating togenerate moire fringes. Two different optical geome-tries have been proposed and used in moire topography;one has some merit over the other as well as some de-merit. In crossed-optical-axes geometry, the opticalaxes of a projector and a camera lie in the same planeand intersect a point near the center of the object. Thisgeometry is easy to construct because both a grating andan image sensor can be placed on the optical axes of theprojector and the camera, respectively, but it givesplanar contours only when the optics are telecentric. 8 9

In parallel-optical-axes geometry, the optical axes of aprojector and a camera lie in the same plane and areparallel. This geometry gives planar contours but issomewhat awkward because the grating must be placedfar off the optical axis of the projector to ensure that thegrating image is formed within the field of view of theobservation camera.2 6 7 These two options of opticalgeometry are also available in FTP, and in addition FTP

15 December 1983 / Vol. 22, No. 24 / APPLIED OPTICS 3977

Fig. 1. Crossed-optical-axes geometry.

can solve the problem of nonplanar contourscrossed-optical-axes geometry.

in

A. Crossed-Optical-Axes GeometryFigure 1 shows a geometry in which the optical axes

E p Ep of a projector lens crosses the other optical axisEc E, of a camera lens at point 0 on a reference planeR, which is a fictitious plane normal to E' E, and servesas a reference from which object height h(x,y) is mea-sured. Grating G has its lines normal to the plane of thefigure, and its conjugate image (with period p) is formedby the projector lens on plane I through point 0; E, andEp denote, respectively, the centers of the entrance andthe exit pupils of the projector lens. The camera lens,with the centers of the entrance and the exit pupils atEc and E', images reference plane R onto the imagesensor plane S. Ep and E, are located at the samedistance lo from plane R. It should be noted that Epand E, are the centers of the pupils,10 not the nodalpoints of the lenses as is so often confused in the liter-ature. 2 '6 ' 8 When the object is a flat and uniform planeon R, i.e., h (xy) = 0, and if Ep is at infinity (as denotedby E for a telecentric projector), the grating imageprojected on the object surface and observed throughpoint E is a regular grating pattern which can be ex-pressed by a Fourier series expansion:

gT(XY) = An ep(2iiinfox), (1)n=-w

where

where so(x) = BC is a function of x and has a positivesign when C is to the right of B as in the figure. For theconvenience of later discussion, we express Eq. (3) as aspatially phase-modulated signal

go(x,y) = EI An expli[2irnfox + n40 (x)fl,n=-X

where

Oo(x) = 27rfoso(x) = 2irfoBC.

(4)

(5)

Since the grating image is deformed and phase-modu-lated even for h(x,y) = 0, the crossed-optical-axes ge-ometry, when used in moire topography, gives nonpla-nar contours unless the pupils are at infinity, i.e., thecase of telecentric optics.9 This has imposed a greatrestriction on the application of the nontelecentriccrossed-optical-axes geometry to moire topography, inspite of its easy-to-construct merit. In FTP, this initialphase modulation is automatically corrected as will beshown in the next section.

For a general object with varying h(x,y), the principalray EpA strikes the object surface at point H, and pointH will be seen to be a point D on plane R when observedthrough Ec. Hence, the deformed grating image for ageneral object is given by

g(x,y) = r(x,y) An expJ27rinfo[x + s(xy)]},n=-

(6)

or

g(x,y) = r(x,y) Z An expli[27rnfox + np(x,y)]J,n=-

where

O(x,y) = 27rfos(xy) = 2rfoBD,

(7)

(8)

and r(x,y) is a nonuniform distribution of reflectivityon the object surface.

B. Parallel-Optical-Axes GeometryFigure 2 shows a geometry in which the optical axis

EEp of a projector lens and that of a camera lens E'ECare parallel and are normal to reference plane R. Theconjugate image of grating G is formed on plane R, andthe three points A, B, and C in Fig. 1 degenerate intopoint C in Fig. 2, so that Eqs. (5) and (8) become

Jo IPo = cosO/p (2)

is the fundamental frequency of the observed gratingimage. The x axis is chosen as in the figure and the yaxis is normal to the plane of the figure. If Ep is at fi-nite distance, we observe on the image sensor plane adeformed grating image with a pitch increasing with x,even for h (x ,y) = 0. Noting that a principal ray througha conjugate image point A strikes reference plane R atpoint B in the telecentric case and at point C in thenontelecentric case, we write the deformed gratingimage for h(x,y) = 0 as

go(x,y) = _ A expI27rinfo[x + so(x)]1,n=--

(3)Fig. 2. Parallel-optical-axes geometry.

3978 APPLIED OPTICS / Vol. 22, No. 24 / 15 December 1983

G-

Oo(x) = 27rfoso(x) = 27rfoBC = 0,

O(x,y) = 2 irfos(x,y) = 27rfoCD.

(9)

(10)

Hence, the grating image projected on the plane h(x,y)= 0 remains as a regular grating pattern regardless ofthe position of the pupil of the projector. This bringsa merit of generating planar contours in moire topog-raphy, but the optical setup becomes more or lessawkward because the grating must be translated far offthe optical axis of the projector lens in order to form itsimage within the field of view of the camera lens.

IlI. Fourier Transform MethodThe deformed grating image given by Eq. (7) can be

interpreted as multiple signals with spatial carrierfrequencies nfo modulated both in phase q(x,y) andamplitude r(x,y). Since the phase carries informationabout the 3-D shape to be measured, the problem is howto obtain (x,y) separately from the unwanted ampli-tude variation r(x,y) caused by nonuniform reflectivityon the object surface. We rewrite Eq. (7) as

g(x,y) = qn(X,Y) *exp(27rinf~x), (1n=--

G (f, Y)J

00

-3f,) 2 f,/.L /L I iL

fto 0 f0 2fo

03

_ - f

3f,

Fig. 3. Spatial frequency spectra of deformed grating image for afixed y value. Only a spectrum Q1 (dotted) is selected by the filtering

operation.

the phase modulation due to the object-height distri-bution. In principle, this operation is not necessary inparallel-optical-axes geometry, but we apply it also toparallel-optical-axes geometry because by phase sub-traction we can cancel errors caused by misalignmentsand/or distortion of the lenses. For example, a mis-alignment which rotates the grating by an angle bacaround the optical axis gives a nonzero initial phasedistribution

where Oo(Y) = 27rfoy tan(6a),

q,,(x,y) = A,,r(x,y) exp[in0(xy)]. (12)

By using a FFT algorithm, we compute the 1-D Fouriertransform of Eq. (12) for the variable x only, with ybeing fixed:

G(f,y) = f g(x,y) exp(-2-rifx)dx

= E Qn(f - nfo,y), (13)

where G(fy) and Qn(f y) are the 1-D Fourier spectraof g(x,y) and qn(xy), respectively, computed with re-spect only to the variable x, and the other variable ybeing treated as a fixed parameter. Since in most casesr(x,y) and O(x,y) vary very slowly compared to thefrequency fo of the grating pattern, all the spectra Qn (f- nfo,y) are separated from each other by the carrierfrequency fo, as shown in Fig. 3. We select only onespectrum Q1(f - fo,y) dotted in the figure and computeits inverse Fourier transform to obtain a complexsignal

g(x,y) = ql(x,y) - exp(27rifox)

= Air(x,y) expli[2-rfox + 0(x,y)]}. (14)

In the crossed-optical-axes case, we do the same filteringoperation for Eq. (4) to obtain

Ro(x,y) = A, expli[27rfox + 0o(x)]¢, (15)

and generate from Eqs. (14) and (15) a new signal

4(xy) .gR(xy) = IA11 2r(x,y) expti[Ao(x,y)]b, (16)

where

AO(x,y) = O(x,y) - 0o(X)

= 2irfo(BD - BC) = 2irfoCD. (17)

Since the initial phase modulation 00(x) for h(x,y) =0 is now subtracted, A\O(x,y) in Eqs. (16) and (17) gives

(18)

which gives rise to a mismatching error in moire to-pography. We therefore use the formulas of Eqs. (16)and (17) in both cases.

Now our task is to obtain the phase distributionAO(x,y) in Eq. (16), separating it from the unwantedamplitude variation r(x,y). Noting that both A 1 -r(x,y) and AO(x,y) in Eq. (16) are real functions, wecompute a complex logarithm of Eq. (16):

log(g(x,y) R;(x,y)] = log[IA1J2r(x,y)J + iAk(x,y). (19)

We obtain the phase distribution AO (x,y) in the imag-inary part, completely separated from the unwantedvariation of reflectivity r(x,y) in the real part. Sincethe phase calculation by computer gives principal valuesranging from -v to 7r, the phase distribution is wrappedinto this range and consequently has discontinuitieswith 27r-phase jumps for variations more than 27r.These discontinuities can be corrected easily by addingor subtracting 27r according to the phase jump rangingfrom 7r to -7r or vice versa. A 2-D phase distributioncan be obtained simply by repeating the same procedurefor the y sections. Details of the phase-unwrappingalgorithm are described in Ref. 5. Since the loci of thediscontinuities with 27r-phase jumps correspond tocontour fringes in moire topography, the phase-un-wrapping process plays the role of the fringe order as-signment algorithm of conventional moire topog-raphy.

IV. Phase-to-Height ConversionIn this section we derive a formula for converting the

measured phase distribution into the physical heightdistribution. Noting that AEPHEC A ACHD in bothFigs. 1 and 2, we can write

CD = -dh(x,y)/[10 - h(x,y)I, (20)

15 December 1983 / Vol. 22, No. 24 / APPLIED OPTICS 3979

= 1, 2, and 3. Substituting Eq. (23) into Eqs. (24) and(25), we have

I-I < nfo+ -.~02o -2 a max 2 r mi n- ( m

(n = 2,3, .. .), (26)

21 (min (27)

fb

Fig. 4. Condition for separating fundamental frequency spectrumQ, (dotted) from other spectra.

where the object height h (x,y) is defined positive whenmeasured upward from reference plane R. SubstitutingEq. (20) into Eq. (17) and solving it for h(xy), we obtainthe conversion formula

h(x,y) = loAq0(x,y)/[A/0(x,y) - 2rfod]. (21)

We can express this formula in another form which canbe directly compared to the well-known formula ofmoire topography. Substituting Eq. (2) into Eq. (21),we have

h(x,y) = lopo[A/\(x,y)/27r]/lpo[A1(x,y)/22r] - d. (22)

When we note that Aq(x,y)/2r gives the number N ofthe fringe order in moire topography, we see that Eq.(22) is exactly the same as the formula of moire topog-raphy.1 However, the difference should be noted that,whereas in the moire technique the height distributioninformation is given only along a discrete set of contourlines, our technique, FTP, gives the height informationat all picture elements regardless of whether A0(xy)/2wris an integer or not. This is the reason that FTP doesnot need fringe interpolation as is necessary in moiretopography.

V. Maximum Range of MeasurementSince FTP is based on filtering for selecting only a

single spectrum of the fundamental frequency compo-nent, the carrier frequency fo must separate this spec-trum from all other spectra. This condition limits themaximum range measurable by FTP. Noting thatr(x,y) varies much slower than o in Eq. (7), we definefor the nth spectrum component a local spatial fre-quency1 f analogous to an instantaneous frequencyof FM signal12:

=*- [27rnfox + n(x,y)]27r axn ak(x,y) (3

= nfo + - x (23)27 ax

For the fundamental spectrum to be separated from allother spectra, it is necessary that

(f)max < (fn)min, (n = 2,3,.... ) (24)

and that

A6 < (fl)min (25)

where fb, (fn)max, and (fn)min are shown in Fig. 4 for n

A safer and more practical condition can be set by

-o -< nf. -- n I-27r ax max 27r |x I max

(n = 2,3, ... ), (28)

Ab <o K - (29)27r ax max

where (0)/(ax) max denotes the maximum absolutevalue which is a larger value of [(a0)/ax)]maxl and

(8k)/(ax)IminI. From Eqs. (28) and (29) we have

< I 2 fo, (n = 2,3 ... ), (30)

< 2r(fo - f). (31)ax max

Since in most cases fb is much smaller than fo/2 and (n- 1)/(n + 1) increases monotonically with n, the limitis set by Eq. (30) for n = 2:

Ja~~l < 2,,fo ~~~(32)ax max 3

When discussing the maximum range of measurement,we can assume that k(x,y) is much larger than 00(x),and from Eq. (21) we can write

O(X,Y) /O(X~Y) -(27rfod/lo) h(xy), (33)

where we have also assumed that lo >> h (x,y). Substi-tuting Eq. (33) into Eq. (32), we finally obtain

(34)

This condition states that the maximum range ofmeasurement is not limited by the height distributionh(x,y) itself but by its derivative in the direction normalto the line of the grating. The maximum range ofmeasurement can be extended by employing a geometryin which 10/d is large to prevent the phase from beingovermodulated. This corresponds to reducing thefringe sensitivity in the case of moire topography, butFTP retains a sensitivity sufficient for most applicationssince it can detect a phase distribution much less than2wr. This will be demonstrated by experiments in Sec.VI.

VI. ExperimentsFigure 5 shows a schematic diagram of the experi-

mental setup. A crossed-optical-axes geometry wasemployed because of its easy-to-construct merit. A300-W slide projector with an 85-mm focal length pro-jecting lens was used to project a Ronchi grating of 150lines/in. onto an object surface. The object is a whiskeybottle embedded in a uniform plane plate which serves

3980 APPLIED OPTICS / Vol. 22, No. 24 / 15 December 1983

0,

( f2)Mi.. 02

03^ a l 0 - w -- u l, z { l \

*f

t f f f( f, ).11. ( f, ). . ( f3 -. ( f2

ah (xy) < , . �,O� .I ax I max 3 d

TV Monitor

Fig. 5. Schematic diagram of experimental setup.

V

I x

Fig. 7. Irradiance profile of deformed grating pattern for a fixed yvalue.

-A- X

Fig. 6. Deformed grating pattern with straight lines in the back-ground for reference signals.

as a reference plane in the background. The deformedgrating pattern was observed by a low distortion TVcamera (Hamamatsu C-1000) with a 55-mm focal lengthMicro-Nikkor lens. An analog video output signal isconverted into an 8-bit digital signal and stored in aframe memory in the form of a picture with 512 X 512pixels which can be monitored through a TV monitor.The picture in the frame memory is DMA transferredto the memory of a Digital LSI-11/23 microcomputerto make a temporal file on a disk which is then trans-ferred through a communication line to a faster DigitalPDP-11/44 minicomputer and processed. The finalresult is sent back and displayed on an X- Y plotter.Figure 6 shows a picture of a deformed or phase-mod-ulated grating pattern, where the straight grating linesin the background serve as reference signals for deter-mining the absolute phase values to be converted intoa height distribution. Figure 7 shows an example of theirradiance profile along a horizontal line in the directionof the x axis. Note that the reflectivity r(x,y) isstrongly nonuniform over the object surface. Figure 8

Fig. 8. Spatial frequency spectra of deformed grating pattern.

Fig. 9. Wrapped phase distribution. The line along discontinuitieswith 27r-phase jumps corresponds to a fringe contour in moire

topography.

shows Fourier spectra of Eq. (13) computed by using aFFT algorithm. In the figure, the large spectrum (n =0) is clipped to get an enhanced view of other spectra.Note that the spectrum (n = 1) is completely separatedfrom other spectra satisfying the condition of Eq. (34).Figure 9 shows a wrapped phase distribution Ao(x,y)computed from the imaginary part of Eq. (19). Noting

15 December 1983 / Vol. 22, No. 24 / APPLIED OPTICS 3981

Fig. 10. Unwrapped phase distribution.

w ~~'* - x

y = 260

(Line No.)

~~~~~~~~E

20 flim

E

. s -~~~~~~X

Fig. 11. Comparison with contact measurement. Solid lines denotemeasurements by FTP, and circles denote measurements by the

contact method.

that from Eq. (22) the line along the discontinuities with2i7r-phase jumps corresponds to one contour fringe inmoire topography, we can see that the height variationless than the amount of one fringe is clearly detected inthe figure. Figure 10 shows the unwrapped phase dis-tribution which has the form of a whiskey bottle. Thisphase distribution is converted into the height distri-bution using the formula of Eq. (21) and compared withthe result of direct measurement by the contact method.Figure 11 shows three examples of the object profiles,where the lines and circles represent the results ob-tained by FTP and the contact method, respectively.

VII. ConclusionWe have proposed a new technique, Fourier trans-

form profilometry, which is suitable for automaticmeasurement of a 3-D object shape. Since FTP doesnot use the moire contouring technique, it is completelyfree from various cumbersome problems associated withmoire topography. For example, FTP can accomplishfully automatic distinction between a depression andan elevation of the object shape, it requires no fringe-order assignments or fringe-center determination, andit needs no interpolation between fringes as it givesheight distribution at all the picture elements over theobject image. Furthermore, FTP has the advantagethat it can detect height variations less than the amountof one fringe in the conventional moire contouringtechnique. Another merit of FTP is that it is perfectlyfree from the effect of unwanted spurious moire fringesgenerated by the higher harmonic components of thegrating pattern, since these components are filtered outin the spectrum domain. Finally, we discussed theapplicability of FTP and proposed a practical criterionthat the maximum slope of the object be less than 10/3d,where o and d are the distances between the cameraand the object and the camera and the projector, re-spectively.

References1. See, for example, D. M. Meadows, W. 0. Johnson, and J. B. Allen,

Appl. Opt. 9, 942 (1970); H. Takasaki, Appl. Opt. 9, 1467(1970).

2. M. Idesawa, T. Yatagai, and T. Soma, Appl. Opt. 16, 2152(1977).

3. T. Yatagai and M. Idesawa, Opt. Laser Eng. 3, 73 (1982).4. D. T. Moore and B. E. Truax, Appl. Opt. 18, 91 (1979).5. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156

(1982).6. Y. Yoshino, Kogaku (Jpn. J. Opt.) 1, 128 (1972).7. M. Suzuki and K. Suzuki, Bull. Jpn. Soc. Precis. Eng. 8, 23

(1974).8. M. Idesawa and T. Yatagai, Sci. Pap. Inst. Phys. Chem. Res. Jpn.

71, 57 (1977).9. J. L. Doty, J. Opt. Soc. Am. 73, 366 (1983).

10. M. Takeda, Opt. Laser Eng. 3, 45 (1982).11. M. Takeda, M. Kawabuchi, and T. Ose, Kogaku (Jpn. J. Opt.) 3,

373 (1974).12. See, for example, J. J. Downing, Modulation Systems and Noise

(Prentice-Hall, Englewood Cliffs, N.J., 1964), pp. 86-112.

The authors thank T. Yatagai of Tsukuba Universityfor his helpful discussions.

3982 APPLIED OPTICS / Vol. 22, No. 24 / 15 December 1983