Surface Measurement

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Optik

Optics

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Surface profile measurement by use of fringe projection method based on Talbot effect C. Quan, M. Thakur Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 4 July 2005; accepted 11 October 2005

Abstract This paper describes a grating projection system (GPS) based on Talbot effect for surface profile measurement. The self-image of the grating is projected on the object surface and the deformed grating image is captured by a charged couple device (CCD) camera for subsequent processing. The phase shifting is achieved by the use of a linear translation stage incorporated to the grating. In this application, two specimens are tested to demonstrate the validity of the method, one is a spherical cap with a height of 4 mm, and the other is a small coin with an uneven surface of less than 0.2 mm. The experimental results are compared with that of mechanical stylus method and grating projection system based on Lau phase interferometry. r 2005 Elsevier GmbH. All rights reserved. Keywords: Surface profile measurement; Fringe projection; Talbot-effect; Grating; Phase shifting

1. Introduction A number of optical techniques are available to obtain topographic information. Among them moire´ technique, proposed by Meadows et al. [1] and Takasaki [2] in early 1970s, has been widely used for threedimensional (3-D) shape measurement and surface contour analysis. According to the optical arrangement of the system it is classified into the following methods [1–9]: projection moire´, shadow moire´ and fringe projection. In projection moire´, the fringes, which contain the information of the surface profile, are generated by projecting a grating onto the object and viewing it through the second grating in front of the viewer. Comparing with the projection moire´ the shadow moire´ is relatively cheap and simple. The Corresponding author. Tel.: +65 6874 8089; fax: +65 6779 1459.

E-mail address: [email protected] (C. Quan). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.10.005

shadow moire´ technique uses the single grating placed close to the object and the oblique light source illumination which casts a shadow of the grating on the object surface. When the shadow is viewed from a different direction through the original grating, the grating and its distorted shadow interfere, generating fringes. Due to progress in computer capacities and image processing techniques in 1990s, different types of phase shifting method [9–18] are applied in moire´ and projection fringe (or projection grating). Hovanesian and Hung [3] proposed a moire´ setup where a grating is projected and tilted with respect to the object under test. A similar approach with coherent illumination is described by Chamg and Lin [4] with a simple projection system using a shear plate and a five phase-shifting technique for contour deformation measurement. Rodriguez-Vera et al. [5] used the Talbot interferometry for the surface contouring of the diffused object. In this study, the grating projection system

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(GPS) based on self-imaging of a grating is used for surface profile measurement. The method is simple and produces a set of images along the optical axis. The positions of the images can be adjusted by moving one optical element and no alignment is needed. At a particular arrangement along the optical axis a set of measurement planes is generated and measurements can be made at a number of points across the duct. Self-imaging properties were first observed in 1836 when Talbot noted the reproduction of a wavefront at certain distances behind a periodic object. It has been used in many applications [5,19–23] such as Fourier spectrometry, interferometry, image processing, distance and velocity measurement, etc. The Talbot effect is the reproduction of exact images of a grating illuminated by a monochromatic plane wave at a distance behind a grating [23]: 2np2 , (1) l where n is an integer, p the pitch of the grating and l the wavelength of the light source. When the grating is illuminated with an incoherent light source the resultant irradiance displays Lau bands [19,24]. In this paper a GPS based on the Talbot effect is described. The object having a diffused surface is placed at one of the self-image plane of the grating. A linear translation stage is used to apply the phase shift of the grating. The distorted fringe patterns caused by the surface profile are captured by a charged couple device (CCD) camera and subsequently processed by using phase shifting algorithm to obtain the surface contour. The experimental results are compared with that of measured by mechanical stylus method and GPS based on Lau effect.

Light source Lc Pinhole Projection grating z

p View point

d

Z


1
2 3-D Object x

h Reference plane

P

C B

A

O

y

Fig. 1. Schematic of the optical setup: Lc , collimating lens; p, pitch of grating; Z, Talbot plane; d, distance between grating and CCD camera; h, height of the surface.

Z¼

imaging axis with the reference plane. A CCD camera is used to measure the intensity at point C on the reference plane and point P on the object. The intensity observed at P modified by the object reflectivity R is the same as that observed at A on the reference plane: I P ¼ R½a þ b cosð2pOA=pÞ
.

The phase difference f for points C and P observed by the CCD camera is related to the geometric distance ACð¼ p=2pÞf and AC is related to the object height h ( ¼ BP) by h ¼ AC=ðtan y1 þ tan y2 Þ,

Fig. 1 shows the optical geometry of fringe projection and imaging system used in the experiment. It consists of a grating and a collimating lens. A collimated beam of laser light illuminates the grating. An object is placed at the self-image plane and the grating transmittance TðxÞ can be expressed by TðxÞ ¼ am ½1 þ cosð2px=pÞ
,

(2)

where am and ‘p’ are the amplitude and spatial pitch of the grating, respectively. When the grating is projected onto the object its intensity distribution at point C is given by [25] I C ¼ a þ b cosð2pOC=pÞ,

(3)

where a is the background intensity, b the intensity modulation, OC is distance from the point O to point C on the reference plane, point O is the intersection of the

(5)

where y1 and y2 represent the directions of projection and viewing, respectively. When the direction of viewing is perpendicular to the reference plane, Eq. (5) becomes h ¼ AC=ðtan y1 Þ ¼ ðp=2pÞf= tan y1 ¼ kf,

2. Theoretical analysis

(4)

(6)

where k ¼ p=ð2p tan y1 Þ is an optical coefficient related to the configuration of the system and can be obtained by calibration. One method of calibration is to shift the test object through known distances and corresponding phase is measured using phase shifting method. In this study, however the calibration is achieved by shifting a reference plane. The distance between the shifted planes is set to be 100 mm that can be used to calculate the constant k. The object is placed at one of the self-image plane of the grating and the profile of object surface is encoded on grating lines. A CCD camera is used to capture the deformed grating. The mathematical representation of the intensity distribution captured by a CCD camera is governed by the following equation: Iðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos½fðx; yÞ þ D
,

(7)

where aðx; yÞ is the average intensity, bðx; yÞ is the intensity modulation, and fðx; yÞ is the phase to be

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determined. If the pitch of the grating is p and one of gratings is moved by a distance jt2  t1 j ¼ p=n (n ¼ 1, 2y), this would represent a phase shift of D ¼ 2p=n for each fringe. To retrieve the phase fðx; yÞ, additional phase shift D in the steps of p=2 are required. In this study, three phase shifted sinusoidal fringe patterns are projected onto an object surface with a phase shift of 0, p=2; p: The following equations represent three corresponding intensity distributions [26]: I 1 ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos fðx; yÞ,

(8)

I 2 ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos f½ðx; yÞ þ p=2
,

(9)

I 3 ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos f½ðx; yÞ þ p
.

(10)

By solving the above equations simultaneously, the phase fðx; yÞ at each point (x, y) on the object can be obtained by   I 1 ðx; yÞ  2I 2 ðx; yÞ þ I 3 ðx; yÞ fðx; yÞ ¼ arctan . (11) I 1 ðx; yÞ  I 3 ðx; yÞ The phase fðx; yÞ obtained from Eq. (11) resulted with the principal value of phase fðx; yÞ in region between p and þp regardless of the actual value of the phase. Phase unwrapping is carried out in order to remove phase ambiguities by adding or subtracting 2p from individual pixel until the phase difference between the adjacent pixel is less then p. Once the value of fðx; yÞ is known the surface profile can be calculated using Eq. (6).

3. Experimental work Fig. 2 shows the experimental setup. A 50 mW He–Ne laser beam (of wavelength 632 nm) is expanded using a 40  microscope objective, 5 mm pinhole and collimating lens ðLc Þ of focal length 100 mm. The collimated laser beam falls on the grating of pitch 0.5 mm and self-

Fig. 2. Experimental arrangement: f, focal length of the collimating lens.

3

image is formed at one of the Talbot plane. The illumination angle used for projecting the grating on the object is 25 . An object is placed at one of the self-image planes of the projection grating. The image of the grating falls on the object surface; the information about the shape of the object is encoded on the grating lines. This image is captured by a CCD camera, which is placed at a normal view. The distorted fringe images on CCD camera are captured at different distances. It is noted that when the distance between the grating [20] and the object is approximately equal to Z ¼ np2 =l (where n is an integer) the phase map of the object which is processed by using the phase shifting algorithm is accurate. The self-image plane is optimized by repeating the experiment a number of times with various pitches of the grating depending on the object size. The measurement of the object size depends upon the pitch of the gratings. The self-image plane is optimized in such a way that diffraction does not affect the result. It should be noted that if the object is placed at higher selfimage plane diffraction will affect the result. Hence, the object is kept at the first image plane to avoid diffraction. For specimens under study, three images are recorded successively. Each subsequent image is recorded with a phase difference of p=2. The phase difference is introduced by a prescribed in-plane displacement of grating which is mounted on a linear translation stage. A linear displacement of 125 mm on the translation stage would translate into a phase step of p=2 for a spherical cap. In the case of coin specimen we use a grating of pitch 0.25 mm. A linear displacement of 63.5 mm on the translation stage would translate into a phase step of p=2.

4. Results and discussion Fig. 3(a) shows the first test specimen without projection of a grating on it. The radius rs , height h and the base radius rb of the spherical cap are, respectively, 14.5, 4 and 10 mm, as shown in Fig. 3(b). The pitch of the grating is 0.5 mm. Fig. 4(a) shows the fringe patterns recorded by the CCD camera for a phase shift of p/2. Figs. 4(b) and (c) show, respectively, a wrapped and the corresponding unwrapped phase maps. The phase value can be converted into a 3-D surface profile as shown in Fig. 4(d). To verify the accuracy of surface profile measurement by the proposed method, a comparison is made with mechanical stylus and GPS based on the Lau effect. In the case of GPS based on Lau effect, we use two gratings which are illuminated by an incoherent light source. A self-image of grating is produced at one of the Lau planes. The advantage of using incoherent light is that it

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C. Quan, M. Thakur / Optik ] (]]]]) ]]]–]]] 5 Mechanical Stylus Data

Data with coherent light

Data with Incohrent light

4

(mm)

3

A

2

A 1

0 0 -1

5

0

(a)

5

10

15

20

25

(mm)

Fig. 5. A comparison of surface profile along section A-A between present and mechanical stylus method.

10 (mm)

rs

h

rb

(b)

Fig. 3. (a) Specimen 1: spherical cap; (b) dimensions of the spherical cap: rs , radius; h, height; rb , base radius of the spherical cap.

Fig. 6. Specimen 2: a 50-cent coin and area of interest.

Fig. 4. (a) Fringe pattern on spherical cap; (b) wrapped phase map; (c) unwrapped phase map; (d) 3-D plot of the spherical cap.

reduces coherent noise in the interferogram and leads to high signal-to-noise ratio, where as in the case of coherent light source filtering is required to remove the noise. Fig. 5 shows the comparison of the profile on

cross section A-A (indicated in Fig. 3(a)). The average discrepancy is 1.5% where as the maximum discrepancy is 4%. The discrepancy is partially due to the nonlinearity of the optical system such as CCD camera and collimating lens. It is shown that the results obtained by using incoherent light source are better than that of using the coherent light source. The discrepancy is large at the corner of the spherical cap; this is due to the shadow effect. The proposed method is also applied on a coin of 24.5 mm diameter and having a diffuse surface (shown in Fig. 6). A small area of interest containing 256  256 pixels (also indicated in the Fig. 6) is cropped. A grating with a density of 100 lines/mm is used. Fig. 7(a) shows the fringe patterns recorded by the CCD camera for a phase shift of p/2. Filtering is applied to remove the noise in this case. Figs. 7(b) and (c) show, respectively, a wrapped and the corresponding unwrapped phase maps. Subsequently the 3-D profile of the interested area is obtained as shown in Fig. 7(d). Fig. 8 shows a

ARTICLE IN PRESS C. Quan, M. Thakur / Optik ] (]]]]) ]]]–]]]

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Fig. 7. (a) Fringe pattern on an area of interest; (b) wrapped phase map; (c) unwrapped phase map; (d) 3-D plot of the surface profile.

Phase shifting method

180

Mechanical stylus method

160

hight (micron)

140 120 100 80 60 40 20 0 -20

0

2

4

6

8

(mm)

Fig. 8. A comparison of surface profile of 50-cent coin on cross-section B-B between present and mechanical stylus method.

comparison of the profile along section B-B (indicated in Fig. 6) with the mechanical stylus method. The average discrepancy is 3% where as the maximum discrepancy is 4% and as mentioned above the discrepancy is due to the non-linearity of the optical system such as CCD camera and collimating lens. As shown in the results we used gratings of different pitch for the different size of objects. If the height of object is in a range of few hundred microns, we need to use a fine grating to achieve an accuracy of 10 mm. The experimental setup is optimized according to the measurement range of the specimens. The limitation of this setup is that as the self-image plane appears at

specific distance the setup can be applied only for a small object. It is noteworthy that the grating displacement required to producing the phase steps are of the same order as the grating pitch. Since the proposed method employs a course grating, it is relatively easy to achieve sufficiently accurate grating translations. The results obtained by this method is also comparable with the one obtained by using the LCD [27] projection method. The LCD projection equipment is very costly whereas the proposed setup is very simple and cost effective. The sensitivity of the system depends upon the illumination angle. The system is less sensitive for smaller illumination angle. As the illumination angle is increased, the system sensitivity is also increased. However, the larger illumination angle can cause a shadow effect. Therefore, the illumination angle is optimized to 25 in order to achieve higher sensitivity.

5. Concluding remarks A phase shifted fringe projection method based on Talbot effect has been presented for measurement of surface profile. The phase variation is achieved by an inplane translation of a projection grating. In the

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proposed technique the optical arrangement is simple and utilizes few optical components. The experimental setup is cost effective and easy to handle with a simple instrument. The system is efficient for the small-size components in industrial applications. The measurement accuracy is reasonably good and the experimental results compared well with that of the mechanical stylus method. With this method, the range of measurement has been extended to sub-millimetre and an accuracy of 10 mm is achieved. A high-resolution contour map of a small coin has been obtained. With further development, the system could be used for industrial quality inspection.

Acknowledgments The authors would like to acknowledge the financial support provided by the National University of Singapore under research project R-265-000-071-112.

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