Spatiotemporal phase unwrapping for the measurement of discontinuous objects in dynamic fringe-projection phase-shifting profilometry Hong Zhang, Michael J. Lalor, and David R. Burton

A spatiotemporal phase-unwrapping method is presented that combines the dynamic fringe-projection method and the phase-shifting technique and extends the phase-unwrapping method, which measures two phase maps at different sensitivities. The most important feature of the method is that it makes possible the automatic three-dimensional shape measurement of discontinuous objects with large dynamic range limits and high precision because the effective wavelength of the fringe-projection profilometry can be continuously varied over several orders of magnitude by rotation of the projection grating in its own plane. Only one grating and several steps of rotating the grating are required; therefore the method is inherently simple, fast, and robust. In the experiment, choosing the rotation angle was crucial for optimizing the measurement speed and the measurement accuracy. A criterion is presented for the choice of the minimum number of rotational steps for a given accuracy. The experimental results demonstrate the validity of the proposed method. © 1999 Optical Society of America OCIS codes: 120.2650, 120.3180, 120.3940, 120.4630, 120.4640, 120.5050.

1. Introduction

Projection of fringes for the measurement of surface shape is a noncontact optical method that has been widely recognized as being potentially useful in the measurement of various diffuse objects. The technique is referred to as fringe-projection profilometry.1–3 This method uses parallel or divergent fringes projected onto the object surface either by a conventional imaging system or by coherent light interference patterns, in which the projection and recording directions are different. The resulting phase distribution of the measured fringe pattern includes information on the surface height variation of the object. An automated analysis of the fringe patterns is normally carried out either by the Fourier-transform method or by the phase-shifting technique.4,5 Both methods produce wrapped phase maps ~i.e., phase values that lie in the range 2p to p!; the 2p phase jumps must then be removed by the process known as phase unwrapping to recover the

The authors are with the Coherent and Electro-Optics Research Group, School of Engineering, Liverpool John Moores University, Byrom Street, Liverpool L3 3AF, United Kingdom. Received 8 October 1998; revised manuscript received 8 February 1999. 0003-6935y99y163534-08$15.00y0 © 1999 Optical Society of America 3534

APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

surface heights. One normally carries out phase unwrapping by comparing the phases at neighboring pixels and adding or subtracting 2p to bring the relative phase between the two pixels into the range 2p to p. This causes problems when the technique is applied to real objects in engineering or medical measurements, because such objects often contain edges or discontinuities that cause phase jumps greater than p.6 – 8 It can then become impossible to unwrap correctly across the image. The problem can be reduced in principle by measurement of several phase maps at different sensitivities.7,8 White-light interferometry can also measure absolute surface height9 but is difficult to apply when height variations exceed the range of a piezoelectric translator or a dc motor; also, it is limited by the measuring aperture. A temporal phase-unwrapping method has been proposed and developed for the measurement of discontinuous objects.10,11 In this method the phase at each pixel is unwrapped along the time axis; because the unwrapping path does not cross object discontinuities, 2p phase errors do not propagate across the image as with spatial-unwrapping approaches. To obtain high-precision results, however, one needs a relatively large number of incremental maps, and there are calculation costs.11 The phaseunwrapping method, which measures the phase maps at different sensitivities ~the algorithm proposed by Zhao et al.8!, and the temporal phase-unwrapping

method ~the algorithm proposed by Huntley and Saldner11! were analyzed by numerical simulation and an experimental technique.12 It was shown that the Zhao et al.8 algorithm unwrapping success rate begins to drop at low noise levels and that the Huntley and Saldner11 algorithm success rate varies according to the position of the object within the measurement volume. This is especially true in the area close to the edge, where the success rate drops dramatically even at quite low noise levels for the method used by Huntley and Saldner11. As a result of this study those authors went on to propose an improved temporal phase unwrapping.12 Another absolute phasemeasurement procedure adopts the temporal phaseunwrapping method, which was based on the derivation of a precision adapted sequence of grating orientations and the stepwise reconstruction of the object phase.13 The first step in the temporal phaseunwrapping procedure is to construct a fringe pattern whose effective wavelength is sufficiently long and covers the whole measurement range including the object height. Therefore the measured phase values will automatically lie in the range 2p to p, and no spatial phase unwrapping is needed. However, for different projection systems and projection strategies this step will introduce high phase-measurement noise and influence the unwrapping success rate. Hence it makes the measurement procedure complex and system adjustment difficult. For instance, Fig. 3 of Ref. 13 showed that a small rotation angle g will lead to a drastic decrease in accuracy. In the first step of the experiment, g~0! 5 0.12°, the phase error obtained was DN 5 2εysing~0! 5 19 ~effective waves! ~ε indicates the phase measurement inaccuracy and equals 1y25 effective wavelength!.13 Recently Xie et al.14 proposed a three-map absolute phase-measurement method based on a shadow moire´ system. Rotating the grating altered the effective wavelength and changed the phase of the moire´ fringe, in which the moire´ fringe signal was assumed to be a sinusoidal waveform. Therefore phase-measurement error occurs when the moire´ fringe signal is not a sinusoidal waveform and there are errors in the grating rotation angles.14 In this paper a novel spatiotemporal phaseunwrapping method is proposed that uses dynamic fringe projection7,14 and the phase-shifting technique to measure discontinuous objects. This novel method extends the phase-unwrapping method to measure two phase maps at different sensitivities.8 The main attribute of fringe projection is the generation of an effective wavelength by projection of a known fringe pattern onto the object under investigation at a certain angle of incidence. Changing the fringe pattern’s pitch or the angle of incidence may alter the effective wavelength. The aim of the dynamic fringe-projection method is to change the effective wavelength continuously, over several orders of magnitude ~to infinity, in principle!, by rotating the projection grating in its own plane. In the first step of the spatiotemporal phase-unwrapping method a large effective wavelength should be chosen such that the phase jump at the discontinuous profile is less

Fig. 1. Optical geometry of the dynamic fringe-projection and -recording system.

than p; then the spatial phase-unwrapping method can be applied.15 Therefore this method avoids the first step of the temporal phase-unwrapping procedure, which requires a large effective wavelength for covering the whole measurement range and the object height, improves the first-step phasemeasurement accuracy, and simplifies the measurement procedure and the system adjustment. After the first step, several intermediate phase maps can be obtained by rotation of the grating in its own plane in several steps, reducing the effective wavelength step by step, and unwrapping each pixel along the time axis. In the final step the grating is in the position in which the effective wavelength is minimum and the precision result can be obtained. Experimental results from measurements of a step object and a human hand are presented. Compared with other methods,14 this novel technique improves the phase-measurement accuracy by employing a phase-shifting technique in which errorcompensating algorithms can eliminate or minimize the effects of the harmonics of nonsinusoidal waveforms and system phase-shift errors.15–18 Furthermore, the errors in grating rotation angles have no effect on the phase-measurement accuracy. 2. Description of the Technique A. Optical Geometry of Dynamic Fringe Projection and Phase Measurement

The optical geometry of the dynamic fringe-projection and -recording system is shown in Fig. 1. The grating is projected in a parallel beam. The x axis is chosen as in the figure, the y axis is normal to the plane of the figure and on the reference plane, and z is the distance between the object surface and the reference plane. The recording direction is perpendicular to the reference plane. Let g~x, y, ti ! be the intensity distribution of the grating pattern recorded by the CCD camera at a designated point ~x, y! on a two-dimensional array as a function of ti , the phaseshift parameter. Here g is a real, periodic function of the array point position ~x, y! and can be written as a Fourier series: `

gi ~x, y, ti ! 5 s0~x, y! 1

( s ~x, y!cos@nf~x, y! 1 nt #, n

n51

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS

i

(1)

3535

of the phase-shifting technique.20 A general expression for an m-sample phase-shifting algorithm is m

(ag i

i

tan f 5 i51 m

(bg i

,

(6)

i

i51

where ai and bi are the filtering amplitudes of the ith sample and gi is the ith sample’s intensity value defined by Eq. ~1!. According to the Fourier description method,16 the filter functions f1~t! and f2~t! can be expressed as m

Fig. 2. Variation of effective wavelength le with rotation angle a.

f1~t! 5

( a d~t 2 t !, i

i

i51 m

where f~x, y! is the phase of the first harmonic of the fundamental frequency of the grating pattern or the phase to be measured at point ~x, y!, sn~x, y! is the amplitude of the nth harmonic of the grating pattern and n is a positive integer. According to Fig. 1, the phase f~x, y! can be written as19 f~x, y! 5 ~2pyp0!@ x cos u cos a 2 y sin a 1 z~x, y!sin u cos a#,

(2)

where p0 is the pitch of the grating pattern on the plane that is perpendicular to the projection direction, u is the incident angle of fringe projection, and a is the rotation angle of the grating in its own plane. A special case a 5 0 is considered when the grating lines are parallel to the y axis. From Eq. ~2!, the phase f~x, y! can be written as f~x, y! 5 ~2pyp0!@ x cos u 1 z~x, y!sin u# 5 ~2pyp!@ x 1 z~x, y!tan u#,

(4)

is obtained for a 5 0. When the grating is rotated in its own plane ~a Þ 0!, from Eq. ~2! the effective wavelength p0 p l0 le 5 5 5 sin u cos a tan u cos a cos a

APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

i

(7)

i

where d~t! is Dirac delta function. System phaseshift errors, as well as the nonsinusoidal waveform of the fringe pattern, are the most common sources of systematic errors in the evaluated phase for the phase-shifting technique. By use of the Fourier description method one can obtain the frequency spectra of filter functions and analyze the sensitivity of the phase-shift algorithms to systematic phase-shift errors and the nonsinusoidal waveform of the fringe pattern. Many studies have been reported of errorcompensating algorithms that can eliminate or minimize the effects of systematic phase-shift errors and harmonics of nonsinusoidal waveforms.15–18 B.

Spatiotemporal Phase Unwrapping Method

From Eqs. ~1! and ~2!, when a grating pattern is projected onto a reference plane the grating pattern’s intensity distribution gri~x, y, ti ! recorded on the reference plane can be written as `

gri~x, y, ti ! 5 s0~x, y! 1

( s ~x, y!cos@nf ~x, y! 1 nt #, n

r

i

n51

(8)

where fr~x, y! 5 ~2pyp0!~ x cos u cos a 2 y sin a!

(9)

is the carrier’s frequency phase. According to Eq. ~6!, the phase functions f~x, y! and fr~x, y! can be obtained as

1 2 12 1 2 12 m

(ag i

(5)

can be obtained. Equation ~5! shows that, by dynamic fringe projection, the effective wavelength can be changed continuously. Figure 2 shows a typical case when l0 5 1.484 mm. It is shown that the effective wavelength can be varied over several orders of magnitude ~to infinity, in principle! by rotation of the projection grating in its own plane. The phase f~x, y! in Eq. ~2! can be retrieved by use 3536

( b d~t 2 t !, i51

(3)

where p 5 p0ycos u is the pitch of the grating pattern on the reference plane. From Eq. ~3!, the effective wavelength l0 5 pytan u

f2~t! 5

f~x, y! 5 arctan

i

i51 m

(

5 arctan

Sm , Cm

(10)

5 arctan

Sr , Cr

(11)

bi gi

i51 m

(ag i

fr~x, y! 5 arctan

ri

i51 m

(

bi gi

i51

where subscripts m and r correspond to the measured object and the reference plane, respectively.

From Eqs. ~10! and ~11!, the superimposed phase caused by the carrier frequency21 can be removed; then the wrapping phase map caused only by the object shape is obtained:

S

f0~x, y! 5 f~x, y! 2 fr~x, y! 5 arctan

D

Sm Cr 2 Cm Sr . Cm Cr 1 Sm Sr (12)

The object phase f0~x, y! calculated according to Eq. ~12! is wrapped in the range 2p and 1p. The true phase of the object is F0~x, y! 5 2n~x, y!p 1 f0~x, y!,

(13)

where n~x, y! is an integer. From Eqs. ~2!, ~9!, and ~13!, the object profile z~x, y! can be expressed as follows: z~x, y! 5

F0~x, y!l0 . 2p cos a

(14)

Equation ~13! shows that phase unwrapping is only a process of the determination of n~x, y!. One normally carries out phase unwrapping by comparing the phases at neighboring pixels and adding or subtracting 2p to bring the relative phase between the two pixels into the range 2p to 1p. This process is known as spatial phase unwrapping11 and causes problems when it is applied to discontinuous objects and the absolute values of the phase jumps are larger than p because of discontinuities. According to Eq. ~5! and Fig. 2, the effective wavelength le can be changed by rotation of the projection grating in its own plane. The sensitivity is reduced with a large effective wavelength at an angle a1, so the absolute values of the phase jumps are less than p at discontinuities. Then one can use spatial phase-unwrapping methods to obtain the true phase F01~x, y! and profile z1~x, y! of the object15: F01~x, y! 5 2n1~x, y!p 1 f01~x, y!, z1~x, y! 5

F01~x, y!l0 . 2p cos a1

(15)

To obtain a high-precision result one should increase sensitivity with a small effective wavelength by reducing the rotation angle a in several steps until a 5 0. In this case we rotate the projection grating in k steps ~a1 . a2 . . . . . ai . . . . . ak, ak 5 0! to obtain the precise and correct result. For a 5 ai ~i 5 2, . . . , k! the intermediate phases of the object F90i and the integer ni ~x, y! can be written as F90i~x, y! 5 F0~i21! ~x,y!

H

ni ~x, y! 5 int

cos ai , cos a~i21!

J

1 @F90i~x, y! 2 f0i~x, y!# , 2p

(16)

Fig. 3. Experimental setup for dynamic fringe-projection phaseshifting profilometry.

where the function int$ % returns the nearest integer to its argument. From Eqs. ~16!, the true object phase F0i~x, y! and the object profile zi ~x, y! are obtained: F0i~x, y! 5 2ni ~x, y!p 1 f0i~x, y!, zi ~x, y! 5

F0i~x, y!l0 . 2p cos ai

(17)

3. Experiment

A dynamic fringe-projection phase-shifting experimental setup with a white-light source and projecting a Ronchi grating was constructed. The experimental setup is shown in Fig. 3. As the light source a 50-W halogen lightbulb was used, whose output was imaged onto a diaphragm. A subsequent achromatic lens ~L1! produced parallel beams, which passed through the Ronchi grating that can be rotated with a polarizer holder with a resolution of 0.1°. Two achromatic lenses ~L2 and L3! imaged the grating onto the three-dimensional ~3-D! object surface and formed parallel beams for projection. The angle of incidence was ;30°. The fringe pattern was captured by a CCD camera and converted into an electrical signal. A frame grabber in a PC computer digitized the images to a 640 3 480 3 8 bit array. The Ronchi grating can be shifted by a precisionmotorized translation stage with a resolution of 0.1 mm. In this way a defined and accurate phase shift can be introduced for phase evaluation of various images. In the experimental setup the CCD camera lens was defocused to yield a quasi-sinusoidal fringe pattern; however, there are still several harmonics in the fringe pattern. In the experiment, the Surrel 10sample phase-shifting algorithm was chosen,18 which suppresses harmonics up to the fourth harmonic and is insensitive to linear phase-shift miscalibration at the first, second, third, and fourth harmonics. Figure 4 shows a fringe pattern and a wrapped phase map of the step object under investigation, in which the rotation angle is a 5 0. The heights of the two steps were both approximately 5 mm. It is impossible to unwrap the phase map in Fig. 4~b! correctly by conventional spatial methods because the 1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS

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Fig. 4. ~a! Fringe pattern projected onto the step object with rotation angle a 5 0, ~b! wrapped phase map with a 5 0, showing discontinuities of the step object. The grayscale represents phases of 2p and 1p.

Fig. 5. ~a! Fringe pattern projected onto a human hand with rotation angle a 5 0, ~b! wrapped phase map with a 5 0, showing discontinuities of the profile of the human hand compared with the reference plane. The grayscale represents phases of 2p and 1p.

phase jumps at the steps are too large ~more than 2p!. Therefore the relative heights of the steps are uncertain to integral multiple of 2p. Figure 5 shows the fringe pattern and the wrapped phase map of a human hand in which the rotation angle was a 5 0 and the hand was on the reference plane. The 3-D profile of the hand contains discontinuities compared with the reference plane. It is therefore impossible to unwrap the phase map in Fig. 5~b! correctly by conventional spatial methods because the phase jumps at the discontinuities are too large ~more than 2p!. There are shadows in the fringe patterns of Figs. 4~a! and 5~a!. For Fig. 4~a!, invalid pixels are masked out by the fringe modulations.22 For Fig. 5~a!, because of the variation in the fringe modulations in the whole image, the invalid pixels are removed by the mask generated with a combined criterion of fringe modulations22 and second differences of the locally unwrapped phase.23

For the first step of the spatiotemporal phaseunwrapping method in Eqs. ~15! the spatial phaseunwrapping method15 used is similar to that described in Refs. 22 and 24. The path of phase unwrapping is always along the direction from the pixel with higher modulation to the pixel with lower modulation. In every step the invalid pixels are masked out. After the valid pixels are unwrapped, the invalid pixels are replaced by the average of their valid neighbors in a 3 3 3 neighborhood. In the final step, the high-precision true phase of the object is obtained and the phase data are converted to height values by being scaled with the effective wavelength l0 in Eq. ~17! ~ai 5 ak 5 0°!. The calibration procedure for the effective wavelength l0 includes two stages. In the first stage the fringe pattern was projected onto the reference surface and 10 fringe samples were recorded with phase

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shift py3. The Surrel 10-sample phase shifting algorithm was used to calculate phase f1 : 10

( a9g i

1i

tan f1 5 i51 10

(

5

b9i g1i

S1 , C1

(18)

i51

where $a9i % 5 ~2=3, 23=3, 23=3, =3, 6=3, 6=3, =3, 23=3 2 3=3 2 =3! and $b9i % 5 ~1, 2 1, 27, 211, 26, 6, 11, 7, 1, 21!. In the second stage the reference surface was moved in the direction of the observation, which is perpendicular to the reference surface. The movement distance Dl was less than half of the effective wavelength l0. Therefore no wrapped phase occurred. Ten fringe samples were recorded with phase shift py3. The phase f2 was calculated by 10

( a9g i

2i

tan f2 5 i51 10

(

b9i g2i

5

S2 . C2

(19)

i51

According to the method presented in Ref. 21, the movement phase of the reference surface Df was calculated by

S

Df 5 f2 2 f1 5 arctan

D

S2 C1 2 C2 S1 , C2 C1 1 S2 S1

(20)

where the carrier frequency was removed automatically. From Eq. ~14!, the effective wavelength l0 can be calculated with a 5 0°: l0 5

2pDl . Df

(21)

Phase values in the points from the horizontal line in the center of the fringe pattern were calculated by Eq. ~20!. We calibrated the effective wavelength l0 from Eq. ~21!, using the average of Df and controlling the movement distance Dl of the reference surface mounted upon a micrometer-driven precision translation stage. The rms phase error was calculated between the phase values and the best-fitting straight line through these points. The first calibration was made for the experimental setup that employed parallel fringes projected onto the step object. The effective wavelength l0 was calibrated as 2.015 mm. The rms phase error was calculated as 7.2 mm, which is 1y280 of the effective wavelength and 1y3470 of the depth of the measurement range ~25 mm!. The second calibration was made for the experimental setup that employed divergent fringes projected onto the human hand. The effective wavelength l0 was calibrated as 14.04 mm. The rms phase error was calculated as 59 mm, which is 1y240 of the effective wavelength and 1y2800 of the depth of the measurement range ~165 mm!.

Fig. 6. Shading display showing the profile of the step object obtained by unwrapping Fig. 4~b! by the spatiotemporal phaseunwrapping method through five intermediate rotation angles.

4. Experimental Results and Discussion

Figure 6 shows the result of unwrapping the phase map of Fig. 4~b! through five intermediate rotation angles, a 5 82°, 75°, 65°, 45°, 0°. The heights between the two steps were successfully measured, which would not have been possible with any of the existing spatial phase-unwrapping methods. Calculated cross sections are shown in Fig. 7 for several values of a. Figure 7~e! is the cross section through the surface plot shown in Fig. 6. The reduction of noise in the profile of the object is significant as the rotation angle is decreased, because the effective wavelength is reduced and the sensitivity is improved. Figure 7~a! is the result that would have been obtained with a conventional phase-shifting fringe-projection method of large fringe spacing with rotation angle a 5 82° and therefore low sensitivity. Although the steps were successfully measured, the signal-to-noise ratio is poor. From Fig. 7~a! to Fig. 7~e! the signal-to-noise ratio is improved step by step as the rotation angle is reduced. The step heights obtained by averaging of the data points within each step from the cross section in Fig. 7~e! are 9.874 and 4.944 mm, respectively and compare well with the average values 9.846 and 4.893 mm measured at the same cross section on the step object with the contact three-axis measuring machine with a resolution of 1 mm. We found that, in the experiment, the ratio cos ai ycos a~i21! should be less than 2 to yield the correct result. Here we define cos ai 5 Ai cos a~i21!

~i 5 2, . . . , k!

(22)

because Ai is an amplification for obtaining the intermediate phase F9i ~x, y! from the true phase of the object F~i21!~x, y! in Eqs. ~16!. Because of the noise in F~i21!~x, y!, if Ai is too large ~ Ai . 2! the noise is amplified as well when F9i ~x, y! is calculated. Therefore the correct integer ni ~x, y! cannot be obtained from Eqs. ~16!. Figure 8 shows the result of unwrapping the phase of Fig. 5~b! through four intermediate rotation angles, a 5 77°, 70°, 50°, 0°. In the case of Fig. 8, lens L3 1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS

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Fig. 7. Cross sections of the step object surface profile ~row 450 of Fig. 6! obtained by the spatiotemporal phase-unwrapping method through ~a! a 5 82°; ~b! a 5 82°, 75°; ~c! a 5 82°, 75°, 65°; ~d! a 5 82°, 75°, 65°, 45°; ~e! a 5 82°, 75°, 65°, 45°, 0° intermediate rotation angles.

was removed from the experimental setup so a divergent projection beam was formed. According to Ref. 21, the additional phase modulation introduced by the divergent projection can be canceled out when Eq. ~12! is used to remove the superimposed 3540

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phase caused by the carrier frequency. The experimental result and the calibration result discussed in Section 3 show that the divergent projection has no influence on the spatiotemporal phaseunwrapping method.

5.

6.

7.

Fig. 8. Shading display showing the profile of the human hand obtained by unwrapping Fig. 5~b! by the spatiotemporal phaseunwrapping method through four intermediate rotation angles.

8.

9.

5. Conclusions

A spatiotemporal phase-unwrapping method has been proposed that combines the dynamic fringe-projection method and the phase-shifting technique and extends the phase-unwrapping method, which measures two phase maps at different sensitivities. The optical geometry of dynamic fringe projection was presented, and the spatiotemporal phase-unwrapping method was analyzed. The important feature of the novel method is that it permits automatic 3-D shape measurement of discontinuous objects with large dynamic range limits and high precision because the effective wavelength can be continuously varied over several orders of magnitude. This novel method avoids the first step of the temporal phase-unwrapping procedure, which requires a large effective wavelength to cover the whole measurement range and the object height, improves the first-step phase-measurement accuracy, and simplifies the measurement procedure and the system adjustment. Only one grating and rotation of the grating in several steps are required in the experiment; the method is therefore inherently simple, fast, and robust. We found that in the experiment the ratio cos ai ycos a~i21! ~i 5 2, . . . , k! should be less than 2 to yield the correct result. This means that the choice of rotation angle a is crucial for optimizing the measurement speed and the measurement accuracy. Generally speaking, the more steps that are chosen, the more accuracy is obtained. In terms of measurement efficiency, however, the minimum steps may be chosen to yield the required accuracy according to the conclusion given above. The experimental results confirm the validity of the proposed method and the conclusion.

10.

11.

12.

13.

14. 15.

16.

17.

18. 19.

20.

21.

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