- Email: [email protected]
Improved composite Fourier transform profilometry
Optics & Laser Technology 44 (2012) 2037–2042
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Improved composite Fourier transform profilometry Ying-chun Wu, Yi-ping Cao n, Zhen-fen Huang, Ming-teng Lu, De-liang Chen Department of Opto-Electronics, Sichuan University, Chengdu 610065, China
a r t i c l e i n f o
abstract
Article history: Received 17 December 2011 Received in revised form 6 March 2012 Accepted 22 March 2012 Available online 11 April 2012
An improved composite Fourier transform profilometry (ICFTP) is proposed through optimizing the composite grating. Compared with the composite Fourier transform profilometry (CFTP), the frequencies of the two sinusoidal images modulated along the orthogonal direction in the optimized composite grating are not equal. The overlapping degree of Fourier spectrum of the optimized grating is lowered. The measuring accuracy is improved because two frequencies are used to calculate phase more accurately. By using a more advantageous flat image extracted ingeniously from the optimized grating, the zero component of the lower frequency sinusoidal image in the Fourier spectrum can be eliminated. So the measuring range is extended the same as that by the p phase shifting technique. Both numerical simulation and experiment demonstrate the feasibility and validity of the proposed method. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Composite grating Measuring range Measuring accuracy
1. Introduction New technology brings about a lot of new methods in optical 3-D sensing [1]. Based on digital fringe projection techniques, fringe projection profilometry has a huge potential for applications in many areas, including on-line inspection, medical sciences, reverse engineering, computer graphics and security [2–4]. With the improvement of parallel operation speed and the development of recent technologies including digital image, digital projection and digital display, people have higher level of requirements in measuring speed and accuracy. Real-time 3-D sharp measurement just as on-line inspection and dynamic profilometry becomes a hot topic. Fourier transform profilometry (FTP) is one of the most used methods in real-time 3-D sharp measurement due to only one sinusoidal image projection [5]. By analyzing the captured image using the Fourier method, FTP can demodulate the object’s shape from the fundamental component in the Fourier spectrum. Because the filtering process is involved, the requirement of preventing spectral components from overlapping influences the maximal slope of height variation of measured object (i.e., measuring range) and measuring accuracy. Su et al. proposed a p phase shifting technique which has successfully tripled the measuring range by eliminating zero component from the Fourier spectrum [6]. But two sinusoidal images projection influences the instantaneous character of FTP. By using Guan’s composite structured pattern [7–9], Yue et al. proposed the composite Fourier transform profilometry (CFTP) by modulating two separate sinusoidal images with a p phase difference along the
n
Corresponding author. E-mail address: [email protected] (Y.-p. Cao).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.03.030
orthogonal direction in a composite grating. Her method can eliminate the zero component but only one image projection [10]. All the methods mentioned above used single frequency image to calculate the phase, the measuring accuracy were limited. Dualfrequency image projection is a common method which can be used to improve the measuring accuracy [11,12]. But if a higher measuring range is required at the same time, the p phase shifting technique is needed, which also results in two dual-frequency images projection and time consuming [13]. An improved composite Fourier transform profilometry (ICFTP) is proposed in the paper. The composite grating is optimized by modulating two sinusoidal images with different frequencies along the orthogonal direction. Compared with Yue’s grating in CFTP, the overlapping degree of the Fourier spectrum in the ICFTP is lowered because the distance between the alternating-current (AC) components of the two carrier channels is increased. Without modulating two sinusoidal images with equal frequency and p phase difference, the measuring range can also be tripled by using a flat image which is extracted from the composite grating. With the two different frequencies, two wrapped phases can be calculated from the optimized composite grating. The measuring accuracy is improved because a more accurate unwrapping phase of higher frequency can obtain with the guidance of the of lower frequency phase.
2. Brief description of the composite Fourier transform profilometry (CFTP) In order to enlarge the measuring range of FTP, Yue et al. proposed the CFTP which adopted the p phase shifting technique in the composite structured light projection system [7,10]. The modulation and demodulation processes are shown in Fig. 1.
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Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
Carrier channels
Modulating images
Composite grating Phase dimension
Object
Orthogonal dimension
Deformed sinusoidal images
Spectrum distribution
Deformed image
Retrieved height Filter, IFFT,
FFT
Modulus
Fig. 1. Schematic diagram of the modulation and demodulation.
Two separate sinusoidal images with a p phase difference are modulated into two carrier waves to form a composite grating. The composite grating is then projected on the measured object. Through demodulating the captured deformed composite image, the two sinusoidal images were extracted to calculate the phase of the object by the FTP. After the phase unwrapping, the height of the object can be restored with the height-phase mapping algorithm [14,15].
filter3 filter1
filter2 3. Improved composite Fourier transform profilometry (ICFTP) 3.1. Principle of the ICFTP
Fig. 2. Spectrum distribution of the optimized composite grating.
In the ICFTP, the composite grating is optimized by replacing one of the sinusoidal images with a higher frequency one. The projected composite grating can be written as Ip ðxp ,yp Þ ¼ C p þDp
2 X
p
cosð2pf n xp ÞIpn ðxp ,yp Þ
ð1Þ
n¼1
where Ipn ðxp ,yp Þ ¼ Ap þ Bp cos 2pF pn yp ðn ¼ 1,2Þ are the two sinusoidal images to be modulated. (xp,yp) are the projector coordinates. Ap and Bp are the background intensity and contrast respectively. p Cp and Dp are projection constants. f n is called the carrier p p p p frequency and f 1 of 2 . F 1 and F 2 are the frequencies of two sinusoidal images, are called the modulation frequencies and are in the phase direction. F p1 oF p2 , F p2 ¼ mF p1 and m is an integer. The yp dimension is in the direction of the depth deformation and is called the phase dimension. On the other hand, xp dimension is perpendicular to the phase dimension, so is called the orthogonal dimension. The captured intensity image reflected by the object surface is " # 2 X c Ic ðxc ,yc Þ ¼ Rðxc ,yc Þ C c þ Dc ðcos 2pf n xc ÞIcn ðxc ,yc Þ ð2Þ n¼1 c
c
c
c
where (x ,y ) are the image coordinates, R(x ,y ) is the reflectance variation and Icn ðxc ,yc Þ ¼ Ac þ Bc cos ð2pF cn yc þ jn ðxc ,yc ÞÞ
n ¼ 1,2
where F(x,Z), G(x,Z), P(x,Z), and Q(x,Z) respectively represent the 2-D Fourier spectrum of Ic(xc,yc), R(xc,yc)Cc, R(xc,yc)DcAc and ð1=2ÞRðxc ,yc ÞDc Bc exp½ijðxc ,yc Þ. Filter3 is used to filter the corresponding component shown in Fig. 2(a) and then the phase of higher frequency jc2 ðxc ,yc Þ caused by the height of the measured object can be calculated by FTP. Filtering the corresponding components by filter1 and filter2 and then applying inverse Fourier transform and Modulus, the sinusoidal image of lower frequency and the flat image can be obtained. Subtracting the flat image from the sinusoidal image, the zero component disappears in the Fourier spectrum and the measuring range is tripled [6]. Then the phase jc1 ðxc ,yc Þ of lower frequency caused by the height of the measured object can be calculated by FTP. If the lower frequency is selected suitably, jc1 ðxc ,yc Þ is easy to unwrap by the spatial approach. And cc1 ðxc ,yc Þ represents the unwrapping phase of jc1 ðxc ,yc Þ. The unwrapping phase of jc2 ðxc ,yc Þ can be calculated as [16] c mc1 ðxc ,yc Þjc2 ðxc ,yc Þ þ jc2 ðxc ,yc Þ cc2 ðxc ,yc Þ ¼ 2pINT ð5Þ 2p where INT() is an integer operation which represents a rounding to the nearest integer, and m ¼ F p2 =F p1 . 3.2. Advantages discussion
ð3Þ
The Fourier spectrum of Eq. (2) is shown in Fig. 2 and can be described as ( 2 1 X c c c ½Pðx þ f n , ZÞ þ Pðxf n , ZÞ þ Q ðx þf n , Z þ F cn Þ Fðx, ZÞ ¼ Gðx, ZÞ þ 2 n¼1 c c c þQ ðxf n , Z þF cn Þ þ Q n ðx þ f n , ZF cn Þ þ Q n ðxf n , ZF cn Þ ð4Þ
Our research focus is to propose a simple composite grating with high measuring accuracy and large measuring range. Compared with CFTP, the proposed method does not increase the complexity of the composite grating, and has the following advantages: (i) Lower overlapping degree of Fourier spectrum: In experiments, the spectrum overlapping is inevitable [9]. Spectrum overlapping
Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
exists between adjacent carrier channels due to divergence characteristic of the light source, the diffusion of digital light field and frequency spectrum leak of the captured image [17], and is more serious between the adjacent AC components of each channel. Fig. 3 shows the grayscale spectrum distributions of the two gratings in the condition of spectrum overlapped. Compared with Yue’s grating, the proposed grating modulates two different frequencies sinusoidal images along the orthogonal direction, which result in a longer distance between the AC components. So the degree of spectrum overlapping between the AC components lowers. (ii) Higher measuring accuracy: In the CFTP, the single frequency sinusoidal images are used to calculate the phase. The measuring accuracy is limited because the frequency selecting must satisfy the spatial phase unwrapping. But in the optimized grating, two different frequencies are involved and two wrapping phases can be calculated. The lower frequency is equal to Yue’s grating frequency and the unwrapped phase can be obtained by the spatial unwrapping approach. The higher frequency is m times bigger than the lower one. When the value of m is determined, the linearity of the two phases c is considered and a more accurate unwrapped phase c2 ðxc ,yc Þ under higher frequency can be obtained. So the measuring accuracy is improved. (iii) Easier intensity calibration: In the CFTP, the intensities of the two sinusoidal images with p phase difference are changed after demodulation, so they must be calibrated [10,18,19]. Accordingly, the intensity of the flat image extracted from the optimized grating must be calibrated. The calibration of the flat image is simpler than the sinusoidal image. Only the background intensity is included in the flat image, so the calibration of mean value is involved and the calibration of contrast can be omitted. Here a simple calibration method is proposed.Assuming that the sinusoidal image of lower
d
d’
Fig. 3. Overlapping spectrum distributions: (a) spectrum distribution of Yue’s composite grating and (b) spectrum distribution of the optimized composite grating.
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frequency and the flat image extracted from the deformed composite image can be simply described as I1 ðx,yÞ ¼ a1 ðx,yÞ þ b1 ðx,yÞcos j1 ðx,yÞ
ð6Þ
I2 ðx,yÞ ¼ a2 ðx,yÞ
ð7Þ
and their size are i j pixels. The mean values of l1 and l2 are supposed to be 1, m1 and m2 respectively. When the size of the composite image in the phase direction can be divided exactly by the spatial frequency of the sinusoidal image, integral periods are obtained in the sinusoidal image. The mean value of I1(x,y) can be approximately calculated as P Pi,j m1 ¼ i,j x ¼ 1,y ¼ 1 a1 ðx,yÞ=ði jÞ for x ¼ 1,y ¼ 1 b1 ðx,yÞcos j1 ðx,yÞ approximately equal to zero. So we can calibrate the flat image as I02 ðx,yÞ ¼ kI2 ðx,yÞ
ð8Þ
where k¼m1/m2. So calculating the frequency of the lower frequency image in the composite image and choosing a suitable size of the composite image which can be divided by the frequency exactly in the phase direction before the demodulation process, the flat image can be directly calibrated by the above method. (iv) Flat image for other uses besides enlarge measuring range: Without p phase shifting technique, the ICFTP can also extend the measuring range by ingeniously extracted a flat image from the optimized grating. Analyzing the flat image extracted from the deformed composite grating, it can be described as IR2 ðxc ,yc Þ ¼
1 Rðxc ,yc ÞDc Ac 2
ð9Þ
and its distribution only depends on the reflectance variation R(xc,yc). The reflectance variation is sensitive in the shadow area and the steep area of the measured object. So the flat image can be used as a reliability-guided parameter map to guide the phase unwrapping as modulation [20].
4. Numerical simulation and experiments To verify the feasibility of the proposed method, numerical simulation is performed. As shown in Fig. 4(a), a ‘‘cone like’’ with a step in the bottom is measured by both the CFTP and the ICFTP. The peak height of the object is 27.075 mm and the height of the step is 5.7 mm. The corresponding surface reflectance variation is shown in Fig. 4(b), and the size of the image is 756 640 pixels. The object is measured firstly by the CFTP. The two carrier p p frequencies are f 1 ¼ 1=18 pixels and f 2 ¼ 1=6 pixels respectively. The modulation frequencies of two sinusoidal images with a p phase difference are chosen to be 1/16 pixels. If a higher frequency is chosen, the stripe modulated by the height of the object is not continuous and the phase unwrapping is easy to make mistake using the spatial approach. Fig. 5(a) is the measurement result. Fig. 5(b) is the corresponding error distribution and the root mean square (RMS) is 0.3658 mm. Then the ICFTP is used to reconstruct the object. In the optimized composite grating, the carrier frequencies are equal to the ones of the CFTP. The modulation frequencies meet F p2 ¼ 4F p1 . F p1 ¼ 1=16 pixels, which is equal to the modulation frequency in CFTP. Being modulated by the height of the object, the deformed composite pattern is shown in Fig. 6(a), and the corresponding spectral distribution is shown in Fig. 6(b). After demodulating, the sinusoidal image of lower frequency and the flat image extracted from Fig. 6(a) are shown in Fig. 6(c). Subtracting the flat image from the sinusoidal image and then applying Fourier transform of the result image, the 1-D frequency spectrum in the y direction is
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Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
Fig. 4. The measured object: (a) height distribution and (b) reflectance variation.
Fig. 5. Numerical simulation of CFTP: (a) retrieved height and (b) error distribution.
Fig. 6. Numerical simulation of ICFTP: (a) deformed pattern, (b) corresponding spectrum distribution, (c) the sinusoidal image of low frequency and the flat image, (d) spectrum distribution, (e) retrieved height and (f) error distribution.
Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
shown in Fig. 6(d). The measuring range of the lower frequency is enlarged obviously because the zero component is successfully eliminated in the Fourier spectrum by the aid of the flat image. The wrapped phases of the two frequencies can be calculated by the FTP method. The lower spatial frequency is selected suitably and the wrapped phase is easy to be unwrapped by the spatial approach. With the guidance of the phase of lower frequency, an accurate phase of higher frequency can be unwrapped and the corresponding height distribution of the measured object is shown in Fig. 6(e). Fig. 6(f) is the error distribution and the RMS is 0.1457 mm. Comparing Fig. 5(b) and Fig. 6(f), the error in the edge of the step in Fig. 6(f) is smaller. It shows that the ICFTP can improve the measuring accuracy notably. In order to verify the practicability of the proposed method, the ‘‘Mikey’’ and a ‘‘cube’’ with height of 15 mm shown in Fig. 7(a) are measured in the experiment. A Hitachi HCP-70X Digital Light Processor (DLP) projector and a Nikon CCD camera (4320 3240 pixels spatial resolution, 8-bit intensity resolution) are used to project and collect the image.
2041
The objects are firstly retrieved by the CFTP and the height distribution is shown in Fig. 7(b). Then the ICFTP is used to retrieve the objects. In the projection grating, the carrier frequencies are equal to the ones of the CFTP, and the modulation frequencies meet F p2 ¼ 2F p1 . F p1 is also equal to the modulation frequency in CFTP. Fig. 8(a) is the deformed composite image and its size is 1479 958 pixels. Fig. 8(b) is the corresponding spectrum distribution. The sinusoidal image of lower frequency and the flat image extracted from the composite image are shown in Fig. 8(c). We can see that the flat image can reflect the reflectance variation of the object, which can be used as a reliability-guided parameter map to guide the phase unwrapping as modulation. Subtracting the flat image form the sinusoidal image and then applying Fourier transform of the result image, the 1-D frequency spectrum in the y direction is shown in Fig. 8(d). The zero component is not completely eliminated because the intensity of the flat image is not equal to the background intensity of the sinusoidal image after demodulating from the composite image. So the flat image must be calibrated
Fig. 7. Experiment of CFTP: (a) measured object and (b) retrieved height.
Fig. 8. Experiment of ICFTP: (a) deformed pattern, (b) spectrum distribution of (a), (c) the sinusoidal image of low frequency and the flat image, (d) spectrum distribution before calibration, (e) spectrum distribution after calibration, and (f) retrieved height.
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Fig. 9. Comparison of the measurement result: (a) cross sections of Fig. 7(b) and Fig. 8(f) in row 400, and (b) cross sections of Fig. 7(b) and Fig. 8(f) in column 1100.
before subtracting operation [10,18,19]. After calibrating, the frequency spectrum is shown in Fig. 8(e). The retrieved height distribution is shown in Fig. 8(f). Comparing the two measuring results, the height retrieved by the ICFTP is clearer than the one by the CFTP in the mutation part of the object. Fig. 9(a) and (b) are the cross sections in row 700 and column 1100 of the two measuring results respectively. The dash line denotes the actual height of the ‘‘cube’’ and the solid line and dotted line denote the retrieved height using ICFTP and CFTP respectively. We can see that the retrieved height of ICFTP is more close to the actual height. The measurement RMS of the CFTP and ICFTP in Fig. 9(b) are 0.8127 mm and 0.6293 mm respectively. So the ICFTP has a higher accuracy than the CFTP.
5. Conclusions With two sinusoidal images of different frequencies modulated along the orthogonal direction, the Fourier spectrum distribution is more reasonable and the overlapping degree of Fourier spectrum is lowered in the optimized composite grating. The measuring range is extended by using a flat image which can be easily extracted from the carrier channel which modulates the higher frequency sinusoidal image. The measuring accuracy is improved because two frequencies (one is equal to Yue’s and the other one is m times bigger than that of Yue’s) are used to calculate two wrapped phases. The higher frequency phase can be unwrapped by the aid of the lower frequency phase. So the proposed ICFTP can enlarge the measuring range and improve the measuring accuracy at the same time. Numerical simulation and experiments have verified the feasibility and validity of the proposed method.
Acknowledgments This work was supported by the 863 National Plan Foundation of China under Grant no. 2007AA01Z333 and Special Grand National Project of China under Grant no. 2009ZX02204-008.
References [1] Blais F. Review of 20 years of range sensor development. Journal of Electronic Imaging 2004;13:231–40. [2] Zhang S. Recent progress on real-time 3D shape measurement using digital fringe projection techniques. Optics and Lasers in Engineering 2010;48: 149–58. [3] Huang L, Qian KM, Pan B, Asundi AK. Comparison of Fourier transform, Windowed Fourier transform, and wavelet transform methods for phase extraction from a fringe pattern in fringe projection profilometry. Optics and Lasers in Engineering 2010;48:141–8. [4] Gorthi SS, Rastogi P. Fringe projection technique: Whither we are? Optics and Lasers in Engineering 2010;48:133–40. [5] Takeda M, Motoh K. Fourier transform profilomerty for the automatic measurement of 3-D object shapes. Applied Optics 1983;22:3977–82. [6] Li J, Su XY, Guo LR. Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes. Optical Engineering 1990;29:1439–44. [7] Guan C, Hassebrook LG, Lau DL. Composite structrured light pattern for threedemensional video. Optics Express 2003;11:406–17. [8] He YH, Cao YP. Three-dimensional measurement method with orthogonal composite grating aided by fringe contrast and background calibration. Optical Engineering 2010;49:073603. [9] He YH, Cao YP. A composite-structured-light 3D measurement method based on fringe parameter calibration. Optics and Lasers in Engineering 2011;49: 773–9. [10] Yue HM, Su XY, Liu YZ. Fourier transform profilometry based on composite structured light pattern. Optics and Laser Technology 2007;39:1170–5. [11] Li JL, Su HJ, Su XY. Two-frequency used in phase-measuring profilometry. Applied Optics 1997;36:277–80. [12] Ikeda Y, Yoneyama S, Fujigaki M. Absolute phase analysis method for threedimensional surface profilometry using frequency-modulated grating. Optical Engineering 2003;42:1249–56. [13] Qiao NS, Cai XH, Yao CM. Measurement of a discontinous object based on a dual-frequency grating. Chinese Physics B 2009;18:4881–6. [14] Zhou WS, Su XY. A direct mapping algorithm for phase-measurement profilometry. Journal of Modern Optics 1994;41:89–94. [15] Li WS, Su XY, Liu ZB. Large-scale three-dimensional object measurement: a practical coordinate mapping and image data-patching method. Applied Optics 2001;40:3326–33. [16] Kinell L. Spatiotemporal approach for real-time absolute shape measurement by use of projected fringes. Applied Optics 2004;43:3018–27. [17] Chen WJ, Su XY, Tan SX. Discussion on phase error caused by frequency leakage in FTP. Acta Optica Sinica 2000;20:1429–34. [18] Li WS, Su XY. Application of improved phase-measuring profilometry in nonconstant environmental light. Optical Engineering 2001;40:478–85. [19] Chen WJ, Su XY, Cao YP, Xiang LQ. Improved Fourier transform profilometry based on bicolor fringe pattern. Optical Engineering 2004;43:192–8. [20] Su XY, Chen WJ. Reliability-guided phase unwrapping algorithm: a review. Optics and Lasers in Engineering 2004;42:245–61.
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Improved composite Fourier transform profilometry Ying-chun Wu, Yi-ping Cao n, Zhen-fen Huang, Ming-teng Lu, De-liang Chen Department of Opto-Electronics, Sichuan University, Chengdu 610065, China
a r t i c l e i n f o
abstract
Article history: Received 17 December 2011 Received in revised form 6 March 2012 Accepted 22 March 2012 Available online 11 April 2012
An improved composite Fourier transform profilometry (ICFTP) is proposed through optimizing the composite grating. Compared with the composite Fourier transform profilometry (CFTP), the frequencies of the two sinusoidal images modulated along the orthogonal direction in the optimized composite grating are not equal. The overlapping degree of Fourier spectrum of the optimized grating is lowered. The measuring accuracy is improved because two frequencies are used to calculate phase more accurately. By using a more advantageous flat image extracted ingeniously from the optimized grating, the zero component of the lower frequency sinusoidal image in the Fourier spectrum can be eliminated. So the measuring range is extended the same as that by the p phase shifting technique. Both numerical simulation and experiment demonstrate the feasibility and validity of the proposed method. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Composite grating Measuring range Measuring accuracy
1. Introduction New technology brings about a lot of new methods in optical 3-D sensing [1]. Based on digital fringe projection techniques, fringe projection profilometry has a huge potential for applications in many areas, including on-line inspection, medical sciences, reverse engineering, computer graphics and security [2–4]. With the improvement of parallel operation speed and the development of recent technologies including digital image, digital projection and digital display, people have higher level of requirements in measuring speed and accuracy. Real-time 3-D sharp measurement just as on-line inspection and dynamic profilometry becomes a hot topic. Fourier transform profilometry (FTP) is one of the most used methods in real-time 3-D sharp measurement due to only one sinusoidal image projection [5]. By analyzing the captured image using the Fourier method, FTP can demodulate the object’s shape from the fundamental component in the Fourier spectrum. Because the filtering process is involved, the requirement of preventing spectral components from overlapping influences the maximal slope of height variation of measured object (i.e., measuring range) and measuring accuracy. Su et al. proposed a p phase shifting technique which has successfully tripled the measuring range by eliminating zero component from the Fourier spectrum [6]. But two sinusoidal images projection influences the instantaneous character of FTP. By using Guan’s composite structured pattern [7–9], Yue et al. proposed the composite Fourier transform profilometry (CFTP) by modulating two separate sinusoidal images with a p phase difference along the
n
Corresponding author. E-mail address: [email protected] (Y.-p. Cao).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.03.030
orthogonal direction in a composite grating. Her method can eliminate the zero component but only one image projection [10]. All the methods mentioned above used single frequency image to calculate the phase, the measuring accuracy were limited. Dualfrequency image projection is a common method which can be used to improve the measuring accuracy [11,12]. But if a higher measuring range is required at the same time, the p phase shifting technique is needed, which also results in two dual-frequency images projection and time consuming [13]. An improved composite Fourier transform profilometry (ICFTP) is proposed in the paper. The composite grating is optimized by modulating two sinusoidal images with different frequencies along the orthogonal direction. Compared with Yue’s grating in CFTP, the overlapping degree of the Fourier spectrum in the ICFTP is lowered because the distance between the alternating-current (AC) components of the two carrier channels is increased. Without modulating two sinusoidal images with equal frequency and p phase difference, the measuring range can also be tripled by using a flat image which is extracted from the composite grating. With the two different frequencies, two wrapped phases can be calculated from the optimized composite grating. The measuring accuracy is improved because a more accurate unwrapping phase of higher frequency can obtain with the guidance of the of lower frequency phase.
2. Brief description of the composite Fourier transform profilometry (CFTP) In order to enlarge the measuring range of FTP, Yue et al. proposed the CFTP which adopted the p phase shifting technique in the composite structured light projection system [7,10]. The modulation and demodulation processes are shown in Fig. 1.
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Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
Carrier channels
Modulating images
Composite grating Phase dimension
Object
Orthogonal dimension
Deformed sinusoidal images
Spectrum distribution
Deformed image
Retrieved height Filter, IFFT,
FFT
Modulus
Fig. 1. Schematic diagram of the modulation and demodulation.
Two separate sinusoidal images with a p phase difference are modulated into two carrier waves to form a composite grating. The composite grating is then projected on the measured object. Through demodulating the captured deformed composite image, the two sinusoidal images were extracted to calculate the phase of the object by the FTP. After the phase unwrapping, the height of the object can be restored with the height-phase mapping algorithm [14,15].
filter3 filter1
filter2 3. Improved composite Fourier transform profilometry (ICFTP) 3.1. Principle of the ICFTP
Fig. 2. Spectrum distribution of the optimized composite grating.
In the ICFTP, the composite grating is optimized by replacing one of the sinusoidal images with a higher frequency one. The projected composite grating can be written as Ip ðxp ,yp Þ ¼ C p þDp
2 X
p
cosð2pf n xp ÞIpn ðxp ,yp Þ
ð1Þ
n¼1
where Ipn ðxp ,yp Þ ¼ Ap þ Bp cos 2pF pn yp ðn ¼ 1,2Þ are the two sinusoidal images to be modulated. (xp,yp) are the projector coordinates. Ap and Bp are the background intensity and contrast respectively. p Cp and Dp are projection constants. f n is called the carrier p p p p frequency and f 1 of 2 . F 1 and F 2 are the frequencies of two sinusoidal images, are called the modulation frequencies and are in the phase direction. F p1 oF p2 , F p2 ¼ mF p1 and m is an integer. The yp dimension is in the direction of the depth deformation and is called the phase dimension. On the other hand, xp dimension is perpendicular to the phase dimension, so is called the orthogonal dimension. The captured intensity image reflected by the object surface is " # 2 X c Ic ðxc ,yc Þ ¼ Rðxc ,yc Þ C c þ Dc ðcos 2pf n xc ÞIcn ðxc ,yc Þ ð2Þ n¼1 c
c
c
c
where (x ,y ) are the image coordinates, R(x ,y ) is the reflectance variation and Icn ðxc ,yc Þ ¼ Ac þ Bc cos ð2pF cn yc þ jn ðxc ,yc ÞÞ
n ¼ 1,2
where F(x,Z), G(x,Z), P(x,Z), and Q(x,Z) respectively represent the 2-D Fourier spectrum of Ic(xc,yc), R(xc,yc)Cc, R(xc,yc)DcAc and ð1=2ÞRðxc ,yc ÞDc Bc exp½ijðxc ,yc Þ. Filter3 is used to filter the corresponding component shown in Fig. 2(a) and then the phase of higher frequency jc2 ðxc ,yc Þ caused by the height of the measured object can be calculated by FTP. Filtering the corresponding components by filter1 and filter2 and then applying inverse Fourier transform and Modulus, the sinusoidal image of lower frequency and the flat image can be obtained. Subtracting the flat image from the sinusoidal image, the zero component disappears in the Fourier spectrum and the measuring range is tripled [6]. Then the phase jc1 ðxc ,yc Þ of lower frequency caused by the height of the measured object can be calculated by FTP. If the lower frequency is selected suitably, jc1 ðxc ,yc Þ is easy to unwrap by the spatial approach. And cc1 ðxc ,yc Þ represents the unwrapping phase of jc1 ðxc ,yc Þ. The unwrapping phase of jc2 ðxc ,yc Þ can be calculated as [16] c mc1 ðxc ,yc Þjc2 ðxc ,yc Þ þ jc2 ðxc ,yc Þ cc2 ðxc ,yc Þ ¼ 2pINT ð5Þ 2p where INT() is an integer operation which represents a rounding to the nearest integer, and m ¼ F p2 =F p1 . 3.2. Advantages discussion
ð3Þ
The Fourier spectrum of Eq. (2) is shown in Fig. 2 and can be described as ( 2 1 X c c c ½Pðx þ f n , ZÞ þ Pðxf n , ZÞ þ Q ðx þf n , Z þ F cn Þ Fðx, ZÞ ¼ Gðx, ZÞ þ 2 n¼1 c c c þQ ðxf n , Z þF cn Þ þ Q n ðx þ f n , ZF cn Þ þ Q n ðxf n , ZF cn Þ ð4Þ
Our research focus is to propose a simple composite grating with high measuring accuracy and large measuring range. Compared with CFTP, the proposed method does not increase the complexity of the composite grating, and has the following advantages: (i) Lower overlapping degree of Fourier spectrum: In experiments, the spectrum overlapping is inevitable [9]. Spectrum overlapping
Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
exists between adjacent carrier channels due to divergence characteristic of the light source, the diffusion of digital light field and frequency spectrum leak of the captured image [17], and is more serious between the adjacent AC components of each channel. Fig. 3 shows the grayscale spectrum distributions of the two gratings in the condition of spectrum overlapped. Compared with Yue’s grating, the proposed grating modulates two different frequencies sinusoidal images along the orthogonal direction, which result in a longer distance between the AC components. So the degree of spectrum overlapping between the AC components lowers. (ii) Higher measuring accuracy: In the CFTP, the single frequency sinusoidal images are used to calculate the phase. The measuring accuracy is limited because the frequency selecting must satisfy the spatial phase unwrapping. But in the optimized grating, two different frequencies are involved and two wrapping phases can be calculated. The lower frequency is equal to Yue’s grating frequency and the unwrapped phase can be obtained by the spatial unwrapping approach. The higher frequency is m times bigger than the lower one. When the value of m is determined, the linearity of the two phases c is considered and a more accurate unwrapped phase c2 ðxc ,yc Þ under higher frequency can be obtained. So the measuring accuracy is improved. (iii) Easier intensity calibration: In the CFTP, the intensities of the two sinusoidal images with p phase difference are changed after demodulation, so they must be calibrated [10,18,19]. Accordingly, the intensity of the flat image extracted from the optimized grating must be calibrated. The calibration of the flat image is simpler than the sinusoidal image. Only the background intensity is included in the flat image, so the calibration of mean value is involved and the calibration of contrast can be omitted. Here a simple calibration method is proposed.Assuming that the sinusoidal image of lower
d
d’
Fig. 3. Overlapping spectrum distributions: (a) spectrum distribution of Yue’s composite grating and (b) spectrum distribution of the optimized composite grating.
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frequency and the flat image extracted from the deformed composite image can be simply described as I1 ðx,yÞ ¼ a1 ðx,yÞ þ b1 ðx,yÞcos j1 ðx,yÞ
ð6Þ
I2 ðx,yÞ ¼ a2 ðx,yÞ
ð7Þ
and their size are i j pixels. The mean values of l1 and l2 are supposed to be 1, m1 and m2 respectively. When the size of the composite image in the phase direction can be divided exactly by the spatial frequency of the sinusoidal image, integral periods are obtained in the sinusoidal image. The mean value of I1(x,y) can be approximately calculated as P Pi,j m1 ¼ i,j x ¼ 1,y ¼ 1 a1 ðx,yÞ=ði jÞ for x ¼ 1,y ¼ 1 b1 ðx,yÞcos j1 ðx,yÞ approximately equal to zero. So we can calibrate the flat image as I02 ðx,yÞ ¼ kI2 ðx,yÞ
ð8Þ
where k¼m1/m2. So calculating the frequency of the lower frequency image in the composite image and choosing a suitable size of the composite image which can be divided by the frequency exactly in the phase direction before the demodulation process, the flat image can be directly calibrated by the above method. (iv) Flat image for other uses besides enlarge measuring range: Without p phase shifting technique, the ICFTP can also extend the measuring range by ingeniously extracted a flat image from the optimized grating. Analyzing the flat image extracted from the deformed composite grating, it can be described as IR2 ðxc ,yc Þ ¼
1 Rðxc ,yc ÞDc Ac 2
ð9Þ
and its distribution only depends on the reflectance variation R(xc,yc). The reflectance variation is sensitive in the shadow area and the steep area of the measured object. So the flat image can be used as a reliability-guided parameter map to guide the phase unwrapping as modulation [20].
4. Numerical simulation and experiments To verify the feasibility of the proposed method, numerical simulation is performed. As shown in Fig. 4(a), a ‘‘cone like’’ with a step in the bottom is measured by both the CFTP and the ICFTP. The peak height of the object is 27.075 mm and the height of the step is 5.7 mm. The corresponding surface reflectance variation is shown in Fig. 4(b), and the size of the image is 756 640 pixels. The object is measured firstly by the CFTP. The two carrier p p frequencies are f 1 ¼ 1=18 pixels and f 2 ¼ 1=6 pixels respectively. The modulation frequencies of two sinusoidal images with a p phase difference are chosen to be 1/16 pixels. If a higher frequency is chosen, the stripe modulated by the height of the object is not continuous and the phase unwrapping is easy to make mistake using the spatial approach. Fig. 5(a) is the measurement result. Fig. 5(b) is the corresponding error distribution and the root mean square (RMS) is 0.3658 mm. Then the ICFTP is used to reconstruct the object. In the optimized composite grating, the carrier frequencies are equal to the ones of the CFTP. The modulation frequencies meet F p2 ¼ 4F p1 . F p1 ¼ 1=16 pixels, which is equal to the modulation frequency in CFTP. Being modulated by the height of the object, the deformed composite pattern is shown in Fig. 6(a), and the corresponding spectral distribution is shown in Fig. 6(b). After demodulating, the sinusoidal image of lower frequency and the flat image extracted from Fig. 6(a) are shown in Fig. 6(c). Subtracting the flat image from the sinusoidal image and then applying Fourier transform of the result image, the 1-D frequency spectrum in the y direction is
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Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
Fig. 4. The measured object: (a) height distribution and (b) reflectance variation.
Fig. 5. Numerical simulation of CFTP: (a) retrieved height and (b) error distribution.
Fig. 6. Numerical simulation of ICFTP: (a) deformed pattern, (b) corresponding spectrum distribution, (c) the sinusoidal image of low frequency and the flat image, (d) spectrum distribution, (e) retrieved height and (f) error distribution.
Ying-chun Wu et al. / Optics & Laser Technology 44 (2012) 2037–2042
shown in Fig. 6(d). The measuring range of the lower frequency is enlarged obviously because the zero component is successfully eliminated in the Fourier spectrum by the aid of the flat image. The wrapped phases of the two frequencies can be calculated by the FTP method. The lower spatial frequency is selected suitably and the wrapped phase is easy to be unwrapped by the spatial approach. With the guidance of the phase of lower frequency, an accurate phase of higher frequency can be unwrapped and the corresponding height distribution of the measured object is shown in Fig. 6(e). Fig. 6(f) is the error distribution and the RMS is 0.1457 mm. Comparing Fig. 5(b) and Fig. 6(f), the error in the edge of the step in Fig. 6(f) is smaller. It shows that the ICFTP can improve the measuring accuracy notably. In order to verify the practicability of the proposed method, the ‘‘Mikey’’ and a ‘‘cube’’ with height of 15 mm shown in Fig. 7(a) are measured in the experiment. A Hitachi HCP-70X Digital Light Processor (DLP) projector and a Nikon CCD camera (4320 3240 pixels spatial resolution, 8-bit intensity resolution) are used to project and collect the image.
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The objects are firstly retrieved by the CFTP and the height distribution is shown in Fig. 7(b). Then the ICFTP is used to retrieve the objects. In the projection grating, the carrier frequencies are equal to the ones of the CFTP, and the modulation frequencies meet F p2 ¼ 2F p1 . F p1 is also equal to the modulation frequency in CFTP. Fig. 8(a) is the deformed composite image and its size is 1479 958 pixels. Fig. 8(b) is the corresponding spectrum distribution. The sinusoidal image of lower frequency and the flat image extracted from the composite image are shown in Fig. 8(c). We can see that the flat image can reflect the reflectance variation of the object, which can be used as a reliability-guided parameter map to guide the phase unwrapping as modulation. Subtracting the flat image form the sinusoidal image and then applying Fourier transform of the result image, the 1-D frequency spectrum in the y direction is shown in Fig. 8(d). The zero component is not completely eliminated because the intensity of the flat image is not equal to the background intensity of the sinusoidal image after demodulating from the composite image. So the flat image must be calibrated
Fig. 7. Experiment of CFTP: (a) measured object and (b) retrieved height.
Fig. 8. Experiment of ICFTP: (a) deformed pattern, (b) spectrum distribution of (a), (c) the sinusoidal image of low frequency and the flat image, (d) spectrum distribution before calibration, (e) spectrum distribution after calibration, and (f) retrieved height.
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Fig. 9. Comparison of the measurement result: (a) cross sections of Fig. 7(b) and Fig. 8(f) in row 400, and (b) cross sections of Fig. 7(b) and Fig. 8(f) in column 1100.
before subtracting operation [10,18,19]. After calibrating, the frequency spectrum is shown in Fig. 8(e). The retrieved height distribution is shown in Fig. 8(f). Comparing the two measuring results, the height retrieved by the ICFTP is clearer than the one by the CFTP in the mutation part of the object. Fig. 9(a) and (b) are the cross sections in row 700 and column 1100 of the two measuring results respectively. The dash line denotes the actual height of the ‘‘cube’’ and the solid line and dotted line denote the retrieved height using ICFTP and CFTP respectively. We can see that the retrieved height of ICFTP is more close to the actual height. The measurement RMS of the CFTP and ICFTP in Fig. 9(b) are 0.8127 mm and 0.6293 mm respectively. So the ICFTP has a higher accuracy than the CFTP.
5. Conclusions With two sinusoidal images of different frequencies modulated along the orthogonal direction, the Fourier spectrum distribution is more reasonable and the overlapping degree of Fourier spectrum is lowered in the optimized composite grating. The measuring range is extended by using a flat image which can be easily extracted from the carrier channel which modulates the higher frequency sinusoidal image. The measuring accuracy is improved because two frequencies (one is equal to Yue’s and the other one is m times bigger than that of Yue’s) are used to calculate two wrapped phases. The higher frequency phase can be unwrapped by the aid of the lower frequency phase. So the proposed ICFTP can enlarge the measuring range and improve the measuring accuracy at the same time. Numerical simulation and experiments have verified the feasibility and validity of the proposed method.
Acknowledgments This work was supported by the 863 National Plan Foundation of China under Grant no. 2007AA01Z333 and Special Grand National Project of China under Grant no. 2009ZX02204-008.
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