- Email: [email protected]
A flexible calculation on improved Fourier transform profilometry
Optik 121 (2010) 2023–2027
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
A flexible calculation on improved Fourier transform profilometry Xianfu Mao, Xianyu Su , Wenjing Chen, Hailong Jin Department of Opto-electronics, Sichuan University, Chengdu, Sichuan 610064, China
a r t i c l e in fo
abstract
Article history: Received 23 February 2009 Accepted 4 July 2009
Here a flexible optical geometry of Fourier Transform Profilometry (FTP) is discussed, and a strict theoretical analysis about an improved phase-height mapping formula based on a new description of a reference fringe and a deformed fringe proposed based on a general case, in which the exit pupil of the projecting lens and the entrance pupil of the imaging lens are not horizontal and do not have the equal distance to the reference plane; moreover, the optical axes of the projecting device and of the imaging device are not coplanar. Compared with the traditional Fourier Transform Profilometry, it is easier for us to obtain the full-field fringe through adjusting the location of either the projector or the imaging device. Compared with the Improved Fourier Transform Profilometry (IFTP), only three length system parameters need to be measured, so the operation is simple and the result is more reliable. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Fourier transform profilometry (FTP) Improved Fourier Transform Profilometry Phase-height mapping
1. Introduction
2. Method
Fourier Transform Profilometry (FTP) [1], an automatic noncontract optical measurement of 3-D shape based on a fringe profilometry system, is very important in machine vision, solid modeling, industrial auto-measuring, etc. In FTP, a grating pattern is projected onto an object and the deformed fringe is Fourier transformed, filtered, and inversely Fourier transformed, and then the height distribution of the surface can be reconstructed. In recent years, increasing number of researchers [2–12] pay more attentions to Fourier transform analysis of projection fringe or interferogram because of its characteristics of needing only one fringe, full-field analysis, and high precision. Now, when the exit pupil of the projecting lens and the entrance pupil of the imaging lens are not horizontal and do not have equal distance to the reference plane, and moreover the optical axis of the projecting device and that of the imaging device are not coplanar, the height distribution of the measured object can be obtained by Improved Fourier Transform Profilometry [12]. However, it is difficult to accurately obtain the four system parameters, specially the angle system parameter. In this paper, a new phase-height mapping formula based on a new description of a reference fringe and a deformed fringe in Improved Fourier Transform Profilometry [12] is proposed. The strict theoretical analysis about the fringe description and the relationship between phase and height are given.
2.1. Fringe analysis and phase calculation The optical geometry of Improved Fourier Transform Profilometry is shown in Fig. 1. I1 is the exit pupil of the projector and I20 is the entrance pupil of the CCD camera. The optical axis I1 O crosses the reference plane R at point O, and the optical axis I20 O1 is vertical to the reference plane and crosses it at another point O1. Note that the connecting line I1 I20 is not parallel to the reference plane as well as I1 O and I20 O1 are not coplanar. D(x,y) and C1 are points on the 3-D object and on the reference plane, respectively, and they are imaged at the same point on the CCD array. I1 A is a ray through point D to the reference plane. I1 K is vertical to reference plane R and crosses it on point K. Given the system parameters: I20 O 1 ¼ L; OK ¼ r; I1 K ¼ m:The fringe pattern on the reference plane may be expressed as [8] I1 ðx; yÞ ¼ I0 ð1þ cos 2pxf ðxÞÞ
ð1Þ
where x is x-axial coordinate of the measured object, I0 is the illumination light intensity, and f(x) is the spatial frequency of the reference fringe along the x-axis, which can be expressed by [8,13] 2x sin y f ðxÞ ¼ f cos y 1 I1 O
ð2Þ
where f is the carry frequency of the grating.
Corresponding author.
E-mail address: [email protected] (X. Su). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.07.005
Compared with the case where the imaging axis crosses the projecting axis at the same point on the reference plane, if the influence on CCD imaging quality is neglected, adjustment of the
2024
X. Mao et al. / Optik 121 (2010) 2023–2027
According to the similar characteristic among some triangles, the following expression can be obtained: BD xðm LÞ CA ¼ r ð7Þ m BD L BD m BD xðm LÞ ‘cðx; yÞ ¼ 2pf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr L BD m2 þ r 2 m BD
! ð8Þ
Substituting Eq. (8) into Eq. (6), 2pfm I2 ðx; yÞ ¼ I0 1 þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þr 2
Fig. 1. The measuring sketch map.
x 1
2xr m2 þ r 2
!!! BD xðm LÞ r m BD L BD
ð9Þ After simplification, the reference fringe and the deformed fringe can be rewritten as the simple forms, respectively, as follows: I1 ðx; yÞ ¼ I0 ð1 þ cosð2pf0 x þ f0 ðx; yÞÞÞ
I2 ðx; yÞ ¼ I0 ð1 þ cosð2pf0 x þ fðx; yÞÞÞ ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where f0 ¼ f cos y ¼ f ðm= m2 þ r 2 Þ, f0 ðx; yÞ and fðx; yÞ are the phase distributions included in the reference fringe and the deformed fringe. I1 ðx; yÞ and I2 ðx; yÞ are Fourier transformed and processed in their spatial frequency domain as well as in their space-signal domain, then f0 ðx; yÞ and fðx; yÞ can be calculated, and the core phase Dfðx; yÞ directly caused by the height variation can be obtained as
Fig. 2. The geometrical sketch map.
CCD position just causes the movement of the image’s location on the CCD array whereas the relationship among the adjusted pixels is unchanged [12]. We add several dashed lines in the figure as assistant lines for our analysis as shown in Fig. 2. Draw a line I2 O through point O, which parallels the optical axis of CCD and the point I20 is rotated to the point I2 around the point D, as well as C1 is rotated to point C, which must be on the lineOA. BD is vertical to plane R, which denotes the height of point D on the measured object. I1 F parallels to the reference plane R and crosses I2 O and the prolongation of BD at point G and point F, respectively. I2 D crosses I1 F at point P. In DOI1 G, r sin y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þr 2
h¼
m cos y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ r 2
ðcðx; yÞðL þmÞ þ rLs sðm LÞxÞ 7
ð3Þ
Dfðx; yÞ ¼ fðx; yÞ f0 ðx; yÞ
In the next section, the conversion formula between the phase and the height will be deduced. Given m BD ¼ h; s ¼ 2pf0 ¼ 2pf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ r 2 deformed Eq. (8), we can obtain ðcðx; yÞ þsrÞh2 ðcðx; yÞðL þ mÞ þ sðLr xðm LÞÞÞhþ mLcðx; yÞ ¼ 0
2ðsr þ cðx; yÞÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ r 2
then
Substituting Eq. (4) into Eq. (2), 2pfxm 2xr I1 ðx; yÞ ¼ I0 1þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2 2 m þr m2 þ r 2
ð4Þ
ð5Þ
when the measured object is put, the deformed fringe pattern can be expressed as 2pfxm 2xr c ðx; yÞ ð6Þ I2 ðx; yÞ ¼ I0 1þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 m þ r2 m2 þ r 2 where [1] cðx; yÞ ¼ 2pf CA cos y
ð13Þ
then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcðx; yÞðL þmÞ þrLs sðm LÞxÞ2 4ðsr þ cðx; yÞÞcðx; yÞmL
2x sin y fm 2xr ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 f ðxÞ ¼ f cos y 1 m þ r2 I1 O m2 þ r 2
ð12Þ
2.2. The phase-height mapping
and I1 O ¼
ð10Þ
Fig. 3. The simulated object.
ð14Þ
X. Mao et al. / Optik 121 (2010) 2023–2027
2025
where cðx; yÞ ¼ Dfðx; yÞ, the system parameters (L, m, r) can be measured directly. Selecting the right radix, Eq. (14) will be the phase-height mapping formula. Given m =L, Eq. (14) can be simplified as h¼
2Lcðx; yÞ þrLs 7 rLs 2sr þ cðx; yÞ
ð15Þ
By selecting the right radix, we can obtain h¼
Lcðx; yÞ 2pf0 r þ cðx; yÞ
ð16Þ
It is the same as the traditional phase-height mapping formula [1]. Compared with the traditional Fourier Transform Profilometry [1], with the new method it is easier for us to obtain the full-field fringe through adjusting the location of either the projector or the imaging device. Compared with the Improved Fourier Transform Profilometry [12], only three length system parameters (L, m, r) need to be measured, especially, the difficult angle measurement [12] would not be needed, so the operation is simple and the result is more accurate.
Fig. 6. The height error distribution when m= L= 1700 mm.
Fig. 7. The height error distribution when m =2000 mm.
Fig. 4. The deformed fringe distribution when m = L= 1700 mm.
Fig. 8. The height error distribution when m =2300 mm.
3. Computer simulations
Fig. 5. The height error distribution when m =1400 mm.
In this section, some computer simulations are used to verify our discussion. The measured object is half an elliptical sphere, as
2026
X. Mao et al. / Optik 121 (2010) 2023–2027
shown in Fig. 3, in which the maximal height is about h= 33.25 mm. The simulated deformed grating with period T= 16, and the size of the simulated image is 512 512 pixel. System parameters are L= 1700 mm, r =400 mm. In our experiments, when m is selected from 1400 to 3500 mm, respectively, some computer simulations are carried out. Fig. 4 shows the deformed fringe corresponding to the case where m= L=1700 mm. From Figs. 5–10, each of them denotes the reconstructed height error distribution corresponding to m= 1400, 1700, 2000, 2300, 2600, and 2900 mm, respectively. From m=1400 to 3500 mm, the mean square deviation and their maximal height distributions of the reconstructed object are shown in Table 1.
Fig. 11. The grating fringe of reference plane.
Fig. 12. The grating fringe modulated by 3-D object.
Fig. 9. The height error distribution when m= 2600 mm.
Fig. 13. The reconstructed height distribution.
4. Experiment
Fig. 10. The height error distribution when m= 2900 mm.
Table 1 Corresponding to different m, the mean square deviations (RMS) and maximal height distribution of the reconstructed object. m (mm) 1400 1700 2000 2300 2600 2900 3200 3500 RMS (mm) 0.5499 0.3855 0.2898 0.2304 0.1851 0.1737 0.1312 0.1191 hmax (mm) 34.74 34.36 34.07 33.88 33.75 33.66 33.59 33.51 The maximal height of the measured object is 33.25 mm.
A primary experiment is used to verify our method, in which a toy model is measured. Projecting fringe pattern with sinusoidal intensities through a projector (PLUS U3-880) on the object, a CCD camera (MTV-1881EX) captured the deformed pattern. The system parameters are r = 38.00 cm, m= 93.00 cm, L= 122.00 cm. The grating period was T= 16, the maximal height of the object is h=2.90 cm, and image size is 256 256 pixels. Figs. 11 and 12 are the reference fringe and the deformed fringe, respectively. Fig. 13 is the reconstruction-height distribution by the new method where the maximal height is 2.95 cm. Compared with Improved Fourier Transform Profilometry method [12], the new phaseheight algorithm gives more reliable height reconstruction.
5. Conclusions A flexible description about a reference fringe and a deformed fringe and a new phase-height mapping formula is deduced.
X. Mao et al. / Optik 121 (2010) 2023–2027
Compared with the traditional Fourier Transform Profilometry, with the new method it is easier for us to obtain the full-field fringe through moving either the projector or the imaging system. Compared with the Improved Fourier Transform Profilometry [12], only three length system parameters need to be measured, so its operation is simple and its result is more reliable.
Acknowledgments The authors wish to acknowledge the support by the National Natural Science Foundation of China (10876021, 60677028). References [1] M. Takeda, K. Mutoh, Fourier transform profilometry for the automatic measurement 3-D object shapes, Appl. Opt. 22 (24) (1983) 3977–3982. [2] Fiona Berryman, Paul Pynsent, James Cubillo, A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise, Opt. Lasers Eng. 39 (1) (2003) 35–50. [3] J. Vanherzeele, P. Guillaume, S. Vanlanduit, Fourier fringe processing using a regressive Fourier-transform technique, Opt. Lasers Eng. 43 (6) (2005) 645–658.
2027
[4] Wenjing Chen, Xianyu Su, Likun Su, et al., Improvement for measuring accuracy of fourier transform profilometry with gray image, Opto-Electron. Eng. 27 (148) (2000) 55–59 (in Chinese). [5] Wenjing Chen, Xianyu Su, Songxin Tan, Basis requirement for measurement system in FTP, J. Optoelectron. Laser 10 (6) (1999) 535–539. [6] Jingang Zhong, Jiawen Weng, Phase retrieval of optical fringe patterns from the ridge of a wavelet transform, Opt. Lett. 30 (2005) 2560–2562. [7] Xianyu Su, Wenjing Chen, Fourier transform profilometry: a review, Opt. Lasers Eng. 35 (5) (2001) 263–284. [8] Giuseppe Schirripa Spagnolo, Giorgio Guattari, Carmine Sapia, et al., Contouring of artwork surface by fringe projection and FFT analysis, Opt. Lasers Eng. 33 (2000) 141–156. [9] Wenjing Chen, Xian-yu Su, A new method for improving measuring accuracy of fourier transform profilometry, Opto-Electron. Eng. 29 (1) (2002) 19–22. [10] Wansong Li, Xianyu Su, Likun Su, Liqun Xiang, Simultaneous calibration algorithm of phase-shifting based on FFT, Acta Opt. Sini. 19 (10) (1999) 1390–1395. [11] Wenjing Chen, Hu Yang, Xianyu Su, Songxin Tan, Error caused by sampling in Fourier transform profilometry, Opt. Eng. 38 (6) (1999) 1029–1034. [12] Xianfu Mao, Wenjing Chen, Xianyu Su, Improved fourier transform profilometry, Appl. Opt. 46 (5) (2007) 664–668. ˚ [13] Kjell J Gasvik, Optical techniques: Toyohiko Yatagai. Intensity based analysis methods, in: D.W. Robinson, G.T. Reid (Eds.), Interferograms Analysis, IOP, Bristol, 1993, pp. 49–93.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
A flexible calculation on improved Fourier transform profilometry Xianfu Mao, Xianyu Su , Wenjing Chen, Hailong Jin Department of Opto-electronics, Sichuan University, Chengdu, Sichuan 610064, China
a r t i c l e in fo
abstract
Article history: Received 23 February 2009 Accepted 4 July 2009
Here a flexible optical geometry of Fourier Transform Profilometry (FTP) is discussed, and a strict theoretical analysis about an improved phase-height mapping formula based on a new description of a reference fringe and a deformed fringe proposed based on a general case, in which the exit pupil of the projecting lens and the entrance pupil of the imaging lens are not horizontal and do not have the equal distance to the reference plane; moreover, the optical axes of the projecting device and of the imaging device are not coplanar. Compared with the traditional Fourier Transform Profilometry, it is easier for us to obtain the full-field fringe through adjusting the location of either the projector or the imaging device. Compared with the Improved Fourier Transform Profilometry (IFTP), only three length system parameters need to be measured, so the operation is simple and the result is more reliable. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Fourier transform profilometry (FTP) Improved Fourier Transform Profilometry Phase-height mapping
1. Introduction
2. Method
Fourier Transform Profilometry (FTP) [1], an automatic noncontract optical measurement of 3-D shape based on a fringe profilometry system, is very important in machine vision, solid modeling, industrial auto-measuring, etc. In FTP, a grating pattern is projected onto an object and the deformed fringe is Fourier transformed, filtered, and inversely Fourier transformed, and then the height distribution of the surface can be reconstructed. In recent years, increasing number of researchers [2–12] pay more attentions to Fourier transform analysis of projection fringe or interferogram because of its characteristics of needing only one fringe, full-field analysis, and high precision. Now, when the exit pupil of the projecting lens and the entrance pupil of the imaging lens are not horizontal and do not have equal distance to the reference plane, and moreover the optical axis of the projecting device and that of the imaging device are not coplanar, the height distribution of the measured object can be obtained by Improved Fourier Transform Profilometry [12]. However, it is difficult to accurately obtain the four system parameters, specially the angle system parameter. In this paper, a new phase-height mapping formula based on a new description of a reference fringe and a deformed fringe in Improved Fourier Transform Profilometry [12] is proposed. The strict theoretical analysis about the fringe description and the relationship between phase and height are given.
2.1. Fringe analysis and phase calculation The optical geometry of Improved Fourier Transform Profilometry is shown in Fig. 1. I1 is the exit pupil of the projector and I20 is the entrance pupil of the CCD camera. The optical axis I1 O crosses the reference plane R at point O, and the optical axis I20 O1 is vertical to the reference plane and crosses it at another point O1. Note that the connecting line I1 I20 is not parallel to the reference plane as well as I1 O and I20 O1 are not coplanar. D(x,y) and C1 are points on the 3-D object and on the reference plane, respectively, and they are imaged at the same point on the CCD array. I1 A is a ray through point D to the reference plane. I1 K is vertical to reference plane R and crosses it on point K. Given the system parameters: I20 O 1 ¼ L; OK ¼ r; I1 K ¼ m:The fringe pattern on the reference plane may be expressed as [8] I1 ðx; yÞ ¼ I0 ð1þ cos 2pxf ðxÞÞ
ð1Þ
where x is x-axial coordinate of the measured object, I0 is the illumination light intensity, and f(x) is the spatial frequency of the reference fringe along the x-axis, which can be expressed by [8,13] 2x sin y f ðxÞ ¼ f cos y 1 I1 O
ð2Þ
where f is the carry frequency of the grating.
Corresponding author.
E-mail address: [email protected] (X. Su). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.07.005
Compared with the case where the imaging axis crosses the projecting axis at the same point on the reference plane, if the influence on CCD imaging quality is neglected, adjustment of the
2024
X. Mao et al. / Optik 121 (2010) 2023–2027
According to the similar characteristic among some triangles, the following expression can be obtained: BD xðm LÞ CA ¼ r ð7Þ m BD L BD m BD xðm LÞ ‘cðx; yÞ ¼ 2pf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr L BD m2 þ r 2 m BD
! ð8Þ
Substituting Eq. (8) into Eq. (6), 2pfm I2 ðx; yÞ ¼ I0 1 þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þr 2
Fig. 1. The measuring sketch map.
x 1
2xr m2 þ r 2
!!! BD xðm LÞ r m BD L BD
ð9Þ After simplification, the reference fringe and the deformed fringe can be rewritten as the simple forms, respectively, as follows: I1 ðx; yÞ ¼ I0 ð1 þ cosð2pf0 x þ f0 ðx; yÞÞÞ
I2 ðx; yÞ ¼ I0 ð1 þ cosð2pf0 x þ fðx; yÞÞÞ ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where f0 ¼ f cos y ¼ f ðm= m2 þ r 2 Þ, f0 ðx; yÞ and fðx; yÞ are the phase distributions included in the reference fringe and the deformed fringe. I1 ðx; yÞ and I2 ðx; yÞ are Fourier transformed and processed in their spatial frequency domain as well as in their space-signal domain, then f0 ðx; yÞ and fðx; yÞ can be calculated, and the core phase Dfðx; yÞ directly caused by the height variation can be obtained as
Fig. 2. The geometrical sketch map.
CCD position just causes the movement of the image’s location on the CCD array whereas the relationship among the adjusted pixels is unchanged [12]. We add several dashed lines in the figure as assistant lines for our analysis as shown in Fig. 2. Draw a line I2 O through point O, which parallels the optical axis of CCD and the point I20 is rotated to the point I2 around the point D, as well as C1 is rotated to point C, which must be on the lineOA. BD is vertical to plane R, which denotes the height of point D on the measured object. I1 F parallels to the reference plane R and crosses I2 O and the prolongation of BD at point G and point F, respectively. I2 D crosses I1 F at point P. In DOI1 G, r sin y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þr 2
h¼
m cos y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ r 2
ðcðx; yÞðL þmÞ þ rLs sðm LÞxÞ 7
ð3Þ
Dfðx; yÞ ¼ fðx; yÞ f0 ðx; yÞ
In the next section, the conversion formula between the phase and the height will be deduced. Given m BD ¼ h; s ¼ 2pf0 ¼ 2pf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ r 2 deformed Eq. (8), we can obtain ðcðx; yÞ þsrÞh2 ðcðx; yÞðL þ mÞ þ sðLr xðm LÞÞÞhþ mLcðx; yÞ ¼ 0
2ðsr þ cðx; yÞÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ r 2
then
Substituting Eq. (4) into Eq. (2), 2pfxm 2xr I1 ðx; yÞ ¼ I0 1þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2 2 m þr m2 þ r 2
ð4Þ
ð5Þ
when the measured object is put, the deformed fringe pattern can be expressed as 2pfxm 2xr c ðx; yÞ ð6Þ I2 ðx; yÞ ¼ I0 1þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 m þ r2 m2 þ r 2 where [1] cðx; yÞ ¼ 2pf CA cos y
ð13Þ
then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcðx; yÞðL þmÞ þrLs sðm LÞxÞ2 4ðsr þ cðx; yÞÞcðx; yÞmL
2x sin y fm 2xr ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 f ðxÞ ¼ f cos y 1 m þ r2 I1 O m2 þ r 2
ð12Þ
2.2. The phase-height mapping
and I1 O ¼
ð10Þ
Fig. 3. The simulated object.
ð14Þ
X. Mao et al. / Optik 121 (2010) 2023–2027
2025
where cðx; yÞ ¼ Dfðx; yÞ, the system parameters (L, m, r) can be measured directly. Selecting the right radix, Eq. (14) will be the phase-height mapping formula. Given m =L, Eq. (14) can be simplified as h¼
2Lcðx; yÞ þrLs 7 rLs 2sr þ cðx; yÞ
ð15Þ
By selecting the right radix, we can obtain h¼
Lcðx; yÞ 2pf0 r þ cðx; yÞ
ð16Þ
It is the same as the traditional phase-height mapping formula [1]. Compared with the traditional Fourier Transform Profilometry [1], with the new method it is easier for us to obtain the full-field fringe through adjusting the location of either the projector or the imaging device. Compared with the Improved Fourier Transform Profilometry [12], only three length system parameters (L, m, r) need to be measured, especially, the difficult angle measurement [12] would not be needed, so the operation is simple and the result is more accurate.
Fig. 6. The height error distribution when m= L= 1700 mm.
Fig. 7. The height error distribution when m =2000 mm.
Fig. 4. The deformed fringe distribution when m = L= 1700 mm.
Fig. 8. The height error distribution when m =2300 mm.
3. Computer simulations
Fig. 5. The height error distribution when m =1400 mm.
In this section, some computer simulations are used to verify our discussion. The measured object is half an elliptical sphere, as
2026
X. Mao et al. / Optik 121 (2010) 2023–2027
shown in Fig. 3, in which the maximal height is about h= 33.25 mm. The simulated deformed grating with period T= 16, and the size of the simulated image is 512 512 pixel. System parameters are L= 1700 mm, r =400 mm. In our experiments, when m is selected from 1400 to 3500 mm, respectively, some computer simulations are carried out. Fig. 4 shows the deformed fringe corresponding to the case where m= L=1700 mm. From Figs. 5–10, each of them denotes the reconstructed height error distribution corresponding to m= 1400, 1700, 2000, 2300, 2600, and 2900 mm, respectively. From m=1400 to 3500 mm, the mean square deviation and their maximal height distributions of the reconstructed object are shown in Table 1.
Fig. 11. The grating fringe of reference plane.
Fig. 12. The grating fringe modulated by 3-D object.
Fig. 9. The height error distribution when m= 2600 mm.
Fig. 13. The reconstructed height distribution.
4. Experiment
Fig. 10. The height error distribution when m= 2900 mm.
Table 1 Corresponding to different m, the mean square deviations (RMS) and maximal height distribution of the reconstructed object. m (mm) 1400 1700 2000 2300 2600 2900 3200 3500 RMS (mm) 0.5499 0.3855 0.2898 0.2304 0.1851 0.1737 0.1312 0.1191 hmax (mm) 34.74 34.36 34.07 33.88 33.75 33.66 33.59 33.51 The maximal height of the measured object is 33.25 mm.
A primary experiment is used to verify our method, in which a toy model is measured. Projecting fringe pattern with sinusoidal intensities through a projector (PLUS U3-880) on the object, a CCD camera (MTV-1881EX) captured the deformed pattern. The system parameters are r = 38.00 cm, m= 93.00 cm, L= 122.00 cm. The grating period was T= 16, the maximal height of the object is h=2.90 cm, and image size is 256 256 pixels. Figs. 11 and 12 are the reference fringe and the deformed fringe, respectively. Fig. 13 is the reconstruction-height distribution by the new method where the maximal height is 2.95 cm. Compared with Improved Fourier Transform Profilometry method [12], the new phaseheight algorithm gives more reliable height reconstruction.
5. Conclusions A flexible description about a reference fringe and a deformed fringe and a new phase-height mapping formula is deduced.
X. Mao et al. / Optik 121 (2010) 2023–2027
Compared with the traditional Fourier Transform Profilometry, with the new method it is easier for us to obtain the full-field fringe through moving either the projector or the imaging system. Compared with the Improved Fourier Transform Profilometry [12], only three length system parameters need to be measured, so its operation is simple and its result is more reliable.
Acknowledgments The authors wish to acknowledge the support by the National Natural Science Foundation of China (10876021, 60677028). References [1] M. Takeda, K. Mutoh, Fourier transform profilometry for the automatic measurement 3-D object shapes, Appl. Opt. 22 (24) (1983) 3977–3982. [2] Fiona Berryman, Paul Pynsent, James Cubillo, A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise, Opt. Lasers Eng. 39 (1) (2003) 35–50. [3] J. Vanherzeele, P. Guillaume, S. Vanlanduit, Fourier fringe processing using a regressive Fourier-transform technique, Opt. Lasers Eng. 43 (6) (2005) 645–658.
2027
[4] Wenjing Chen, Xianyu Su, Likun Su, et al., Improvement for measuring accuracy of fourier transform profilometry with gray image, Opto-Electron. Eng. 27 (148) (2000) 55–59 (in Chinese). [5] Wenjing Chen, Xianyu Su, Songxin Tan, Basis requirement for measurement system in FTP, J. Optoelectron. Laser 10 (6) (1999) 535–539. [6] Jingang Zhong, Jiawen Weng, Phase retrieval of optical fringe patterns from the ridge of a wavelet transform, Opt. Lett. 30 (2005) 2560–2562. [7] Xianyu Su, Wenjing Chen, Fourier transform profilometry: a review, Opt. Lasers Eng. 35 (5) (2001) 263–284. [8] Giuseppe Schirripa Spagnolo, Giorgio Guattari, Carmine Sapia, et al., Contouring of artwork surface by fringe projection and FFT analysis, Opt. Lasers Eng. 33 (2000) 141–156. [9] Wenjing Chen, Xian-yu Su, A new method for improving measuring accuracy of fourier transform profilometry, Opto-Electron. Eng. 29 (1) (2002) 19–22. [10] Wansong Li, Xianyu Su, Likun Su, Liqun Xiang, Simultaneous calibration algorithm of phase-shifting based on FFT, Acta Opt. Sini. 19 (10) (1999) 1390–1395. [11] Wenjing Chen, Hu Yang, Xianyu Su, Songxin Tan, Error caused by sampling in Fourier transform profilometry, Opt. Eng. 38 (6) (1999) 1029–1034. [12] Xianfu Mao, Wenjing Chen, Xianyu Su, Improved fourier transform profilometry, Appl. Opt. 46 (5) (2007) 664–668. ˚ [13] Kjell J Gasvik, Optical techniques: Toyohiko Yatagai. Intensity based analysis methods, in: D.W. Robinson, G.T. Reid (Eds.), Interferograms Analysis, IOP, Bristol, 1993, pp. 49–93.