- Email: [email protected]
Fourier transform profilometry based on composite structured light pattern
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 1170–1175 www.elsevier.com/locate/optlastec
Fourier transform profilometry based on composite structured light pattern Hui-Min Yuea,b,, Xian-Yu Sub, Yong-Zhi Liua a
School of Optic-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China b Department of Opto-Electronics, Sichuan University, Chengdu 610064, China Received 28 March 2006; received in revised form 2 August 2006; accepted 18 August 2006 Available online 23 October 2006
Abstract In Fourier transform profilometry (FTP), the zero frequency of the imaged patterns will influence the measurement range and precision. The p phase shifting technique is usually used to eliminate the zero order component, but this method requires the capture of two fringe patterns with a p phase difference between them, which will impede the real time application of the method. In this paper, a novel method is proposed, in which a composite structured light pattern is projected onto the object. The composite structured light pattern is formed by modulating two separate fringe patterns with a p phase difference along the orthogonal direction of the two distinct carrier frequencies. This method can eliminate the zero frequency by using only one fringe pattern. Experiments show that there is no decrease in the precision of this novel method compared with the traditional p phase shifting technique. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fourier transform profilometry; Composite structured light pattern; p phase shifting technique
1. Introduction The Fourier transform profilometry (FTP), method based on fringe projection has several advantages including its high-resolution, one frame capture, and whole-field depth reconstruction. Since Takeda and Motoh [1] proposed the FTP method, it has been extensively studied [2–10], focusing on improving the measurement range and precision. FTP is one of the most widely used 3-D sensing methods, in which a fringe pattern is projected onto an object. The deformed fringe pattern formed by imaging the object onto a CCD camera is Fourier transformed and processed in the spatial frequency domain. The fundamental frequency component, which includes information on the height information of the measured object, is filtered in the frequency domain. This is followed by an inverse Fourier transform to determine the reconstruction depth. So the influence of frequency spectrum aliasing on the Corresponding author. School of Optic-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China. E-mail address: [email protected] (H.-M. Yue).
0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.08.014
performance of this technique cannot be neglected. A sinusoidal grating projection combined with a p phase shifting technique has been used to eliminate zero and higher order components and improve the measured precision and range. However, to facilitate this two images of the fringe patterns must be captured, which influences the instantaneous characteristic of FTP. Recently, a new phase measurement profilometry called composite phase measurement profilometry has been reported [11], in which only one composite fringe pattern is used to recover the object surface. This method uses at least three carrier frequencies, which increase the problems associated with aliasing and reduce the measuring precision. The literature [12] presents a novel method, in which a bicolor sinusoid fringe pattern (that consists of two interlaced RGB format base color fringe patterns with p phase difference) is projected onto the object via a digital light projector and the deformed color pattern is captured by a color digital camera. In this paper, a novel method is proposed, in which a composite structured light pattern is projected to recover the object surface. The method eliminates the zero frequency by using only one fringe pattern. Theoretical
ARTICLE IN PRESS H.-M. Yue et al. / Optics & Laser Technology 39 (2007) 1170–1175
analysis, numerical simulation and experimental results obtained by employing the new method are presented in this paper. The experimental results indicate that there is no perceivable decrease in the precision of this method compared with the traditional p phase shifting technique. The new method allows for practical real time, high-speed implementation.
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2.2. Composite fourier transform profilometry (CFTP)
2. Theoretical analysis
Two fringe patterns with p phase difference are separately modulated along the orthogonal directions with a distinct carrier frequency and then summed together into one single composite image, as shown in Fig. 1. Therefore, each channel in the composite image along the orthogonal direction represents the individual pattern used in p phase shifting FTP for the phase calculation. Similar to the patterns projected in the p phase shifting FTP as in Eq. (1), the image patterns to be modulated are
2.1. Traditional p phase shifting FTP
I pn ¼ c þ cosð2pf f yp pnÞ.
In p phase shifting FTP, sinusoid patterns are projected as:
A constant c is used here to offset I pn to be non-negative values. After modulated by cosine carrier frequency, the composite structured pattern is combined
I pn ðxp ; yp Þ ¼ Ap þ Bp cosð2pf f yp pnÞ, p
(1)
I p ¼ Ap þ Bp
p
where A and B are the projection constants and ðxp ; yp Þ is the projector coordinate. The yp dimension is in the direction of the depth distortion and is called as the phase dimension. The xp dimension is perpendicular to the phase dimension, and is called as the orthogonal dimension. f f denotes the frequency of the sinusoid wave in the phase direction. The frequency of the fringe pattern is named as the carrier frequency. The subscript n represents the phase shift index and n ¼ 0, 1. The reflected intensity images from the object surface after two successive projections: I n ðx; yÞ ¼ Rðx; yÞ½A þ B cosð2pf f yp þ fðx; yÞ pnÞ,
I pn cosð2pf pn xp Þ,
(4)
n¼1
where f pn are the carrier frequencies along the orthogonal direction and n is the shift index from 0 to 1. The projection constants Ap and Bp are calculated as [11]: " # N X p p p p p I n cosð2pf n x Þ , A ¼ I min B min n¼1
,( p
B ¼ ðI max I min Þ
(2) min
where (x, y) is the image coordinate and R(x, y)is the reflectance variation. After eliminating the first item Rðx; yÞA by the p phase shifting technique, the Fourier transform is followed. Then the base frequency is filtered from the frequency domain, and the inverse Fourier transform is performed to get the phase 2pf f yp þ fðx; yÞ. The carrier frequency 2pf f yp is removed by spectrum shift center to get the phase modulated by the height. The phase calculation gives principal values ranging from p to p, and has discontinuities with 2p phase jumps. By use of a phase unwrapping algorithm, the continuous phase distribution can be obtained. The height distribution can be computed by the phase-height mapping technique.
N X
(3)
" N X
" max
N X n¼1
I pn
cosð2pf pn xp Þ
# I pn
cosð2pf pn xp Þ
#) .
ð5Þ
n¼1
The projection intensity range of the composite pattern falls into [Imin, Imax], which matches the intensity capacity of the projector. The reflected composite pattern image on the measured object captured by the camera is " # N X 0 I CP ðx; yÞ ¼ Rðx; yÞ A þ B I n ðx; yÞ cosð2pf n xÞ , n¼1
I 0n ðx; yÞ ¼ c þ cosð2pf f yp þ fðx; yÞ pnÞ.
(6)
An appropriate carrier frequency fn has to be carefully assigned. The selection of fn is highly dependent on the
Fig. 1. A composite pattern formed by modulating traditional p phase shifting FTP patterns along the orthogonal direction (for explicit show, fringe number of the modulated FTP pattern is five).
ARTICLE IN PRESS H.-M. Yue et al. / Optics & Laser Technology 39 (2007) 1170–1175
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2-D filtering Captured image
Fringe pattern
Gray adjustment Conventional FTPmethod
2-D Fourier transform
2-D filtering
Wrapped phase
Π-phase-shifting Gray fringe pattern adjustment
Fig. 2. Block diagram of the phase decoding process.
projector and camera qualities. Basically, on the premise of satisfying the Niquist sampling theorem, to minimize the carrier frequency spectrum leakage, the adjacent fn should be spread out and kept away from zero frequency as much as possible. The phase decoding process of the composite FTP is shown in Fig. 2. First, the captured image is processed by two-dimensional Fourier transformation. Then two-dimensional band-pass filter are used to separate out each channel. The band-pass filters are centered at the maximum point of spectrum intensity of the two carrier frequencies. After filtering, the inverse Fourier transform is followed, and the module of this result is just the two fringe patterns with p phase shifting. Commonly, the two peak values of the spectrum of the carrier frequency are usually not uniform, which will lead to different intensity value of the decoded two fringe patterns with p phase difference. Therefore, to get the accurate wrapped phase, we must calibrate the intensity of the two fringe patterns. Calibration of fringe intensity has two steps [13]. One is the calibration of the mean value, and the other is the calibration of the contrast. Assuming that the two fringe patterns to be adjusted are fringe (a) and fringe (b), and that fringe (b) is adjusted to be the same as fringe (a), the adjustment formula of mean value is I 0b ðx; yÞ ¼ I b ðx; yÞ þ ma mb ,
(7)
where ma and mb are separately the mean values of the fringe (a) and fringe (b), and I 0b is the intensity of fringe (b) after adjustment. After this step, the mean values of the two fringe patterns are even. The adjustment formula of the contrast is I 00b ðx; yÞ ¼ ½I 0b ðx; yÞ m0b d a =d b þ m0b ,
(8)
where I 0b and m0b are separately the intensity and mean value of fringe (b) that is after adjustment, and d a and d b are separately the root mean square of fringe (a) and (b), and I 00b ðx; yÞ is the intensity of fringe (b) after adjustment. So if the background of the fringes does not vary with time, after the two adjustment steps, the two decoded fringe patterns with p phase difference will have the same background and contrast distribution. After adjustment, we can calculate the wrapped phase as the traditional process of p phase shifting FTP. Therefore, the proposed method can eliminate zero frequency in FTP by projecting only one composite fringe pattern.
3. Numerical simulation In numerical simulation, the two carrier frequencies f1 and f2 of CFTP are separately three lines per 20 pixels, four lines per 20 pixels, and fringe frequency in the phase direction is 512 lines per 20 pixels. The simulated image is shown in Fig. 3(a). The deformation is controlled by h(x, y), which is expressed as hðx; yÞ ¼ 5f3ð1 xÞ2 exp½x2 ðy þ 1Þ2 10ðx=5 x3 y5 Þ expðx2 y2 Þ 1=3 exp½ðx þ 1Þ2 y2 g.
ð9Þ
The deformed fringe pattern is shown in Fig. 3(b). According to the above process of phase decoding, the composite deformed fringe pattern can be decoded into two individual fringe patterns with p phase difference, as shown in Fig. 3(c) and (d). The Fourier spectrum of one of the two fringe patterns along phase direction is shown in Fig. 3(e), which is with zero component. After the calibration of mean and contrast, two fringe patterns are subtracted. The Fourier spectrum of result fringe pattern is shown in Fig. 3(f), and the zero component has been eliminated. The retrieved surface by using CFTP is shown in Fig. 3(g). 4. Experiments Fig. 4 shows the block diagram of the experimental setup. The composite fringe pattern is produced by a computer and projected by a digital light projector (DLP, PLUS U3-880). The image size is 800*600 pixels. The deformed fringe pattern is captured by a digital camera (Cannon A80). The two carrier frequencies f1 and f2 of CFTP are separately three lines per 40 pixels, six lines 40 pixels. The Fringe frequency in the phase direction is 600 lines per 32 pixels. To verify the method, we apply the measurement both using traditional p phase shifting FTP and the proposed method in this paper under the same experimental conditions. The Fringe frequency in the phase direction of the traditional p phase shifting FTP is also 600 lines per 32 pixels. In the experiments, the composite fringe pattern is projected on the cone-shaped object and captured by the camera, and the captured image is shown in Fig. 5(a). The gray image of two-dimensional spectrum of the captured image is shown in Fig. 5(b). Two decoded fringe patterns
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Fig. 3. Numerical simulation of CFTP: (a) simulation object (sub-sampling), (b) simulation composite deformed fringe pattern, (c) decoded fringe pattern, (d) decoded p phase shifting fringe pattern, (e) normalized frequency spectrum distribution along phase directions with the zero component, (f) normalized frequency spectrum distribution along phase directions after eliminating the zero component, and (g) the retrieved height by using CFTP (sub-sampling).
Monitor Object Digital camera DLP projector Fig. 4. Schematic diagram of setup.
IBM PC
with p phase shifting are shown in Fig. 5(c) and (d). According to the above intensity calibration, the two fringe patterns with p phase shifting are adjusted, and then they are subtracted to eliminate zero frequency component. The frequency spectrum of the subtracted fringe is shown in Fig. 5(e). The figure shows that there is no zero frequency component. Thus it can be proved that the CFTP method can realize p phase shifting FTP and eliminate zero frequency by projecting only one composite fringe pattern. The reference plane is measured by FTP and CFTP separately. Least square fitting phase value subtracted from
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Fig. 5. Experiments of CFTP and FTP: (a) the composite fringe patterns captured from experiments, (b) the gray image of two-dimensional frequency spectrum of the captured image in CFTP, (c) and (d) the two fringe patterns with p phase difference attained by filtering in CFTP experiment, (e) spectrum of extracted fringes after gray revision and eliminating the zero frequency component in CFTP, (f) the phase error of one of rows of the reference, the solid line denoted FTP, the doted line denoted CFTP, (g) sections of recovered phase of the cone-shaped object measured by FTP and CFTP separately, the solid line denoted FTP, the doted line denoted CFTP.
the measured phase value gives the phase measurement error. The phase error of a certain row of the reference is shown in Fig. 5(f). The standard deviation of this row is 0.0884 and 0.0881 rad for FTP and CFTP respectively. Fig. 5(g) shows the sections of recovered phase of the coneshaped object measured by FTP and CFTP separately. The experimental results indicate that the measurement precision of CFTP does not decrease because of the multiplication of carrier frequencies.
5. Conclusions In this paper a novel method called as composite FTP, is proposed, in which a composite fringe pattern is used. The composite structured light pattern is formed by modulating two separate fringe patterns with p phase difference along the orthogonal direction of the two distinct carrier frequencies. The theoretical analysis and experimental results show that the CFTP method can eliminate the zero
ARTICLE IN PRESS H.-M. Yue et al. / Optics & Laser Technology 39 (2007) 1170–1175
frequency component by using only one fringe pattern. The ultimate advantage of the proposed method is that only one composite fringe pattern is needed for surface reconstruction. The disadvantage is that the steps of phase decoding are more than the traditional method. But along with the development of computer hardware and software techniques this will be not a problem. So when it comes to high-speed implementation, the new method is better than the traditional method, and otherwise the traditional method is better. References [1] Takeda M, Mutoh K. Fourier transform profilometry for the automatic measurement 3-D object shapes. Appl Opt 1983;22(24):3977–82. [2] Chen W-J, Su X-Y. Error caused by sampling in Fourier transform profilometry. Opt Eng 1999;38(6):1029–34. [3] Li J, Su X-Y, Guo L-R. Improved Fourier transform profilometry of the automatic measurement of three-dimensional object shapes. Opt Eng 1990;29(12):1439–44. [4] Su X-Y, Sajan MR, Asundi A. Fourier transform profilometry for 360-degree shape using TDI camera. International conference on experimental mechanics advances and applications Singapore, 4–6 December 1996.
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[5] Yi J, Huang S. Modified Fourier transform profilometry for the measurement of 3-D steep shapes. Opt Lasers Eng 1997;27(5): 493–505. [6] Su X-Y, Chen W-J. Fourier transfrom profilometry: a review. Opt Lasers Eng 2001;35:263–84. [7] Takeda M, Gu Q, Kinoshita M, et al. Frequency-multiplex Fouriertransform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations. Appl Opt 1997;36(22):5347–54. [8] Kujawinska M, Wojciak J. High accuracy Fourier transform fringe pattern analysis. Opt Lasers Eng 1991;l4:325–39. [9] Su X-Y, Su L-K, Li W-S. A new Fourier transform profilometry based on modulation measurement, ICO-18, 1999. [10] Burton DR, Goodall AJ, Atkinson JT. The use of carrier frequencyshifting for the elimination of phase discontinuities in Fouriertransform profilometry. Opt Laser Eng 1995;23(4):245–57. [11] Guan C, Hassebrook LG, Lau DL. Composite structured light pattern for three-dimensional video, optics express 2003;11(5): 406–17. [12] Chen W-J, Su X-Y, Cao Y-P. Improving Fourier transform profilometry based on bicolor fringe pattern, Opt. Engineering 2004;43(1):192–8. [13] Li W-S, Su X-Y. Application of improved phase-measuring profilometry in nonconstant environmental light. Opt Eng 2001;40(3):478–85.
Optics & Laser Technology 39 (2007) 1170–1175 www.elsevier.com/locate/optlastec
Fourier transform profilometry based on composite structured light pattern Hui-Min Yuea,b,, Xian-Yu Sub, Yong-Zhi Liua a
School of Optic-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China b Department of Opto-Electronics, Sichuan University, Chengdu 610064, China Received 28 March 2006; received in revised form 2 August 2006; accepted 18 August 2006 Available online 23 October 2006
Abstract In Fourier transform profilometry (FTP), the zero frequency of the imaged patterns will influence the measurement range and precision. The p phase shifting technique is usually used to eliminate the zero order component, but this method requires the capture of two fringe patterns with a p phase difference between them, which will impede the real time application of the method. In this paper, a novel method is proposed, in which a composite structured light pattern is projected onto the object. The composite structured light pattern is formed by modulating two separate fringe patterns with a p phase difference along the orthogonal direction of the two distinct carrier frequencies. This method can eliminate the zero frequency by using only one fringe pattern. Experiments show that there is no decrease in the precision of this novel method compared with the traditional p phase shifting technique. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fourier transform profilometry; Composite structured light pattern; p phase shifting technique
1. Introduction The Fourier transform profilometry (FTP), method based on fringe projection has several advantages including its high-resolution, one frame capture, and whole-field depth reconstruction. Since Takeda and Motoh [1] proposed the FTP method, it has been extensively studied [2–10], focusing on improving the measurement range and precision. FTP is one of the most widely used 3-D sensing methods, in which a fringe pattern is projected onto an object. The deformed fringe pattern formed by imaging the object onto a CCD camera is Fourier transformed and processed in the spatial frequency domain. The fundamental frequency component, which includes information on the height information of the measured object, is filtered in the frequency domain. This is followed by an inverse Fourier transform to determine the reconstruction depth. So the influence of frequency spectrum aliasing on the Corresponding author. School of Optic-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China. E-mail address: [email protected] (H.-M. Yue).
0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.08.014
performance of this technique cannot be neglected. A sinusoidal grating projection combined with a p phase shifting technique has been used to eliminate zero and higher order components and improve the measured precision and range. However, to facilitate this two images of the fringe patterns must be captured, which influences the instantaneous characteristic of FTP. Recently, a new phase measurement profilometry called composite phase measurement profilometry has been reported [11], in which only one composite fringe pattern is used to recover the object surface. This method uses at least three carrier frequencies, which increase the problems associated with aliasing and reduce the measuring precision. The literature [12] presents a novel method, in which a bicolor sinusoid fringe pattern (that consists of two interlaced RGB format base color fringe patterns with p phase difference) is projected onto the object via a digital light projector and the deformed color pattern is captured by a color digital camera. In this paper, a novel method is proposed, in which a composite structured light pattern is projected to recover the object surface. The method eliminates the zero frequency by using only one fringe pattern. Theoretical
ARTICLE IN PRESS H.-M. Yue et al. / Optics & Laser Technology 39 (2007) 1170–1175
analysis, numerical simulation and experimental results obtained by employing the new method are presented in this paper. The experimental results indicate that there is no perceivable decrease in the precision of this method compared with the traditional p phase shifting technique. The new method allows for practical real time, high-speed implementation.
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2.2. Composite fourier transform profilometry (CFTP)
2. Theoretical analysis
Two fringe patterns with p phase difference are separately modulated along the orthogonal directions with a distinct carrier frequency and then summed together into one single composite image, as shown in Fig. 1. Therefore, each channel in the composite image along the orthogonal direction represents the individual pattern used in p phase shifting FTP for the phase calculation. Similar to the patterns projected in the p phase shifting FTP as in Eq. (1), the image patterns to be modulated are
2.1. Traditional p phase shifting FTP
I pn ¼ c þ cosð2pf f yp pnÞ.
In p phase shifting FTP, sinusoid patterns are projected as:
A constant c is used here to offset I pn to be non-negative values. After modulated by cosine carrier frequency, the composite structured pattern is combined
I pn ðxp ; yp Þ ¼ Ap þ Bp cosð2pf f yp pnÞ, p
(1)
I p ¼ Ap þ Bp
p
where A and B are the projection constants and ðxp ; yp Þ is the projector coordinate. The yp dimension is in the direction of the depth distortion and is called as the phase dimension. The xp dimension is perpendicular to the phase dimension, and is called as the orthogonal dimension. f f denotes the frequency of the sinusoid wave in the phase direction. The frequency of the fringe pattern is named as the carrier frequency. The subscript n represents the phase shift index and n ¼ 0, 1. The reflected intensity images from the object surface after two successive projections: I n ðx; yÞ ¼ Rðx; yÞ½A þ B cosð2pf f yp þ fðx; yÞ pnÞ,
I pn cosð2pf pn xp Þ,
(4)
n¼1
where f pn are the carrier frequencies along the orthogonal direction and n is the shift index from 0 to 1. The projection constants Ap and Bp are calculated as [11]: " # N X p p p p p I n cosð2pf n x Þ , A ¼ I min B min n¼1
,( p
B ¼ ðI max I min Þ
(2) min
where (x, y) is the image coordinate and R(x, y)is the reflectance variation. After eliminating the first item Rðx; yÞA by the p phase shifting technique, the Fourier transform is followed. Then the base frequency is filtered from the frequency domain, and the inverse Fourier transform is performed to get the phase 2pf f yp þ fðx; yÞ. The carrier frequency 2pf f yp is removed by spectrum shift center to get the phase modulated by the height. The phase calculation gives principal values ranging from p to p, and has discontinuities with 2p phase jumps. By use of a phase unwrapping algorithm, the continuous phase distribution can be obtained. The height distribution can be computed by the phase-height mapping technique.
N X
(3)
" N X
" max
N X n¼1
I pn
cosð2pf pn xp Þ
# I pn
cosð2pf pn xp Þ
#) .
ð5Þ
n¼1
The projection intensity range of the composite pattern falls into [Imin, Imax], which matches the intensity capacity of the projector. The reflected composite pattern image on the measured object captured by the camera is " # N X 0 I CP ðx; yÞ ¼ Rðx; yÞ A þ B I n ðx; yÞ cosð2pf n xÞ , n¼1
I 0n ðx; yÞ ¼ c þ cosð2pf f yp þ fðx; yÞ pnÞ.
(6)
An appropriate carrier frequency fn has to be carefully assigned. The selection of fn is highly dependent on the
Fig. 1. A composite pattern formed by modulating traditional p phase shifting FTP patterns along the orthogonal direction (for explicit show, fringe number of the modulated FTP pattern is five).
ARTICLE IN PRESS H.-M. Yue et al. / Optics & Laser Technology 39 (2007) 1170–1175
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2-D filtering Captured image
Fringe pattern
Gray adjustment Conventional FTPmethod
2-D Fourier transform
2-D filtering
Wrapped phase
Π-phase-shifting Gray fringe pattern adjustment
Fig. 2. Block diagram of the phase decoding process.
projector and camera qualities. Basically, on the premise of satisfying the Niquist sampling theorem, to minimize the carrier frequency spectrum leakage, the adjacent fn should be spread out and kept away from zero frequency as much as possible. The phase decoding process of the composite FTP is shown in Fig. 2. First, the captured image is processed by two-dimensional Fourier transformation. Then two-dimensional band-pass filter are used to separate out each channel. The band-pass filters are centered at the maximum point of spectrum intensity of the two carrier frequencies. After filtering, the inverse Fourier transform is followed, and the module of this result is just the two fringe patterns with p phase shifting. Commonly, the two peak values of the spectrum of the carrier frequency are usually not uniform, which will lead to different intensity value of the decoded two fringe patterns with p phase difference. Therefore, to get the accurate wrapped phase, we must calibrate the intensity of the two fringe patterns. Calibration of fringe intensity has two steps [13]. One is the calibration of the mean value, and the other is the calibration of the contrast. Assuming that the two fringe patterns to be adjusted are fringe (a) and fringe (b), and that fringe (b) is adjusted to be the same as fringe (a), the adjustment formula of mean value is I 0b ðx; yÞ ¼ I b ðx; yÞ þ ma mb ,
(7)
where ma and mb are separately the mean values of the fringe (a) and fringe (b), and I 0b is the intensity of fringe (b) after adjustment. After this step, the mean values of the two fringe patterns are even. The adjustment formula of the contrast is I 00b ðx; yÞ ¼ ½I 0b ðx; yÞ m0b d a =d b þ m0b ,
(8)
where I 0b and m0b are separately the intensity and mean value of fringe (b) that is after adjustment, and d a and d b are separately the root mean square of fringe (a) and (b), and I 00b ðx; yÞ is the intensity of fringe (b) after adjustment. So if the background of the fringes does not vary with time, after the two adjustment steps, the two decoded fringe patterns with p phase difference will have the same background and contrast distribution. After adjustment, we can calculate the wrapped phase as the traditional process of p phase shifting FTP. Therefore, the proposed method can eliminate zero frequency in FTP by projecting only one composite fringe pattern.
3. Numerical simulation In numerical simulation, the two carrier frequencies f1 and f2 of CFTP are separately three lines per 20 pixels, four lines per 20 pixels, and fringe frequency in the phase direction is 512 lines per 20 pixels. The simulated image is shown in Fig. 3(a). The deformation is controlled by h(x, y), which is expressed as hðx; yÞ ¼ 5f3ð1 xÞ2 exp½x2 ðy þ 1Þ2 10ðx=5 x3 y5 Þ expðx2 y2 Þ 1=3 exp½ðx þ 1Þ2 y2 g.
ð9Þ
The deformed fringe pattern is shown in Fig. 3(b). According to the above process of phase decoding, the composite deformed fringe pattern can be decoded into two individual fringe patterns with p phase difference, as shown in Fig. 3(c) and (d). The Fourier spectrum of one of the two fringe patterns along phase direction is shown in Fig. 3(e), which is with zero component. After the calibration of mean and contrast, two fringe patterns are subtracted. The Fourier spectrum of result fringe pattern is shown in Fig. 3(f), and the zero component has been eliminated. The retrieved surface by using CFTP is shown in Fig. 3(g). 4. Experiments Fig. 4 shows the block diagram of the experimental setup. The composite fringe pattern is produced by a computer and projected by a digital light projector (DLP, PLUS U3-880). The image size is 800*600 pixels. The deformed fringe pattern is captured by a digital camera (Cannon A80). The two carrier frequencies f1 and f2 of CFTP are separately three lines per 40 pixels, six lines 40 pixels. The Fringe frequency in the phase direction is 600 lines per 32 pixels. To verify the method, we apply the measurement both using traditional p phase shifting FTP and the proposed method in this paper under the same experimental conditions. The Fringe frequency in the phase direction of the traditional p phase shifting FTP is also 600 lines per 32 pixels. In the experiments, the composite fringe pattern is projected on the cone-shaped object and captured by the camera, and the captured image is shown in Fig. 5(a). The gray image of two-dimensional spectrum of the captured image is shown in Fig. 5(b). Two decoded fringe patterns
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Fig. 3. Numerical simulation of CFTP: (a) simulation object (sub-sampling), (b) simulation composite deformed fringe pattern, (c) decoded fringe pattern, (d) decoded p phase shifting fringe pattern, (e) normalized frequency spectrum distribution along phase directions with the zero component, (f) normalized frequency spectrum distribution along phase directions after eliminating the zero component, and (g) the retrieved height by using CFTP (sub-sampling).
Monitor Object Digital camera DLP projector Fig. 4. Schematic diagram of setup.
IBM PC
with p phase shifting are shown in Fig. 5(c) and (d). According to the above intensity calibration, the two fringe patterns with p phase shifting are adjusted, and then they are subtracted to eliminate zero frequency component. The frequency spectrum of the subtracted fringe is shown in Fig. 5(e). The figure shows that there is no zero frequency component. Thus it can be proved that the CFTP method can realize p phase shifting FTP and eliminate zero frequency by projecting only one composite fringe pattern. The reference plane is measured by FTP and CFTP separately. Least square fitting phase value subtracted from
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Fig. 5. Experiments of CFTP and FTP: (a) the composite fringe patterns captured from experiments, (b) the gray image of two-dimensional frequency spectrum of the captured image in CFTP, (c) and (d) the two fringe patterns with p phase difference attained by filtering in CFTP experiment, (e) spectrum of extracted fringes after gray revision and eliminating the zero frequency component in CFTP, (f) the phase error of one of rows of the reference, the solid line denoted FTP, the doted line denoted CFTP, (g) sections of recovered phase of the cone-shaped object measured by FTP and CFTP separately, the solid line denoted FTP, the doted line denoted CFTP.
the measured phase value gives the phase measurement error. The phase error of a certain row of the reference is shown in Fig. 5(f). The standard deviation of this row is 0.0884 and 0.0881 rad for FTP and CFTP respectively. Fig. 5(g) shows the sections of recovered phase of the coneshaped object measured by FTP and CFTP separately. The experimental results indicate that the measurement precision of CFTP does not decrease because of the multiplication of carrier frequencies.
5. Conclusions In this paper a novel method called as composite FTP, is proposed, in which a composite fringe pattern is used. The composite structured light pattern is formed by modulating two separate fringe patterns with p phase difference along the orthogonal direction of the two distinct carrier frequencies. The theoretical analysis and experimental results show that the CFTP method can eliminate the zero
ARTICLE IN PRESS H.-M. Yue et al. / Optics & Laser Technology 39 (2007) 1170–1175
frequency component by using only one fringe pattern. The ultimate advantage of the proposed method is that only one composite fringe pattern is needed for surface reconstruction. The disadvantage is that the steps of phase decoding are more than the traditional method. But along with the development of computer hardware and software techniques this will be not a problem. So when it comes to high-speed implementation, the new method is better than the traditional method, and otherwise the traditional method is better. References [1] Takeda M, Mutoh K. Fourier transform profilometry for the automatic measurement 3-D object shapes. Appl Opt 1983;22(24):3977–82. [2] Chen W-J, Su X-Y. Error caused by sampling in Fourier transform profilometry. Opt Eng 1999;38(6):1029–34. [3] Li J, Su X-Y, Guo L-R. Improved Fourier transform profilometry of the automatic measurement of three-dimensional object shapes. Opt Eng 1990;29(12):1439–44. [4] Su X-Y, Sajan MR, Asundi A. Fourier transform profilometry for 360-degree shape using TDI camera. International conference on experimental mechanics advances and applications Singapore, 4–6 December 1996.
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