a sequence pulse counting method for shape measurement in dual-beam speckle interferometry

SCIENCE CHINA Physics, Mechanics & Astronomy

© Science China Press and Springer-Verlag Berlin Heidelberg 2011 phys.scichina.com www.springerlink.com

*Corresponding author (email: [email protected])

• Research Paper • April 2011 Vol.54 No.4: 640–646

Special Issue: Forward for the Department of Engineering Mechanics, Tsinghua University doi: 10.1007/s11433-011-4285-1

A sequence pulse counting method for shape measurement in dual-beam speckle interferometry

LIU Liang & LI XiDe*

Department of Engineering Mechanics, Key Laboratory of Applied Mechanics, Tsinghua University, Beijing 100084, China

Received December 6, 2010; accepted January 12, 2011; published online February 22, 2011

A temporal approach to fast shape measurement is presented. In principle, the rotational object method is used in combination with the sequence pulse counting method (SPCM) to determine the height of the object through calculating the related phase. Two specimens are tested to demonstrate the validity of the approach. One is an object covered by a Chinese character (tea) with a height variety of 0.3 mm, and the other is an object surface with a relatively large fluctuation of 3.5 mm. The experi-mental results are compared with mechanical measurements. An axis shifting method is also proposed to determine shapes with relatively large fluctuations. Effects of such parameters on the height measurement as incident angle of the dual light beams, tilting angle of the object, and azimuth angle of the measured point are discussed as well.

shape measurement, profilometry, temporal speckle pattern interferometry, phase measurement

PACS: 07.60.Ly, 46.80.+j

1 Introduction

Shape measurement is most important for many applica-tions, including machine vision, biomedicine and dimension measurement in mechanical and electronic manufacture. Various optical techniques have been developed for shape measurement, such as scanning methods [1–4], struc-tured-light methods [5–10], stereo measurement methods [11] and interferometric profilometry [12–18]. The scanning methods possess relatively low measurement speed due to the scanning procedure. Structured-light and stereo meas-urement methods can provide a large measuring depth but their accuracy is determined by the distinguishable gra- tings or markers projected on or attached to the tested ob-jects. The interferometric methods are highly accurate full-field techniques, which measure the shape by variation of the sensitivity matrix that relates the geometric shape of an object to the phase of the interferometric field [19].

Three variables, wavelength, refractive index, and illumina-tion and observation directions, are involved in the matrix and develop different profile measurement methods, such as two- or multiple-wavelength, refractive-index-change, illu-mination-direction-variation, and two-source methods [20]. But in the traditional interferometric profilometry, high ac-curacy, large measuring depth of the shape and large field of view of the imaging system are often conflict.

In recent years, a new branch of speckle interferometry, called temporal speckle pattern interferometry (TSPI), has been developed, which uses different time sequence analy-sis methods to demodulate the object’s deformations or dis-placements from temporal speckle patterns [21–23]. This technique provides us an opportunity to measure the static and dynamic deformation or displacement with both a large range and high accuracy. Joenathan et al. first reported a method for measuring the shape and the step height of ob-jects using Fourier-transform TSPI [24]. The absolute height of the object or its shape was extracted by Fourier analysis of the temporal speckle patterns recorded during rotation of the tested object. The height variations ranging

Liu L, et al. Sci China Phys Mech Astron April (2011) Vol. 54 No. 4 641

from a few hundreds of micrometers to a couple of centi-meters were determined. However, the measurable minimal height is limited by the requirement of a considerable num-ber of speckle patterns in the Fourier-transform TSPI. In the present work, we will apply our previously developed sequence pulse counting method (SPCM) [25]—a special kind of TSPI methods—to directly extract the object shape with the help of the conventional dual-beam interferometric system.

2 Principle

The fundamental principle of SPCM shape measurement is to detect the change of the phase difference between the two beams caused by the position- or shape-change of the tested object. The interference signal ( , , )I x y t of a dual-beam

interferometric setup shown in Figure 1 can be expressed as:

0 m( , , ) ( , ) ( , ) cos ( , , ),I x y t I x y I x y x y t (1)

where 0 ( , )I x y and m ( , )I x y are the average intensity

and the modulation factor of the temporal interference signal, respectively, and ( , , )x y t the time-dependent

phase function related to the temporal position and height of the object. Actually, 0 ( , )I x y and m ( , )I x y are also func-

tions of time. However, their change rates are much lower than that of the phase function. Therefore, the time factor in them is omitted.

In SPCM’s phase demodulation procedure, the temporal phase ( , , )x y t caused by a monotone displacement or

deformation is resolved as [25]:

1 2( , , ) ( , , ) ( , , ) ( , , ) ,x y t m x y t m x y t m x y t (2)

Figure 1 Schematic diagram of the experimental arrangement of the temporal speckle pattern interferometry (TSPI) system. Top view (a) and side view (b).

where ( , , )m x y t is the half-period number of the temporal

speckle patterns (integer), and 1( , , )m x y t and

2 ( , , )m x y t , the fractional half-period numbers (positive

decimals). Detailed expressions of 1( , , )m x y t and

2 ( , , )m x y t are given in ref. [25]. Thus, the in-plane dis-

placement related to the position- or shape-change of the object can be expressed as:

1 2( , , ) ( , , ) ( , , )

( , , ) ,4sin

m x y t m x y t m x y tU x y t

(3)

where is the wavelength of the incident light and , the incident angle between the incident light beam and the z axis (Figure 1).

With the geometric relation in Figure 2 taken into ac-count, the in-plane displacement caused by the position change, such as tilting the object, can be expressed by [24]

2 ( , )sin ( ) / 2 cos ( , ) ( ) / 2

( , , ) ,cos ( , )

h x y t x y tU x y t

x y

(4)

where ( , )h x y is the height of the object from the reference

plane, ( , )x y , the angle between the z axis and the line

connecting point ( , )x y and the origin O of coordinates,

and ( )t , the tilting angle of the object. Eq. (4) indicates that the in-plane displacement of a point ( , )x y caused by

the object tilting depends on both angles and ( )t if

( , )x y is of the same order of magnitude. But if the tilting

angle ( )t is much smaller than ( , ),x y the in-plane

displacement will be proportional to the height of the object from the reference plane and the tilting angle . Thus, eq. (4) can be simplified as:

( , , ) 2 ( , )sin ( ) / 2 .U x y t h x y t (5)

Consequently, the height of the object at the point ( , )x y

is then extracted from the measured in-plane displacement sequence as:

Figure 2 The geometric relation between h(x,y) the height of the object from the reference plane, α(x,y) the angle between the z axis and the line connecting point (x,y) and origin O of the coordinates, and ω(t) tilting angle of the object.

642 Liu L, et al. Sci China Phys Mech Astron April (2011) Vol. 54 No. 4

1 2( , , ) ( , , ) ( , , )( , )

8sin sin ( ) / 2

( , , ).

8sin sin ( ) / 2

m x y t m x y t m x y th x y

t

M x y t

t

(6)

Eq. (6) indicates that the greater the tilting angle is, the higher the resolution of the height measurement is. On the other hand, in SPCM the measurable lowest phase value is smaller than , i.e., the half period of the interferometric pattern ( , , ) 1M x y t [25]. Therefore, the resolution of the

height measurement can be estimated as:

( , ) / sin ( ) / 2 .

8sin sin ( ) / 2h x y C t

t

(7)

Here C can be considered as the measurement constant of the system. Therefore, by changing the tilting angle, the method yields different resolutions in the height measure-ment.

3 Experimental results

For a display of the performance of the method, a dual- beam interferometric system is set up as shown in Figure 1. A He–Ne laser beam was first split into two branches and then expanded and collimated separately to illuminate the tested object from two sides at an angle of 31 to the normal of its surface. The measurement constant of this configura-tion is C=0.1536 m. The object was imaged by a normal CCD camera (25 f/s) connected to an imaging system. A rotational stage was used to tilt the object. Two objects were tested. One was a Chinese character (tea) carved in a metal plate with a size of 40 mm×35 mm and an averaged height fluctuation of 0.30 mm. The other was a cup side surface with a size of 60 mm×5 mm and a maximum height fluctua-tion about 3.5 mm. Fifty images were taken with the total of 0.05 degree of rotation implemented uniformly in 2 seconds. In order to get good demodulation results for the tested point (x,y) by SPCM, 4–6 cycles of the intensity fluctua-tions were recorded during the rotation.

In the first experiment, a Chinese character tea was placed 1.5 mm behind the reference plane in the measure-ment. Figure 3(a) shows its 3-D plot by using the proposed method. A section view of the 3-D plot is also shown in Figure 3(b), which agrees well with the mechanical meas-urement result from using a Vernier caliper with a sharp probe except parts around the sharp corners. In the second experiment, the side surface of a cup was measured to demonstrate the method’s performance for measuring the surface with a relatively large fluctuation. Two steps were needed because the surface fluctuation plus the minimum distance from the reference plane exceeds the measureable range of the test system. This minimum distance was about

Figure 3 (a) Experimental result of a Chinese character (tea) carved in a metal 40 mm×35 mm base with an absolute height about 0.3 mm; (b) a section of the 3-D plot in the horizontal direction compared with a me-chanical measurement.

1.4 mm in our experiment (refer to sect. 4.3 below). In the first step, the surface of the cup was placed 3 mm in front of the reference plane and the temporal speckle patterns were acquired while the object was rotated for the central part of the profile. In the second step, the cup was moved 2 mm forward from the reference plane and then the temporal speckle patterns were recorded for the left and right parts of the profile. Figure 4 shows the combined 3-D plot and con-tours of the side surface of the cup. An appointed horizontal section line of the 3-D plot is also shown in Figure 4(c) and compared with the mechanical measurement. The maximum difference is about 0.15 mm, 4% of the total height 3.5 mm.

4 Discussion

4.1 Resolution and range of height measurement with SPCM

Rotation of the tested object produces monotonic and tem-poral in-plane displacements, making it possible to extract phases of the interference fields from only a few cycles of the interference intensity by using SPCM. This improves the resolution of the height measurement on the one hand, and reduces the processing time on the other hand. In our measurement the total time for acquiring the temporal speckle patterns was approximately 2 seconds, and the pa-rameter sin / 2 0.1536C h m, smaller than the

value of 2.2 m reported in ref. [24]. By selecting a suitable tilting angle , a different resolution of the height meas-urement can be obtained. Considering the correlation of the temporal speckle pattern sequence, the upper limit of height measurement is determined by the maximum in-plane dis-


Figure 4 (a) Experimental results of a segment of a cup’s side surface with a size about 60 mm×3 mm and an absolute height about 3.5 mm; (b) corresponding contours of the shape; (c) the section view of the black line in the 3-D plot along the horizontal direction, compared with a mechanical measurement.

placement which cannot exceed the transverse speckle size. In our experiments, this value is about 40 m. Thus, the maximum number of intensity cycles in one speckle pattern sequence is 2 sin / 2 40 0.52 / 0.63 65,U which yields 46 mm upper limit of the height measurement if the factor of the transverse speckle size is considered in eq. (6).

4.2 Measurement of a relatively large surface fluctua-tion

High accuracy is a challenge to height measurement for objects with a large scale and a large surface fluctuation. With incoherent methods high spatial resolution conflicts with a large depth of field and a large range of the field of view of the imaging system, while with interferometric methods the field of view illuminated by the laser source is limited. For TSPI technique, limitations include the size of the illumination field of the object and the measurement range of the methods. Improvements can be made by in-creasing the acquiring speed of the CCD camera during the tilting period or enhancing the power of the laser source,

both demanding more expensive hardware. In this work, an axis shifting method was proposed to realize the large scale and large fluctuation surface measurement, in which the tested surface was divided into several regions whose sizes and heights were all in the range of the field of view and SPCM height measurement range. Each region was respec-tively measured at a properly shifted rotation axis, and the results were combined together with the axis shifts taken into account to get the entire surface shape of the object. Figure 5(a) shows separate measurements of the cup whose shapes are shown in Figure 4. Because its surface fluctua-tion plus the minimum distance (1.4 mm) away from the reference plane needed by SPCM exceeds the measureable range of the test system, only the central part of the surface was correctly reconstructed (Figure 5(a)) when the surface was placed 3 mm in front of the reference plane. Then the left and right parts of the surface were not far away enough from or even below the reference plane, leading to incorrect

Figure 5 (a) 3-D plot of the whole surface recovered before axis shifting; (b) 3-D plot of the left margin with axis shifting; (c) 3-D plot of the right margin with axis shifting.


or reversed profiles [24]. To obtain the correct surface shapes of those parts, we moved the cup forward by 2 mm with the profiles of the left and right parts retrieved as shown in Figure 5(b) and (c). The integrated surface shape was shown in Figure 4(a). The rotation stage was placed on a 2-D precision translation stage whose displacement reso-lution was 0.5 m, far higher than the height resolution of the measurement.

4.3 Effects of some parameters on SPCM shape meas-urement

From eq. (4) to eq. (6) we found that the measured height range and resolution are determined by three parameters, angle of incident lights, tilting angle and cycle number M of the temporal speckle patterns. To investigate the effects of these parameters on height measurement for small tilting angles, we calculated changes of the measurable height with these parameters. Figure 6 shows the height vs. parameter , where 2 65M (65 intensity cycles), =

0.05°–0.5°, and 0.6328 m. The curves indicate the smaller the incident angle, the larger the range of measurable heights with the effect especially obvious for a smaller than 20 degrees. These results also show that if we want to measure a large height but with a relatively low resolution, a small incident angle may be the choice. The accuracy of the measured height, however, is sensitive to the error of the incident angle in this situation. For the prac- tical measurement system, a range of 30 to 45 degrees is suitable for the incident angle of the light beams. In this angle range, the measurable height may reach 8 mm or more if the tilting angle 0.2°. Figure 7 shows the relation of the measurable range of the height and tilting angle. Obvious changes of the height measurement range are ob- served for 0.2°, and the range may be larger than 8 mm when 45°. Thus from the results of Figures 6 and 7 we

Figure 6 The relation between measurable range of the height and pa-rameter , while M=2×65 (65 intensity cycles), =0.05°0.5°, and =0.6328 m.

Figure 7 The relation between measurable range of the height and tilting angle , while M=2×65 (65 intensity cycles), =15°75°, and =0.6328 m.

conclude that the upper limit of the height measurement can reach 8 mm when 45°, 0.2°, with the in-plane displacement taking the size of the transverse speckle. With the resolution as presented in sect. 4.1 taken into account, the lower limit of the height measurement will be minh

352.2 m (for 31°, 0.05°). Another factor that has to be discussed is the acquiring

frequency of the system in measurements. The frequency of the sequence speckle intensity fluctuation can be expressed by

2 sin ( ) / ,f h t (8)

where ( )t is the angular tilting velocity. If we substitute

the values max 46h mm, 31°, 0.6328 m, and

( )t 40.05 ( /180) / 2 4.4 10 rad/s into eq. (8), we

have 33f Hz. Because the temporal frequency shouldn’t

be greater than Nyquist frequency (half of the frame rate of the camera), the acquiring frequency of the recording sys-tem should be 2 66f Hz at least. In our measurement

system, the video acquiring frequency is about 0 25f Hz.

Therefore the upper limit of the height measurement must be smaller than max 46h mm. Substituting 0 / 2f f

12.5 Hz into eq. (8), we obtain max 17.5h mm. In practi-

cal measurements, the measurable height was even smaller than 17.5 mm owing to A/D transforming and image storing times necessary for acquiring the sequence speckle patterns. It depends on the hardware components of the image system. In our analog image system, the sample frequency in ac-quiring and storing mode (storing the images into the inter-nal memory) is about the eighth of the video acquiring fre-quency. This leads to the final upper limit 4.4 mm of the height measurement, approximately equivalent to 4–6 cy-cles of image acquirement in our experiments. In fact, if we substitute 2 6M (6 intensity cycles) into eq. (6), and


let 31°, 0.6328 m, and 0.05°, the value of

max 4.2h mm is obtained. Although the maximum height

of the measured cup surface is only 3.5 mm, the object has to be located 1.4 mm away from the reference plane, for at least two intensity cycles are needed in SPCM. Thus the total height is 4.9 mm, which is already greater than maxh

of our system. This difficulty has been overcome by use of the axis shifting method as previously described.

Finally, we discuss the effect of angle ( , )x y , which de-

termines the position of the measured point on the surface of the object. From eq. (4) and phase-displacement relation of SPCM, we have

cos ( ) / 2 ( , , )( , )

cos( ) 8sin sin ( ) / 2

( , , ),

8sin sin ( ) / 2

t M x y th x y

t

M x y tk

t

(9)

where cos ( ) / 2 / cos( )k t is a ratio factor de-

pending on angle ( , )x y . Figure 8 shows k changing with

parameters and . We find that 1k always exists when 45° and 0.2°, and that a corresponding error no more than 0.4% will be caused by use of the approximate eq. (6) without the effect of α taken into account.

5 Conclusions

The shapes of fluctuated surfaces have been determined by dual-beam interferometry with rotation of the tested objects. The temporal speckle patterns caused by the rotation were recorded through a standard CCD camera, and the sinusoi-dal signal for each pixel of the temporal speckle patterns was demodulated with the sequence pulse counting method (SPCM), which can quickly analyze the monotonic intensity fluctuation of temporal speckle patterns according to the

Figure 8 A plot of k changing with and , where k=cos(+(t)/2)/ cos().

acquired cycles from a half to the upper limit. This brings advantages of a wide range and high accuracy in shape measurement. Also, an axis shifting strategy is proposed for measuring shapes with a relatively large height fluctuation. Effects of such parameters on the height measurement, as incident angle of the light beams, tilting angle, and azimuth angle of the measured point, were discussed. Two sample objects were tested to validate the principle and perform-ance of the method with large-range height changes of the objects’ surfaces conveniently measured with a high preci-sion.

This work was supported by the National Natural Science Foundation of China (Grants Nos. 10972113, 10732080), the National Basic Research Program of China (Grant Nos. 2007CB936803, 2010CB631005) and SRFDP (Grant No. 20070003053).

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a sequence pulse counting method for shape measurement in dual-beam speckle interferometry

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