On-line automated phase-measuring profilometry

Optics and Lasers in Engineering 15 (1991) 127-139

On-Line Automated Phase-Measuring Profdometry

Ming Chang

Department of Mechanical Engineering, Chung-Yuan Christian University, Chung Li, Taiwan 32023, Taiwan

&

Der-Shen Wan

Chung-Shan Institute, PO Box 90008-15-6, Lung Tan, Taiwan 32526, Taiwan

(Received 30 August 1989; revised version received 9 November 1990; accepted 14 November 1990)

ABSTRACT

A simple phase reduction algorithm is used for on-line automated phase measuring profilometry of 30 diffuse objects through a projection moire’ method and image processing techniques. The error of the algorithm is within &h wavelength of the projected fringe. In contrast to phase shifting interferometry, this method eliminates the need for an accurate phase shifter and isolation table, thus an on-line or in-process measurement can be executed more easily. Experimental results show that a surface height resolution of better than 60 urn has been attained as the phase of four deformed grating images with random phase shifts are averaged.

INTRODUCTION

Noncontact profilometry of 3D diffuse objects is important in automated manufacturing, component quality control, robotics, and solid modeling applications. Projection moire topography’-3 is a well-known method for the noncontact profilometric measurement of diffuse objects. When a sinusoidal intensity distribution is projected onto an object, the mathematical representation of the deformed grating image intensity distribution is similar to that encountered in conventional optical interferometry. It is shown that the surface height distribution is translated to a phase distribution, and thus the method of phase

127

Optics and Lasers in Engineering 0143-8166/91/$03.50 KJ 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland

128 Ming Chang, Der-Shen Wan

modulation interferometry can be used for a quantitative analysis of the surface topography.

Phase shifting interferometry (PSI)4,5 is a general automatic phase measurement technique which introduces the discrete stepping or ramping phase modulation into an interferogram, using the synchronous detection method, three or four measurements of the intensity distribution across the pupil for some certain amount of phase shift, thus measuring the phase of the interferogram. However, for the technique of projection moire, the projected fringe is insensitive to vibration, and a simpler phase reduction algorithm instead of PSI is more appropriate in the reduction phase so the on-line profilometry can be implemented more easily.

In this paper, a simple automatic phase reduction algorithm”.’ is applied, which eliminates the need for an accurate phase shifter (generally a piezoelectric transducer), synchronous detection interface and isolation table. The algorithm that was first used for interferometry testing of large optics is based on the idea that a phase can be retrieved as a quantity proportional to the intensity of the interferogram if the phase of the interferogram increases montonically, and this limitation is matched on projection moire topography measurements. The deformed grating image intensity distribution is recorded by a CCD camera and transferred digitally to a personal computer through a frame grabber. The phase acquisition algorithm is simple so it can generate high resolution phase maps using a high resolution area detector array in a short time. The error of algorithm can be shown to be within &th wavelength in the wavefront, which can be reduced further as the phase of several interferograms with random phase shifts are averaged. Thus this method is immune to the natural vibration of the testing environ- ment for generating the projection fringe since it can be an advantage in the production of random phase shifts. Due to the advantages of high spatial resolution, simple instrumentation, fast measurement speed and an insensitivity to air turbulence, an online automated phase measuring profilometer of 3D diffuse objects can be developed.

PROFILOMETER SYSTEM AND IMAGE ANALYSIS

Figure 1 is a schematic of the projected moire profilometer set-up. The working system consists of a Twyman-Green interferometric projector (a 3 mW laser was used) to project the sinusoidal grating, an interferogram recording CCD camera, an image processing interface, and a personal computer. When the sinusoidal grating is projected onto a 3D

On-line automated phase-measuring profilometry 129

Twyman-Green Interferometer

Image processing computer system

TV monitor

Fig. 1. Schematic of the phase-measuring surface profilometer.

diffuse object, the mathematical representation of the deformed grating (which is deformed by the surface curve) image intensity distribution is similar to that encountered in conventional optical interferometry. As shown in Fig. 2, the object surface height distribution is translated to a phase distribution, and can be written as3

h = BD = PO tan &,&,/2n(l + tan &/tan 0,) (1)

where h is the surface height relative to an arbitrary chosen reference

Fig. 2. Optical geometry for deformed grating image analysis.


plane, PO is the grating pitch in the direction parallel to the reference plane, &n is the phase difference between point C on the reference plane and point D on the object, 8,, and 8, are the grating projected angle and receiving angle, respectively.

Assuming that 8, is nearly 90” as in the case of a practical system with large demagnification, the object shape h(x, y) at each detection point (x, y) can be determined by

W, Y> = (6 tan %/23+#& y> (2)

where @(x, y) is the phase difference between object surface and reference plane at each point (x, y).

Since the surface height distribution is measured from an arbitrary reference plane, the decision of the initial phase data is also arbitrary, and the measured phase difference 4(x, y) can be directly translated onto the surface topography of the test object. Here, the deformed grating image is recorded by a 530 x 490 CCD area array camera; the camera being connected to a digital image processing interface and an IBM AT personal computer for phase calculation, tilt subtraction, 2D phase display, 3D profile display, and printed output.

AUTOMATIC PHASE REDUCTION ALGORITHM

A brief review of the phase reduction algorithm is presented here. For this algorithm, the interferogram must result from the interference of a tilted wavefront whose phase increases monotonically with respect to the reference wavefront. This requirement is just satisfied at the resulting straight fringes of a projected moire system. The intensity distribution of the deformed fringes are recorded by the CCD video camera in $ s and then transferred digitally to the IBM AT computer through a frame grabber. The recorded intensity Z is then normalized to lie between a minimum of 0 and a maximum of 1. The algorithm tests, pixel by pixel, along each scan line of the fringe pattern for the maximum and minimum of Z successively.

Starting on the left-hand side, the reconstructed phase is set equal to Z3t. The phase pattern of the interferogram can be obtained through searching the consecutive maximum and minimum intensities. Once a maximum is recorded, phase is set equal to 7c(2n - Z), where it is the nth maximum, until a minimum is found, the phase value is then changed to n(2m + Z), where m is the mth minimum. The process is shown in Fig. 3. Figure 3(a) shows the normalized intensity of two-beam interference versus distance perpendicular to fringes. Figure


1.00 .

A 5 E 0.50_

IE

0.00 0.40 0.60 1.20 1.60 2.00

12.56

9.42

j 6.28 P

distance perpendicular to fringes

(4

J 0.00 o_ 40 0.80 1.20 1.60 2.00

dlstonce perpendicular to fringes

@I Fig. 3. Phase reduction algorithm: (a) the normalized intensity of two beam . . ^ I.~ . interference versus distance perpendicular to trmges; (b) phase reduced by the

algorithm from the interference data shown in (a).

3(b) displays the phase (the curve line) reduced by the algorithm from the interference data of Fig. 3(a), and the straight line represents the true phase. Phase reconstructed from uniform sinusoidal fringes should increase linearly, but is approximated here by a series of sine wave segments. The maximum error introduced by the approximation is within &th wavelength. Considering a segment over the range of phase 0 < # < JC the error is given by # - n(l - cos $)/2, where the intensity is scaled between 0 and 1. The largest values of the above error occur when sin @ = 2/n, and are f0.33 radians. The inherent error in a reconstructed wavefront is thus 58 wavelength. Averaging N interferograms with random piston error differences between the wavefronts reduces the error as l/a.

Considering the problem of a nonuniform distribution of intensity over the interferogram, maximum and minima are required as


functions of position. These can be found by recording several interferograms with random piston phase differences. Maxima and minima for each pixel are found by comparing the intensities at that pixel of N interferograms with fringes randomly shifted, and these intensity extrema are used to scale the intensity pixel by pixel of the separate interferogram as

I(& Y) = &(x7 Y) + zmin(X, Y>

&11ax(x7 Y > - zmin(Xt Y 1

where &(x7 Y), L&, Y) and Zmin(X, y) are the original intensity, local intensity maximum and minimum at each pixel (x, y), respectively. After the intensity distribution of an interferogram is scaled using eqn (3), the normalized interferogram with more uniform amplitude will be used for the phase reduction. Since the random phase shift is needed in the phase reduction algorithm, the natural vibration of the environ- ment could be an advantage in the measurement, and the isolation table is naturally eliminated.

The normalization process in the algorithm is very helpful to enhance the fringe contrast. Through this process, not only can the intensity amplitude be more uniform, which is very important if the phase reduction error is to be reduced, but also the fixed pattern noise is subtracted.

COMPUTER PROGRAM

This technique can use computer control to take data, calculate phase, and display. A series of software programs using C language was developed to execute measurement and analysis. The program main operations are described as follows:

(1) When a sinusoidal intensity distribution is projected onto a workpiece, four continuous frames of real-time deformed grating image intensity data under natural vibration are recorded.

(2) The grey levels are compared for each pixel of these four interferograms, finding the maxima and minima for each pixel and saving these two intensity extrema files.

(3) Using eqn (3) and intensity extrema files, the four frames of initial image data are normalized, and then the four initial files and the two extrema files are deleted.

(4) A series of maximum and minimum grey levels for each normalized fringe image file are searched for, reproducing phase


distribution using the phase reduction algorithm, averaging the four frames of phase data and removing tilt.

(5) A phase map is displayed in several ways: a cursor-controlled one-dimensional slice through any section of the phase map, a 2D grey scale or pseudo-color contour map, of a high resolution two-sided with hidden-line, 3D representation.

(4 (b)

(4 Fig. 4. Four randomly shifted interferograms of the deformed grating on a cap-like

surface.


EXPERIMENTAL RESULTS

The phase reduction algorithm has been applied to reduce phase and map 3D profile in several practical cases. The measured results of two examples are demonstrated here. The first object to be profiled is a cap-like surface. Under natural vibration disturbance, each frame of initial deformed grating image was recorded with random phase shift and then normalized. Figure 4 shows the four interferograms of the deformed grating caused by projecting the sinusoidal intensity distribution on the surface and taken randomly shifted to each other. The four projected fringes are used for reducing the error of phase reduction algorithm. The straightness of the undeformed grating is very important so as to make sure that the deformed part is completely caused by surface curve.

Figure 5 shows the local intensity maxima and minima of every pixel, found by comparing four interferograms of Fig. 4. The normalized results are shown in Fig. 6, where the intensity distributions are more uniform and can be used for the phase reduction. The direct conversion results of intensity to phase of the four interferograms are shown in Fig. 7. The 3D display of the surface profile from the average of the four phase maps is shown in Fig. 8, where tilt has been removed by means of computer programming. The peak-to-valley value was read first as

Fig. 5. Local intensity (a) maxima and (b) minima of every pixel.

On-line automated phase-measuring projilometry

(4 (b)

(4 Fig. 6. Normalized interferograms reduced from the initial interferograms of Fig. 4.

15.39 mm by the mechanical contact method. The error obtained from the phase reduction algorithm is about 0.06 mm.

From eqn (2), the effective wavelength of the projected moire system is equivalent to PO tan oO. The measurement sensitivity is mainly dependent on the choice of effective wavelength. A smaller system wavelength is usually needed if the demand of accuracy is important.


(b)

Cc) (4

Fig. 7. Phase maps reduced from the normalized interferograms shown in Fig. 6.

An effective wavelength of 5.12mm was used, i.e. the peak-to-valley measurement with an accuracy of 0.01 wavelength wavefront error was achieved.

The second object to be profiled is a small bottle-gourd. Similarly, Fig. 9(a) (b) are the interferograms of the initial deformed grating intensity distribution and the 3D surface profile, respectively. Com-

On-line automated phase-measuring projilometry 137

Fig. 8. Three-dimensional display of the measured profile of the cap-like surface.

pared with measurement using a contact profilometer, the accuracy of peak-to-valley measuring was found to be about 0.07 mm.

The single surface profile measurement can be taken in as short a time as O-1 s with a graphics display of the surface profile in several seconds. Owing to the fact that the on-line surface profile measurement of engineering surfaces is becoming increasing important, future work will strive to improve the overall system and software to effectively execute a high speed on-line observation.

In the case of large objects, divergent illumination may be necessary to avoid use of large collimating optics. Because of the diverging nature of the illimination, a grating with a nonuniform pitch will be induced on the object. The analysis is more complicated than the previous case and requires a different approach.8 However, the height distribution can still be obtained from phase measurement on the object and a reference plane by the phase reduction algorithm.

CONCLUSIONS

An automatic 3D diffuse object profile measurement using projected moire with random phase shift algorithm is presented. The phase


(4

(b) Fig. 9. The profile measurement of a bottle-gourd: (a) one of the initial deformed

gratings; (b) three-dimensional surface profile display.


acquisition algorithm developed here is based on the assumption that the phase is proportional to the intensity of the interferogram. The direct conversion of intensity to phase makes the algorithm fast and allows it to operate with no smoothing at all. Since the random phase shift is helpful in the algorithm, the fixed pattern noise is not important, and the need for a calibration process, which is usually needed in an experimental technique, is also eliminated. Due to the advantages of fast measurement speed and insensitivity to environmental disturbance, the random phase shift algorithm will be especially valuable for the on-line phase measuring profilometry of 3D objects.

ACKNOWLEDGEMENT

This work was supported by the National Science Council of the Republic of China.

REFERENCES

1. Takasaki, H., Moire topography. Appl. Opt., 9 (1970) 1457-72. 2. Idesawa, M. & Yatagai, T., General theory of projection-type moire

topography. Sci. Papers I. P. C. R., 71 (1977) 57-70. 3. Srinivasan, V., Liu, H. C. & Halioua, M., Automated phase-measuring

profilometry of 3-D diffuse objects. Appl. Opt., 23 (1984) 3105-08. 4. Koliopoulos, C. L., Interferometric optical phase measurement techniques.

PhD dissertation, University of Arizona, 1981. 5. Chang, M., Hu, C. P., Lam, P. & Wyant, J. C., High precision

deformation measurement by digital phase shifting holographic interferometry. Appf. Opt., 24 (1985) 3780-3.

6. Angel, R. & Wan, D. S., A new algorithm for rapid automatic reduction of test interferogram. Workshop on Optical Fabrication and Testing, Optical Society of America, Seattle, 1986.

7. Wan, D. S. & Lin, D. T., Runchi test and a new phase reduction algorithm. Appl. Opt., 29 (1990) 3255-65.

8. Srinivasan, V., Liu, H. C. & Halioua, M., Automated phase-measuring profilometry: a phase mapping approach. Appl. Opt., 24 (1985) 185.

On-line automated phase-measuring profilometry

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