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Non-phase-shifting Fourier transform profilometry using single grating pattern and object image
Optik 127 (2016) 7565–7571
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Non-phase-shifting Fourier transform profilometry using single grating pattern and object image Jaroon Wongjarern, Joewono Widjaja ∗ , Porntip Chuamchaitrakool, Panomsak Meemon School of Physics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand
a r t i c l e
i n f o
Article history: Received 24 March 2016 Accepted 26 May 2016 Keywords: Fourier transform profilometry Background elimination Object image 3-D shape measurement Non-phase shifting method
a b s t r a c t A new white light non-phase-shifting method for eliminating unwanted background in Fourier transform profilometry (FTP) is proposed by using an object image being measured and a single grating image deformed by this object. The background signal of the deformed grating image can be eliminated by using the object image scaled by a contrast ratio of the two images. The proposed method has an advantage over color-encoded FTPs in that the contrast value can be simply calculated from the image itself, regardless of image sensors. Experimental verifications of the proposed method are presented. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction Fourier transform profilometry (FTP) is one of the useful three-dimensional (3-D) shape measurement methods [1–3]. Unlike moiré topography which suffers from fixed 2 sensitivity and requirement of scanning components [4], the FTP is free from these limitations and can perform full field measurement by using simple white-light based setup. When a sinusoidal grating or a Ronchi grating is projected onto an object surface being studied, phase of the projected grating pattern is modulated by spatial profile of the object. This phase modulation is encoded into fundamental frequency spectra of the grating pattern. By recording the deformed grating pattern with an image acquisition sensor, this phase information can be retrieved from the fundamental spectrum by using Fourier transformations. The retrieved phase information is then employed for reconstructing 3-D object surface profile. However besides fundamental components, deformed grating images may also contain lower and higher orders of spectra. When the fundamental component has broad bandwidth, it may be corrupted by the other spectra. This is the inherent drawback of the conventional FTP. In the past decades, several methods for solving this drawback have been reported [5–12]. In the case of the FTP by using the Ronchi grating projection, elimination of unwanted spectra higher than the fundamental one has been proposed by defocusing the projected pattern [5,6]. Spatial translation of the grating plane or manual adjustment of a focus of projector’s lens defocuses the Ronchi grating, yielding quasi-sinusoidal light distribution. However, this method suffers from a phase shift error which may arise from imperfect defocusing process. Consequently, complete spectral elimination is hardly achieved. Improved third-order spectral suppression was reported by area-modulating a square grating into a triangular grating [7]. This method works well, provided a triangular grating to be defocused has short spatial period.
∗ Corresponding author. E-mail addresses: [email protected], joe [email protected] (J. Widjaja). http://dx.doi.org/10.1016/j.ijleo.2016.05.107 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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J. Wongjarern et al. / Optik 127 (2016) 7565–7571
LCD projector
y
CCD camera
θ x
Computer system
Object plane
Fig. 1. A schematic diagram of an optical setup for implementing the proposed FTP.
Although high spectral problem may also be solved by utilizing a perfect sinusoidal grating projection, an existence of the zero-order spectrum still limits the FTP performance. An excellent method for removing the zero-order spectrum of the background by using -phase shifting was reported by Li et al. [8]. This background elimination in the space domain gives better accuracy and three times higher measurement range than the conventional FTP. However, it is not suitable for dynamic object reconstructions. This is because four deformed grating patterns, that are the grating images deformed by the object and the reference plane and their -phase-shifted images, are required to reconstruct the 3-D object profile. Alternatively, the zero-order spectrum removal can be accomplished in the frequency domain. Tavares et al. reported a new method for the background elimination by using two orthogonal sinusoidal gratings [9]. The gratings are subsequently projected onto the object under test to generate the deformed grating patterns. The background is eliminated by subtracting two spectra calculated from the two deformed gratings which will result in four fundamental frequencies. To retrieve the desired phase from one of the four fundamental frequency spectra, the unwanted ones are removed by applying a zero threshold. As a result, negative amplitude of the desired fundamental frequencies may be corrupted. Filtering of the zeroorder spectrum by using 2-D continuous wavelet transform has also been reported [10]. On the basis of its multi-resolution property, a bank of wavelet filters needs to be generated from a unique function which has a response of band-pass filter by translation, dilations and rotations. A main concern of the frequency domain processing is that precise extraction of the fundamental frequency is hardly achieved when bandwidth of the deformed grating image is broad. Recently, a projection of two-interlaced color gratings, red and green, with different frequency and one uniform blue color pattern has been proposed for solving this time delay problem [11]. After capturing the deformed pattern by using a color image sensor, three images that are two deformed grating images and one object image are digitally separated. Since a mean intensity value of each pattern is not the same and each color grating has different contrast value, the background information is hardly eliminated without calibrating these values. The calibrations of the mean and the contrast values is done by measuring a transfer function of each encoding color of the image sensor [12]. After the calibrations, the three images are converted into grayscale format. The background elimination can be finally accomplished by subtracting the object image from the grating images. Not only does this method suffer from lengthy processes of measuring the sensor’s color transfer function, but, it also suffers from the image calibrations and separations prior to the background elimination. Furthermore, besides it is not cost effective due to the need of color image sensors, this background elimination method cannot be applied to coherent structured illumination systems. In this paper, white-light non-phase-shifting method for eliminating the unwanted background signal of the FTP by using a single deformed grating pattern and an object image is proposed. In our proposed method, the mean intensity and the contrast values of the deformed grating and the object images are directly calculated from the recorded images. This is the main difference between the proposed method with the color-encoded FTPs described above. After removing digitally the mean intensity from its corresponding image, the resultant object image scaled by its contrast is subtracted from the deformed grating. This eliminates totally the non-uniform background signal. By taking its Fourier transform, a phase information can be retrieved from the fundamental spectrum and is then used to reconstruct the object height in accordance with the conventional FTP. The proposed method has advantages over the previous works in that firstly, uses of a white light illumination and a monochrome image sensor results in low-cost system. Secondly, the calibration process of the mean and the contrast values is simpler and independent upon characteristics of the image sensors. Thirdly, the use of a single grating pattern minimizes simultaneously projection and image acquisition times and phase error caused by abrupt change in amplitude or timing of light projector’s synchronization signals known as jitters [13,14]. Finally, the proposed method can also be applied to coherent structured illumination, because it does not require color encoding. 2. Proposed FTP Fig. 1 shows a schematic diagram of an optical setup for implementing the proposed method which consists of an LCD projector and an image camera aligned in accordance with a crossed-optical-axes geometry [1]. The projector is used to project the white light and the sinusoidal grating pattern to the object being studied, while the monochrome CCD camera captures its corresponding images. The camera and the projector entrance pupils are located at the same distance l0 from a
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Fig. 2. (a) Image of the isosceles prism under white light illumination at the projection angle of 28◦ and (b) its corresponding intensity scanned at the 100th row.
Fig. 3. (a) Image of the grating pattern deformed by the same prism and (b) its corresponding intensity scanned at the 100th row.
zero height reference plane, while their spatial separation is d. Due to this geometry, the angle formed by the two axes is given by arctan (d/l0 ). Consider a test object that is an isosceles prism embedded in a uniform plane plate serving as the zero-height reference plane in the above setup. Under white light illumination from the projector, its captured image shown in Fig. 2(a) can be mathematically expressed as g1 (x, y) = g1dc + o(x, y).
(1)
Here, g1dc stands for the global mean intensity value of the image which is given by gmdc
1 = A
gm (x, y)dA
(2)
R
with A is the area of the mth image region R. o(x,y) corresponds to the irradiance caused by non-uniform light reflection of the object. A black area appeared on top of the prism is a shadow. Due to its projection angle, the light is incident at a bigger angle on the left side of the prism than that on the right side. Fig. 2(b) plots the 1-D cross sectional intensity scanned at the 100th row of the image. The most left and right signals are the irradiances reflected by the uniform plane plate, while a step-shaped signal in the middle of the plot is caused by the irradiance of the prism reflection. When the same prism object is illuminated by a sinusoidal grating pattern, the object profile deforms the grating pattern. Fig. 3(a) shows the image of the deformed grating pattern captured by the camera, while Fig. 3(b) plots its 1-D cross sectional intensity scanned at the same 100th row. The projection angle now causes elongation of the grating pattern incident on the left side of the prism and compression on the right side. Comparison between Figs. Fig. 22(b) and Fig. 33(b) reveals that the irradiance and the pitch of the deformed gratings on the left side become lower and broader than those on the other side, respectively. Since the grating pattern is used in the illumination, the signal intensity exhibits firstly, lower global mean value. Secondly, besides differences in amplitude and frequency, the sinusoidal signals in the left and the right sides have their local mean intensities which vary according to the step-shaped reflection of the prism. The value of this local mean intensity is lower than that of the step-shaped signal in Fig. 2(b), because contrast of the deformed grating image is
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Fig. 4. (a) Plot of a subtraction of the object signal shown in Fig. 2(b) from the grating signal shown in Fig. 3(b) and (b) its corresponding power spectrum.
higher than that of the prism image. By taking the above characteristics into account, the deformed grating image can be mathematically expressed as g2 (x, y) = g2dc + o (x, y) + bo (x, y) cos [2f0 x + (x, y)] ,
(3)
where g2dc is the global mean intensity value of the deformed grating image. b stands for the modulation factor, while f0 corresponds to the fundamental frequency of the observed grating image on the zero height surface. (x,y) is the phase modulation arising from the height profile of the object and the reference plane. o (x,y) represents the irradiance from the grating reflection which is related to the irradiance caused by the non-uniform light reflection by o (x, y) =
c1 o (x, y) . c2
(4)
c1 and c2 are the contrasts of the object and the deformed grating images, respectively. In image science, this image contrast measures a relative strength of an image information and a background defined by [15]
+∞
|G(fx , fy )|dfx dfy −∞
c=
|G(0, 0)|
(5)
,
with G(fx , fy ) corresponds to the amplitude of the frequency spectrum of the image, while G(0,0) is the zero-frequency component. Since the frequency spectrum of the deformed grating image is broader that of the prism, its contrast is higher. The unwanted background of the deformed grating image in Eq. (3) can be eliminated by a simple arithmetic operation
g (x, y) = g2 (x, y) − g2dc (x, y) − [g1 (x, y) − g1dc (x, y)] /(c2 /c1 ) = bo (x, y) cos [2f0 x + (x, y)].
(6)
In Eq. (6), the global mean intensities and the contrast of the two images are calculated by using Eqs. (2) and (5), respectively. The division of the last term by the contrast ratio c2 /c1 minimizes the intensity difference between o(x,y) and o (x,y). Therefore, the third term gives the object reflection. In order to have better insight into Eq. (6), a comparison with the object image subtraction is presented. Fig. 4(a) shows the result of subtraction of the object image g1 (x,y) from the deformed grating g2 (x,y) shown in Figs. Fig. 22(b) and Fig. 33(b), respectively. It is clear the intensity variation of the resultant signal cannot become symmetrical with respect to a zero level intensity and it has negative value because the object image has higher intensity image. Therefore, there is unbalanced local mean intensities in the grating and the object images. In Fig. 4(b), the spatial-frequency domain representation reveals that this local mean intensity gives very strong low-frequency components around the zero frequency. The presence of this very low components may degrade broad fundamental spectra. In the case of calculating Eq. (6) by using signals shown in Figs. Fig. 22(b) and Fig. 33(b), the resultant signal amplitude plotted in Fig. 5(a) varies almost in symmetry about the spatial position x. The elimination of the local mean intensity can be confirmed from Fig. 5(b) which shows that the low-frequency components around the zero frequency vanishes. Therefore, our proposed method ensures that the fundamental spectra shown in Fig. 5(b) can be extracted without degradation. Next, the calculation of Eq. (6) is repeated for the rows corresponding to the reference plane plate of the two images, such as the 450th rows of Figs. Fig. 22(a) and Fig. 33(a). This calculation gives
g (x, y) = br (x, y)cos[2f0 x + 0 (x, y)],
(7)
where r (x,y) is the irradiance from the grating reflection of the reference plane, while 0 (x,y) is the phase modulation arising from the zero-height plane. In contrast to our preliminary work which required additional images of the whole reference plane and its deformed grating for retrieving this phase information [16], the proposed method reduces the projection and
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Fig. 5. (a) Plot of Eq. (6) by using signals illustrated in Figs. Fig. 22(b) and Fig. 33(b) and (b) its corresponding power spectrum.
90 80 70
Height (mm)
60 50 40 30 20
Digital height gauge Conventional FTP π-phase shifting FTP Proposed FTP
10 0 -10 0
80
160
240 320 400 x (pixels)
480
560
640
Fig. 6. Height profiles of the prism reconstructed by using the conventional, the -phase shifting and the proposed FTPs compared with the digital height gauge.
the image acquisition times. The phase difference (x,y) − 0 (x,y) is then extracted by using the algorithm of the conventional FTP and is used to reconstruct the 3-D object profile by [1] h (x, y) =
l0 [ (x, y) − 0 (x, y)] . [ (x, y) − 0 (x, y)] − 2f0 d
(8)
3. Experimental verifications In order to verify feasibility of our proposed method, the isosceles prism with a dimension of 133.12 mm × 70.1 mm × 81.24 mm and a 600 ml plastic water bottle were employed as the test objects. The sinusoidal grating with the pitch p0 (=1/f0 ) of 5.48 mm and the uniform white patterns generated by using the LCD projector (Toshiba TLP-X2000) with resolution 1024 × 768 pixels were subsequently projected onto the objects. The CCD camera (Hamamatsu C5948) with 50-mm focal length Nikkor lens was used to capture the images of the object and the deformed grating on an image sensor. The captured images were saved into tiff file format with resolution of 640 × 480 pixels. The lens magnification and the distance l0 were about 19 times and 1050 mm, respectively. The setup was calibrated by using this isosceles prism. All computations were done by using Matlab program (R2009a). The resultant heights reconstructed by using the proposed method were compared with the ones obtained by using the conventional and the –phase shifting FTPs together with a digital height gauge (Moore and Wright MW190-30DBL) having an accuracy of 0.01 mm. The first experimental verification was the height reconstruction of the isosceles prism. The images of the test prism and its deformed grating obtained at the projection angle of 28◦ are shown in Figs. Fig. 22(a) and Fig. 33(a), respectively. Fig. 6 plots the height profiles of this prism reconstructed by using the proposed, the conventional and the -phase shifting FTPs. They are compared with the measurements done by using the digital height gauge. The phase modulation 0 (x,y) was
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Fig. 7. Images of (a) the plastic bottle and (b) its deformed grating obtained at the projection angle of 8◦ .
Fig. 8. 3-D surface profile of the test bottle reconstructed by using the proposed method.
calculated from the 450th rows of the reference plane of the two images. The results were the average values of the 63 reconstructed heights. The circle symbol denotes the height measured by the height gauge. The dot, the broken and the solid lines represent the heights obtained by the conventional, the -phase shifting and the proposed FTPs, respectively. It can be seen that the conventional FTP gives considerable reconstruction errors. This is because the first-order and the zero-order spectra overlap. On the other hand, by eliminating the unwanted background via the -phase shifting and the proposed FTPs, the two slopes of the prism and the zero-height reference plane can be correctly reconstructed. The reconstructed heights are in good agreement with the measurement by the height gauge. The maximum reconstruction errors of the two FTP methods that are about 0.75% occurs at the prism peak, because of discontinuous frequency change of the sinusoidal signal. The result verifies that our proposed method has the same measurement accuracy as the -phase shifting FTP although it utilizes less number of grating projections. To further verify feasibility of the proposed method, the surface profile of the plastic water bottle was reconstructed. Fig. 7(a) and (b) show the images of the bottle and its deformed grating obtained at the projection angle of 8◦ , respectively. The phase modulation ϕ0 (x,y) was extracted from the 20th row of the two images. Fig. 8 plots the 3-D surface profile of the bottle reconstructed by using our proposed method. For the sake of clarity, the number of pixels in the horizontal and the vertical axes are downsampled. It is obvious that although the neck and the shoulder areas of the bottle consist of multiple curved elements, the bottle profile can be correctly reconstructed by using the images of the deformed grating and the object. 4. Conclusions We have proposed and verified experimentally a new white light non-phase-shifting method for eliminating the unwanted background signal in the FTP by using a single deformed grating pattern and the object image. Our study reveals that the grating image deformed by the object contains the global and the local mean intensities which are associated with the dc and the object reflection signals, respectively. The object reflection is determined from the object image scaled by the contrast ratio of the two images. In our proposed FTP, the contrast value can be simply calculated from the image itself. Simple subtraction of the object reflection from the deformed grating eliminates the unwanted background signal. The proposed method has advantages in that firstly, the measuring system and the calibrations of the mean and the contrast values are more cost effective and simpler than the color-encoded FTPs, respectively. Secondly, its measurement performance is as accurate as that obtained by using the -phase shifting FTP method although it employs a single deformed grating image. Thirdly, it suffers less from phase errors caused by the jitter of the projected grating. Finally, the proposed method can also be applied to coherent structured illumination, because it does not require color encoding.
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Acknowledgement This research was supported by Suranaree University of Technology and the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. References [1] M. Takeda, K. Mutoh, Fourier transform profilometry for the automatic measurement of 3-D object shapes, Appl. Opt. 22 (1983) 3977–3982. [2] M. Takeda, H. Ina, S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. 72 (1982) 156–160. [3] S.S. Gorthi, P. Rastogi, Fringe projection techniques: whither we are? Opt. Lasers Eng. 48 (2009) 133–140. [4] K. Creath, J. Schmit, J.C. Wyant, Optical metrology of diffuse surfaces, in: D. Malacara (Ed.), Optical Shop Testing, 3rd ed., John Wiley and Sons, New York, 2007, pp. 756–807. [5] J. Li, X.Y. Su, L. Guo, Improved Fourier-transform profilometry for automatic measurement of three- dimensional object shapes, Opt. Eng. 29 (1990) 1439–1444. [6] Y. Fu, J. Wu, G. Jiang, Fourier transform profilometry based on defocusing, Opt. Laser Technol. 44 (2012) 727–733. [7] W. Lohry, S. Zhang, Fourier transform profilometry using a binary area modulation technique, Opt. Eng. 51 (2012) 113602. [8] J. Yi, S. Huang, Modified Fourier transform profilometry for the measurement of 3-D steep shapes, Opt. Laser Eng. 27 (1997) 493–505. [9] P.J. Tavares, M.A. Vaz, Orthogonal projection technique for resolution enhancement of the Fourier transform fringe analysis method, Opt. Commun. 266 (2006) 465–468. [10] M.A. Gdeistat, D.R. Burton, M.J. Lalor, Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform, Opt. Commun. 266 (2006) 482–489. [11] F. Da, H. Huang, A novel color fringe projection based Fourier transform 3D shape measurement method, Optik 123 (2012) 2233–2237. [12] W. Chen, X. Su, Y. Cao, L. Xiang, Improving Fourier transform profilometry based on bicolor fringe pattern, Opt. Eng. 43 (2004) 192–198. [13] K.J. Hsiao, T.C. Lee, The design and analysis of a fully integrated multiplying DLL with adaptive current tuning, IEEE J. Solid-State Circuits 43 (2008) 1427–1435. [14] K. Yamaguchi, Y. Hori, K. Nakajima, K. Suzuki, M. Mizuno, H. Hayama, A 2.0 Gb/s lock-embedded interface for full-HD 10-Bit 120 Hz LCD drivers with 1/5-rate noise-tolerant phase and frequency recovery, IEEE J. Solid-State Circuits 44 (2009) 3560–3567. [15] R.F. Hess, A. Bradley, L. Piotrowski, Contrast-coding in amblyopia. I. Differences in the neural basis of human amblyopia, Proc. Soc. London Ser. B 127 (1983) 309–330. [16] J. Wongjarern, J. Widjaja, W. Sangpech, N. Thongdee, P. Santisoonthornwat, O. Traisak, P. Chuamchaitrakool, P. Meemon, Fourier transform profilometry by using digital dc subtraction, Proc. SPIE 9234 (2014) 923412.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Non-phase-shifting Fourier transform profilometry using single grating pattern and object image Jaroon Wongjarern, Joewono Widjaja ∗ , Porntip Chuamchaitrakool, Panomsak Meemon School of Physics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand
a r t i c l e
i n f o
Article history: Received 24 March 2016 Accepted 26 May 2016 Keywords: Fourier transform profilometry Background elimination Object image 3-D shape measurement Non-phase shifting method
a b s t r a c t A new white light non-phase-shifting method for eliminating unwanted background in Fourier transform profilometry (FTP) is proposed by using an object image being measured and a single grating image deformed by this object. The background signal of the deformed grating image can be eliminated by using the object image scaled by a contrast ratio of the two images. The proposed method has an advantage over color-encoded FTPs in that the contrast value can be simply calculated from the image itself, regardless of image sensors. Experimental verifications of the proposed method are presented. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction Fourier transform profilometry (FTP) is one of the useful three-dimensional (3-D) shape measurement methods [1–3]. Unlike moiré topography which suffers from fixed 2 sensitivity and requirement of scanning components [4], the FTP is free from these limitations and can perform full field measurement by using simple white-light based setup. When a sinusoidal grating or a Ronchi grating is projected onto an object surface being studied, phase of the projected grating pattern is modulated by spatial profile of the object. This phase modulation is encoded into fundamental frequency spectra of the grating pattern. By recording the deformed grating pattern with an image acquisition sensor, this phase information can be retrieved from the fundamental spectrum by using Fourier transformations. The retrieved phase information is then employed for reconstructing 3-D object surface profile. However besides fundamental components, deformed grating images may also contain lower and higher orders of spectra. When the fundamental component has broad bandwidth, it may be corrupted by the other spectra. This is the inherent drawback of the conventional FTP. In the past decades, several methods for solving this drawback have been reported [5–12]. In the case of the FTP by using the Ronchi grating projection, elimination of unwanted spectra higher than the fundamental one has been proposed by defocusing the projected pattern [5,6]. Spatial translation of the grating plane or manual adjustment of a focus of projector’s lens defocuses the Ronchi grating, yielding quasi-sinusoidal light distribution. However, this method suffers from a phase shift error which may arise from imperfect defocusing process. Consequently, complete spectral elimination is hardly achieved. Improved third-order spectral suppression was reported by area-modulating a square grating into a triangular grating [7]. This method works well, provided a triangular grating to be defocused has short spatial period.
∗ Corresponding author. E-mail addresses: [email protected], joe [email protected] (J. Widjaja). http://dx.doi.org/10.1016/j.ijleo.2016.05.107 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
7566
J. Wongjarern et al. / Optik 127 (2016) 7565–7571
LCD projector
y
CCD camera
θ x
Computer system
Object plane
Fig. 1. A schematic diagram of an optical setup for implementing the proposed FTP.
Although high spectral problem may also be solved by utilizing a perfect sinusoidal grating projection, an existence of the zero-order spectrum still limits the FTP performance. An excellent method for removing the zero-order spectrum of the background by using -phase shifting was reported by Li et al. [8]. This background elimination in the space domain gives better accuracy and three times higher measurement range than the conventional FTP. However, it is not suitable for dynamic object reconstructions. This is because four deformed grating patterns, that are the grating images deformed by the object and the reference plane and their -phase-shifted images, are required to reconstruct the 3-D object profile. Alternatively, the zero-order spectrum removal can be accomplished in the frequency domain. Tavares et al. reported a new method for the background elimination by using two orthogonal sinusoidal gratings [9]. The gratings are subsequently projected onto the object under test to generate the deformed grating patterns. The background is eliminated by subtracting two spectra calculated from the two deformed gratings which will result in four fundamental frequencies. To retrieve the desired phase from one of the four fundamental frequency spectra, the unwanted ones are removed by applying a zero threshold. As a result, negative amplitude of the desired fundamental frequencies may be corrupted. Filtering of the zeroorder spectrum by using 2-D continuous wavelet transform has also been reported [10]. On the basis of its multi-resolution property, a bank of wavelet filters needs to be generated from a unique function which has a response of band-pass filter by translation, dilations and rotations. A main concern of the frequency domain processing is that precise extraction of the fundamental frequency is hardly achieved when bandwidth of the deformed grating image is broad. Recently, a projection of two-interlaced color gratings, red and green, with different frequency and one uniform blue color pattern has been proposed for solving this time delay problem [11]. After capturing the deformed pattern by using a color image sensor, three images that are two deformed grating images and one object image are digitally separated. Since a mean intensity value of each pattern is not the same and each color grating has different contrast value, the background information is hardly eliminated without calibrating these values. The calibrations of the mean and the contrast values is done by measuring a transfer function of each encoding color of the image sensor [12]. After the calibrations, the three images are converted into grayscale format. The background elimination can be finally accomplished by subtracting the object image from the grating images. Not only does this method suffer from lengthy processes of measuring the sensor’s color transfer function, but, it also suffers from the image calibrations and separations prior to the background elimination. Furthermore, besides it is not cost effective due to the need of color image sensors, this background elimination method cannot be applied to coherent structured illumination systems. In this paper, white-light non-phase-shifting method for eliminating the unwanted background signal of the FTP by using a single deformed grating pattern and an object image is proposed. In our proposed method, the mean intensity and the contrast values of the deformed grating and the object images are directly calculated from the recorded images. This is the main difference between the proposed method with the color-encoded FTPs described above. After removing digitally the mean intensity from its corresponding image, the resultant object image scaled by its contrast is subtracted from the deformed grating. This eliminates totally the non-uniform background signal. By taking its Fourier transform, a phase information can be retrieved from the fundamental spectrum and is then used to reconstruct the object height in accordance with the conventional FTP. The proposed method has advantages over the previous works in that firstly, uses of a white light illumination and a monochrome image sensor results in low-cost system. Secondly, the calibration process of the mean and the contrast values is simpler and independent upon characteristics of the image sensors. Thirdly, the use of a single grating pattern minimizes simultaneously projection and image acquisition times and phase error caused by abrupt change in amplitude or timing of light projector’s synchronization signals known as jitters [13,14]. Finally, the proposed method can also be applied to coherent structured illumination, because it does not require color encoding. 2. Proposed FTP Fig. 1 shows a schematic diagram of an optical setup for implementing the proposed method which consists of an LCD projector and an image camera aligned in accordance with a crossed-optical-axes geometry [1]. The projector is used to project the white light and the sinusoidal grating pattern to the object being studied, while the monochrome CCD camera captures its corresponding images. The camera and the projector entrance pupils are located at the same distance l0 from a
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Fig. 2. (a) Image of the isosceles prism under white light illumination at the projection angle of 28◦ and (b) its corresponding intensity scanned at the 100th row.
Fig. 3. (a) Image of the grating pattern deformed by the same prism and (b) its corresponding intensity scanned at the 100th row.
zero height reference plane, while their spatial separation is d. Due to this geometry, the angle formed by the two axes is given by arctan (d/l0 ). Consider a test object that is an isosceles prism embedded in a uniform plane plate serving as the zero-height reference plane in the above setup. Under white light illumination from the projector, its captured image shown in Fig. 2(a) can be mathematically expressed as g1 (x, y) = g1dc + o(x, y).
(1)
Here, g1dc stands for the global mean intensity value of the image which is given by gmdc
1 = A
gm (x, y)dA
(2)
R
with A is the area of the mth image region R. o(x,y) corresponds to the irradiance caused by non-uniform light reflection of the object. A black area appeared on top of the prism is a shadow. Due to its projection angle, the light is incident at a bigger angle on the left side of the prism than that on the right side. Fig. 2(b) plots the 1-D cross sectional intensity scanned at the 100th row of the image. The most left and right signals are the irradiances reflected by the uniform plane plate, while a step-shaped signal in the middle of the plot is caused by the irradiance of the prism reflection. When the same prism object is illuminated by a sinusoidal grating pattern, the object profile deforms the grating pattern. Fig. 3(a) shows the image of the deformed grating pattern captured by the camera, while Fig. 3(b) plots its 1-D cross sectional intensity scanned at the same 100th row. The projection angle now causes elongation of the grating pattern incident on the left side of the prism and compression on the right side. Comparison between Figs. Fig. 22(b) and Fig. 33(b) reveals that the irradiance and the pitch of the deformed gratings on the left side become lower and broader than those on the other side, respectively. Since the grating pattern is used in the illumination, the signal intensity exhibits firstly, lower global mean value. Secondly, besides differences in amplitude and frequency, the sinusoidal signals in the left and the right sides have their local mean intensities which vary according to the step-shaped reflection of the prism. The value of this local mean intensity is lower than that of the step-shaped signal in Fig. 2(b), because contrast of the deformed grating image is
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Fig. 4. (a) Plot of a subtraction of the object signal shown in Fig. 2(b) from the grating signal shown in Fig. 3(b) and (b) its corresponding power spectrum.
higher than that of the prism image. By taking the above characteristics into account, the deformed grating image can be mathematically expressed as g2 (x, y) = g2dc + o (x, y) + bo (x, y) cos [2f0 x + (x, y)] ,
(3)
where g2dc is the global mean intensity value of the deformed grating image. b stands for the modulation factor, while f0 corresponds to the fundamental frequency of the observed grating image on the zero height surface. (x,y) is the phase modulation arising from the height profile of the object and the reference plane. o (x,y) represents the irradiance from the grating reflection which is related to the irradiance caused by the non-uniform light reflection by o (x, y) =
c1 o (x, y) . c2
(4)
c1 and c2 are the contrasts of the object and the deformed grating images, respectively. In image science, this image contrast measures a relative strength of an image information and a background defined by [15]
+∞
|G(fx , fy )|dfx dfy −∞
c=
|G(0, 0)|
(5)
,
with G(fx , fy ) corresponds to the amplitude of the frequency spectrum of the image, while G(0,0) is the zero-frequency component. Since the frequency spectrum of the deformed grating image is broader that of the prism, its contrast is higher. The unwanted background of the deformed grating image in Eq. (3) can be eliminated by a simple arithmetic operation
g (x, y) = g2 (x, y) − g2dc (x, y) − [g1 (x, y) − g1dc (x, y)] /(c2 /c1 ) = bo (x, y) cos [2f0 x + (x, y)].
(6)
In Eq. (6), the global mean intensities and the contrast of the two images are calculated by using Eqs. (2) and (5), respectively. The division of the last term by the contrast ratio c2 /c1 minimizes the intensity difference between o(x,y) and o (x,y). Therefore, the third term gives the object reflection. In order to have better insight into Eq. (6), a comparison with the object image subtraction is presented. Fig. 4(a) shows the result of subtraction of the object image g1 (x,y) from the deformed grating g2 (x,y) shown in Figs. Fig. 22(b) and Fig. 33(b), respectively. It is clear the intensity variation of the resultant signal cannot become symmetrical with respect to a zero level intensity and it has negative value because the object image has higher intensity image. Therefore, there is unbalanced local mean intensities in the grating and the object images. In Fig. 4(b), the spatial-frequency domain representation reveals that this local mean intensity gives very strong low-frequency components around the zero frequency. The presence of this very low components may degrade broad fundamental spectra. In the case of calculating Eq. (6) by using signals shown in Figs. Fig. 22(b) and Fig. 33(b), the resultant signal amplitude plotted in Fig. 5(a) varies almost in symmetry about the spatial position x. The elimination of the local mean intensity can be confirmed from Fig. 5(b) which shows that the low-frequency components around the zero frequency vanishes. Therefore, our proposed method ensures that the fundamental spectra shown in Fig. 5(b) can be extracted without degradation. Next, the calculation of Eq. (6) is repeated for the rows corresponding to the reference plane plate of the two images, such as the 450th rows of Figs. Fig. 22(a) and Fig. 33(a). This calculation gives
g (x, y) = br (x, y)cos[2f0 x + 0 (x, y)],
(7)
where r (x,y) is the irradiance from the grating reflection of the reference plane, while 0 (x,y) is the phase modulation arising from the zero-height plane. In contrast to our preliminary work which required additional images of the whole reference plane and its deformed grating for retrieving this phase information [16], the proposed method reduces the projection and
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Fig. 5. (a) Plot of Eq. (6) by using signals illustrated in Figs. Fig. 22(b) and Fig. 33(b) and (b) its corresponding power spectrum.
90 80 70
Height (mm)
60 50 40 30 20
Digital height gauge Conventional FTP π-phase shifting FTP Proposed FTP
10 0 -10 0
80
160
240 320 400 x (pixels)
480
560
640
Fig. 6. Height profiles of the prism reconstructed by using the conventional, the -phase shifting and the proposed FTPs compared with the digital height gauge.
the image acquisition times. The phase difference (x,y) − 0 (x,y) is then extracted by using the algorithm of the conventional FTP and is used to reconstruct the 3-D object profile by [1] h (x, y) =
l0 [ (x, y) − 0 (x, y)] . [ (x, y) − 0 (x, y)] − 2f0 d
(8)
3. Experimental verifications In order to verify feasibility of our proposed method, the isosceles prism with a dimension of 133.12 mm × 70.1 mm × 81.24 mm and a 600 ml plastic water bottle were employed as the test objects. The sinusoidal grating with the pitch p0 (=1/f0 ) of 5.48 mm and the uniform white patterns generated by using the LCD projector (Toshiba TLP-X2000) with resolution 1024 × 768 pixels were subsequently projected onto the objects. The CCD camera (Hamamatsu C5948) with 50-mm focal length Nikkor lens was used to capture the images of the object and the deformed grating on an image sensor. The captured images were saved into tiff file format with resolution of 640 × 480 pixels. The lens magnification and the distance l0 were about 19 times and 1050 mm, respectively. The setup was calibrated by using this isosceles prism. All computations were done by using Matlab program (R2009a). The resultant heights reconstructed by using the proposed method were compared with the ones obtained by using the conventional and the –phase shifting FTPs together with a digital height gauge (Moore and Wright MW190-30DBL) having an accuracy of 0.01 mm. The first experimental verification was the height reconstruction of the isosceles prism. The images of the test prism and its deformed grating obtained at the projection angle of 28◦ are shown in Figs. Fig. 22(a) and Fig. 33(a), respectively. Fig. 6 plots the height profiles of this prism reconstructed by using the proposed, the conventional and the -phase shifting FTPs. They are compared with the measurements done by using the digital height gauge. The phase modulation 0 (x,y) was
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Fig. 7. Images of (a) the plastic bottle and (b) its deformed grating obtained at the projection angle of 8◦ .
Fig. 8. 3-D surface profile of the test bottle reconstructed by using the proposed method.
calculated from the 450th rows of the reference plane of the two images. The results were the average values of the 63 reconstructed heights. The circle symbol denotes the height measured by the height gauge. The dot, the broken and the solid lines represent the heights obtained by the conventional, the -phase shifting and the proposed FTPs, respectively. It can be seen that the conventional FTP gives considerable reconstruction errors. This is because the first-order and the zero-order spectra overlap. On the other hand, by eliminating the unwanted background via the -phase shifting and the proposed FTPs, the two slopes of the prism and the zero-height reference plane can be correctly reconstructed. The reconstructed heights are in good agreement with the measurement by the height gauge. The maximum reconstruction errors of the two FTP methods that are about 0.75% occurs at the prism peak, because of discontinuous frequency change of the sinusoidal signal. The result verifies that our proposed method has the same measurement accuracy as the -phase shifting FTP although it utilizes less number of grating projections. To further verify feasibility of the proposed method, the surface profile of the plastic water bottle was reconstructed. Fig. 7(a) and (b) show the images of the bottle and its deformed grating obtained at the projection angle of 8◦ , respectively. The phase modulation ϕ0 (x,y) was extracted from the 20th row of the two images. Fig. 8 plots the 3-D surface profile of the bottle reconstructed by using our proposed method. For the sake of clarity, the number of pixels in the horizontal and the vertical axes are downsampled. It is obvious that although the neck and the shoulder areas of the bottle consist of multiple curved elements, the bottle profile can be correctly reconstructed by using the images of the deformed grating and the object. 4. Conclusions We have proposed and verified experimentally a new white light non-phase-shifting method for eliminating the unwanted background signal in the FTP by using a single deformed grating pattern and the object image. Our study reveals that the grating image deformed by the object contains the global and the local mean intensities which are associated with the dc and the object reflection signals, respectively. The object reflection is determined from the object image scaled by the contrast ratio of the two images. In our proposed FTP, the contrast value can be simply calculated from the image itself. Simple subtraction of the object reflection from the deformed grating eliminates the unwanted background signal. The proposed method has advantages in that firstly, the measuring system and the calibrations of the mean and the contrast values are more cost effective and simpler than the color-encoded FTPs, respectively. Secondly, its measurement performance is as accurate as that obtained by using the -phase shifting FTP method although it employs a single deformed grating image. Thirdly, it suffers less from phase errors caused by the jitter of the projected grating. Finally, the proposed method can also be applied to coherent structured illumination, because it does not require color encoding.
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Acknowledgement This research was supported by Suranaree University of Technology and the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. References [1] M. Takeda, K. Mutoh, Fourier transform profilometry for the automatic measurement of 3-D object shapes, Appl. Opt. 22 (1983) 3977–3982. [2] M. Takeda, H. Ina, S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. 72 (1982) 156–160. [3] S.S. Gorthi, P. Rastogi, Fringe projection techniques: whither we are? Opt. Lasers Eng. 48 (2009) 133–140. [4] K. Creath, J. Schmit, J.C. Wyant, Optical metrology of diffuse surfaces, in: D. Malacara (Ed.), Optical Shop Testing, 3rd ed., John Wiley and Sons, New York, 2007, pp. 756–807. [5] J. Li, X.Y. Su, L. Guo, Improved Fourier-transform profilometry for automatic measurement of three- dimensional object shapes, Opt. Eng. 29 (1990) 1439–1444. [6] Y. Fu, J. Wu, G. Jiang, Fourier transform profilometry based on defocusing, Opt. Laser Technol. 44 (2012) 727–733. [7] W. Lohry, S. Zhang, Fourier transform profilometry using a binary area modulation technique, Opt. Eng. 51 (2012) 113602. [8] J. Yi, S. Huang, Modified Fourier transform profilometry for the measurement of 3-D steep shapes, Opt. Laser Eng. 27 (1997) 493–505. [9] P.J. Tavares, M.A. Vaz, Orthogonal projection technique for resolution enhancement of the Fourier transform fringe analysis method, Opt. Commun. 266 (2006) 465–468. [10] M.A. Gdeistat, D.R. Burton, M.J. Lalor, Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform, Opt. Commun. 266 (2006) 482–489. [11] F. Da, H. Huang, A novel color fringe projection based Fourier transform 3D shape measurement method, Optik 123 (2012) 2233–2237. [12] W. Chen, X. Su, Y. Cao, L. Xiang, Improving Fourier transform profilometry based on bicolor fringe pattern, Opt. Eng. 43 (2004) 192–198. [13] K.J. Hsiao, T.C. Lee, The design and analysis of a fully integrated multiplying DLL with adaptive current tuning, IEEE J. Solid-State Circuits 43 (2008) 1427–1435. [14] K. Yamaguchi, Y. Hori, K. Nakajima, K. Suzuki, M. Mizuno, H. Hayama, A 2.0 Gb/s lock-embedded interface for full-HD 10-Bit 120 Hz LCD drivers with 1/5-rate noise-tolerant phase and frequency recovery, IEEE J. Solid-State Circuits 44 (2009) 3560–3567. [15] R.F. Hess, A. Bradley, L. Piotrowski, Contrast-coding in amblyopia. I. Differences in the neural basis of human amblyopia, Proc. Soc. London Ser. B 127 (1983) 309–330. [16] J. Wongjarern, J. Widjaja, W. Sangpech, N. Thongdee, P. Santisoonthornwat, O. Traisak, P. Chuamchaitrakool, P. Meemon, Fourier transform profilometry by using digital dc subtraction, Proc. SPIE 9234 (2014) 923412.