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The online measurement of optical distortion for glass defect based on the grating projection method
Optik 127 (2016) 2240–2245
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
The online measurement of optical distortion for glass defect based on the grating projection method Yong Jin, Zhen Wang ∗ , Youxing Chen, Zhaoba Wang National Key Lab for Electronic Measurement Technology, North University of China, Taiyuan 030051, China
a r t i c l e
i n f o
Article history: Received 13 March 2015 Accepted 3 November 2015 Keywords: Optic distortion Online measurement One-dimensional Fourier transform Fringe image
a b s t r a c t In order to measure the optical distortion of float glass defect and obtain more accurate online measurement result, a novel online measuring method based on digital raster projection is proposed in the present paper. In this novel method, the deformation degree of optical distortion is represented as the diopter; thus a mathematical model is derived to describe the relationship between diopter and phase of fringe image. The image processing algorithm based on one-dimensional Fourier transform and elimination method of edge effect based on spliced removal method are used to process the fringe image. Finally, a diopter distribution of fringe image is obtained. The result shows that this novel method is able to fulfill the online measurement of the optical distortion with a higher accuracy of diopter which is up to 0.04 D. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The optical distortion is a main defect in glass products, which is also an important technical index in quality test of float glass [1]. The irregular surface and non-uniform refractive index of optical distortion causes the deformation when people look at scenery through the glass. The quality of glass products can be controlled by online measurement of optical distortion; higher accuracy of measurement will give rise to the final quality of glass. Therefore, in recent years, more and more attentions were paid on the optical distortion measurement. Concerning the optical distortion measurement, Qin Wang et al. [2] used Moiré deflection technology for realizing the measurement of lenses’ power. However, this method is not suitable for the measurement in production field due to the high mechanical stability standard required for measuring equipment. The Moiré fringe method is based on the interference of two precision grids at a defined distance. The glass is placed between the two precision grids, and an optical distortion between the grids changes the imaging of the grid on each other and produces intensity variations [3]. This method can calculate the intensity of optical distortion in glass ribbon with high sensitivity, but the dust and vibration in the environment have a great influence on the detection results, which can easily lead to false result.
∗ Corresponding author. Tel.: +86 15034158237. E-mail address: [email protected] (Z. Wang). http://dx.doi.org/10.1016/j.ijleo.2015.11.111 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Meyer and Surrel [4] used phase stepped deflectometry to measure the optical distortion, which was composed of imaging system including fringe display screen, glass holder and camera. The phase stepping method was implemented to calculate the map of optical distortion expressed in millidipoters reciprocal of focal length lens in meter, and the measurement of optical distortion in car windshield is realized. Unfortunately, multi fringe image with equally spaced phase is required to collect simultaneously. Therefore, it is unable to implement the online measurement with this method. As mentioned in [5], the linear Light Emitting Diode (LED) and grating are adopted to generate the fringe, and the online inspection of the glass defect is realized. But the inconsistent light intensity of the linear LED results in the inconsistent fringe intensity, which increases the complexity of fringe image processing. This method measures the optical deformation as a glass defect without quantifying the degree of optical distortion. In recent years, with the remarkable improvement of hardware performance and imaging precision, the digital grating projector has been widely applied in the optical precise measurement [6,7]. The digital grating is well-known for its uniform light distribution, high contrast, long operating life and variable grating pitch compared to grating. The theoretical model and processing algorithm of grating projection method have been improved by specialist in related field [8,9]. In this paper, the online measuring method is proposed based on digital grating projection. The imaging optical structure of online measurement is designed, and the mathematical model between diopter and phase of fringe image is derived. The one-dimensional
Y. Jin et al. / Optik 127 (2016) 2240–2245
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Fourier transform is adopted to process the collected fringe image. The first-order spectrum is extracted by band-pass filter in frequency domain. The phase map of fringe image is achieved by Inverse Fourier Transform. The elimination method of edge effect based on spliced removal method is developed. The diopter map of fringe image is figured out. The online measurement of optical distortion in glass ribbon is realized. The structure of the paper is organized as follows. Section 2 presents the measurement scheme. Section 3 explicates the processing and analyzing of fringe image. Section 4 presents the elimination of edge effect and phase unwrapping. Section 5 gives experiment and result analysis and Section 6 concludes the paper.
the defective modulation will cause the fringe moving from A to B, and the offset between A and B contains the information of defect density or surface curvature. The C is the symmetric point of C relative to the reference plane, H is the intersection point of BC and PC, H H is the vertical through H, the two triangles AHB and PHC are similar. According to the principle of similar triangles, Eq. (1) can be shown as:
2. The measurement scheme
h AB = d l−h
The transmissive grating projection method based on digital grating projector is designed to measure the optical distortion. The measurement principle is shown in Fig. 1. The linear CCD and digital grating projector are placed on the top and bottom sides of reference plane respectively, both of them locate on the same plane and focus on the reference plane. The glass ribbon parallels to the reference plane, and is located below the reference plane. Through the glass ribbon, the fringe from the digital grating projector forms a sinusoidal distribution on the reference plane. The glass ribbon moves at a certain constant speed, the linear CCD captures the fringe on the reference plane with uniform space, and transmits it to the computer, and thus the fringe image is captured. The optical distortion is a defect of glassy state compared with defect-free glass; therefore, the structure of optical path is simplified as the refraction in defect-free glass, as shown in Fig. 2. In which, P and C are optical center of grating projector and linear CCD respectively. The vertical dimension between P and reference plane is l, it is also the vertical dimension between C and the reference plane. The horizontal distance between P and C is d. If there is no defect, the fringe, which is generated from grating projector and captured by the linear CCD, will intersect at A with reference plane. In addition, the glass ribbon containing defect affected by
AB PC PC
=
HH
(1)
HH
The out of plane displacement is set as HH = h, HH = l − h, and = d can be seen in the figure, Eq. (1) can be converted to Eq. (2). (2)
Eq. (3) can be calculated by Eq. (2). h=
AB · l
(3)
d + AB
The point A is located at (x, y) in the fringe image captured by linear CCD. The image point (x, y) moves to B from A on reference plane because of the defective modulation, the corresponding phase of fringe image moves to B (x, y) from A (x, y), the distance of moving is: AB =
B (x, y) − A (x, y) (x, y) = 2f0 2f0
(4)
where f0 is the spatial frequency of the grating which is determined by the grid pitch of the grating. Eq. (5) can be calculated by substituting Eq. (4) to Eq. (3), h(x, y) =
l · (x, y) (x, y) + 2df0
(5)
According to Eq. (5), the relation between the out of plane displacement h(x, y) and phase difference (x, y) is derived. To x, taking the first and second order derivative to h(x, y) respectively, the h (x, y) and h (x, y) are achieved. According to the curvature formula, the curvature radius of image point (x, y) is given by: R(x, y) =
[1 + h (x, y)2 ] h (x, y)
3/2
(6)
According to the relation between diopter D(x, y) and curvature radius R(x, y), the diopter of the glass ribbon can be expressed as: D(x, y) =
(n − 1) = (n − 1) R(x, y)
h (x, y) [1 + h (x, y)2 ]
3/2
(7)
In which the refractive index of glass is n = 1.532. Therefore, the key of measuring optical distortion is to figure out the phase difference map (x, y) = B (x, y) − A (x, y) from the defect-free and defective fringe image. Fig. 1. The online measurement scheme of optical distortion.
3. The processing and analyzing of fringe image
Fig. 2. Optical path of measurement.
Because of the fast production and high measurement resolution, the multi uniformly spaced fringe images are difficult to be captured simultaneously in product line, and the phase shift method is not applicable for online processing of fringe image. For the single fringe image, the one-dimensional Fourier transform is used to solve the phase in this paper. One big fringe image in memory is divided into the multi-equally sized sub-blocks. The multi-core computer and multi-threading are employed to accelerate the processing for meeting the real-time measuring. Fig. 3 shows the sub-block of fringe image with a size of 200 × 200 pixels and resolution of 0.1 mm/pixel × 0.1 mm/pixel, Fig. 3(a) is the defect-free sub-block, and Fig. 3(b) is the ream defect.
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Fig. 3. The defective fringe sub-block: (a) defect-free; (b) ream.
Fig. 5. The position selected fringe curve in fringe image.
Fig. 6. The defect-free curve of fringe and phase.
Fig. 4. The spectrum of fringe image.
The grayscale distribution of the fringe image in Fig. 3 is given as: I(x, y) = a(x, y) + b(x, y)cos(2f0 x +
(x, y))
(8)
where I(x, y) is the fringe intensity, a(x, y) is the background intensity, b(x, y) is the fringe amplitude and (x, y) is the fringe phase; f0 is the spatial frequency of the grating, x and y are the spatial coordinates along the horizontal and vertical directions, respectively. According to the Euler formula, Eq. (9) can be expressed as: I(x, y) = a(x, y) + c(x, y)ej2f0 x + c ∗ (x, y)e−j2f0 x
(9)
Fig. 7. The fringe curve and phase curve of the defect in sub-block.
j (x,y)
where c(x, y) = b(x,y)e2 , c ∗ (x, y) = conjugate complex of c(x, y). To x, taking 1D Fourier transform on both sides of the Eq. (9), we have: F[I(x, y)] = A(f, y) + C(f − f0 , y) + C ∗ (f − f0 , y)
(10)
where A(f, y) denotes the zero-order and noise spectrum, C(f − f0 , y) and C*(f − f0 , y) denote the first-order spectrum. Fig. 4 shows the spectrum for a row of the fringe image. The band-pass filter H(f, y) can be designed to retain the first-order spectrum. According to the convolution theorem, the first-order spectrum is given by: 2C(f − f0 , y) = H(f, y) · F(I(x, y)) The phase map form for C(f, y):
(11)
(x, y) can be achieved by inverse Fourier trans-
Im(b(x, y)e Im{F −1 [C(f, y)]} = arctan (x, y) = arctan Re{F −1 [C(f, y)]} Re(b(x, y)e
(x,y) ) (x,y) )
(12)
Three rows in the sub-block of fringe image are selected: defectfree, defect in middle position, and defect in edge, which are shown in Fig. 5. Figs. 6–8 show the fringe curves at three positions and the corresponding phase curves obtained by inverse Fourier transforms.
Fig. 8. The fringe curve and phase curve of the defect on the edge of sub-block.
The fringe image is cut off in spatial domain for sub-block division. The truncation of fringe curve is processed by Fourier transform. It generates the Gibbs phenomenon in phase curve, which is also known as edge effect, and it is responsible for the both ends’ increase or decrease of phase values. As it is shown in Figs. 6–8, the defect-free phase suggests flat curve, but the phase value demonstrates dramatic changes at defective region. Apart from the changes caused by the defect, the edge effect, which occurs on both sides of the edge of the phase curve in the above three lines, leads to unreliable result occurring near the edge. Furthermore, the edge effect is in particular apparent when the defect is located at the edge of the sub-block. It is of great significance to eliminate
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Fig. 9. The sketch of spliced removal method.
the edge effect for the benefits of guaranteeing the reliability of the edge value. Fig. 12. The wrapped phase map: (a) defect-free; (b) ream.
4. Elimination of edge effect and phase unwrapping In this work, spliced removal method is proposed to eliminate errors caused by edge effects. The realization of this method is described as follows: assuming row coordinate of any rows in subblock is y, the fringe curve with length 200 + 2 M pixels can be obtained by combining the both ends with M pixels at row coordinate y in the left adjacent sub-block, which is shown in Fig. 9. After the Fourier transform of the spliced fringe curve, the spliced M pixels at the left and right ends of the phase curve are removed, and the phase curve with effective length 200 pixels is obtained. The radix-2 FFT algorithm is used to speed up Fourier transform in the concrete implementation. The spliced length M is set as 28 pixels and the length of fringe curve which carries on radix-2 FFT is 256 pixels. Two fringe curves with length 200 pixels in Figs. 6 and 8 are spliced to length 256 pixels, respectively. The length of phase curve obtained by radix-2 FFT is 256 pixels, the effective phase curves with length 200 pixels are achieved by removing the spliced 28 pixels at the left and right ends. The results are shown in Figs. 10 and 11, respectively. Figs. 10 and 11 show that: (1) The edge effect of phase curve is eliminated in effective length, which ensures the reliability of phase value in fringe image subblock. (2) The jump occurs in phase curve, due to the phase value is calculated by the arctan function, and is wrapped in (− , ], which is known as wrapped phase are shown in Fig. 12.
In order to obtain the real phase, the wrapped phase should be unwrapped owing to the interrupted and jumped phase. A pointto-point unwrapping method with highest efficiency is adopted in this paper due to the small fringe deformation caused by optical distortion. It is a process to determine the integer n(x, y) in Eq. (13): (x, y) =
(x, y) + 2n(x, y)
(13)
where (x, y) is wrapped phase, (x, y) is unwrapped phase and n(x, y) is an integer. For each pixel in the wrapped phase map, the phase value between two adjacent pixels along the row direction is compared as follows: n(x, y) =
⎧ ⎨ 0 ⎩
−1 1
| (x, y + 1) − (x, y + 1) − (x, y + 1) −
(x, y)| < (x, y) ≥
(14)
(x, y) ≤ −
According to Eq. (14), the value of next pixel is added or subtracted 2 when the difference between two adjacent pixels exceeds threshold until all pixels in every row are completely unwrapped. The defect-free and defective unwrapping phase map A (x, y) and B (x, y) are shown in Fig. 13. 5. Experiment and result analysis The phase difference map (x, y) is obtained by (x, y) = B (x, y) − A (x, y). Where, A (x, y) is defect-free unwrapping phase map and B (x, y) is defective unwrapping phase map. The distance from
Fig. 10. The defect-free result of spliced removal method: (a) fringe curve; (b) spliced phase curve; (c) effective phase curve.
Fig. 11. The result of defect on the edge of image sub-block: (a) fringe curve; (b) spliced phase curve; (c) effective phase curve.
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Y. Jin et al. / Optik 127 (2016) 2240–2245 Table 1 The 20 diopter values from both methods.
Fig. 13. The unwrapped phase map: (a) defect-free; (b) ream.
Fig. 14. The map of out-of-plane displacement.
either linear CCD or grating projector to reference plane is set as l = 1000 mm. The horizontal distance between linear CCD and grating projector is d = 250 mm. The out-of-plane displacement h(x, y) is calculated by Eq. (5), as shown in Fig. 14. The diopter distribution map D(x, y) is calculated by Eq. (7). Where h (x, y) and h (x, y) are the first and second order difference of h(x, y) in x-direction. h (x, y) = h(x + 1, y) − h(x, y)
(15)
h (x, y) = h (x + 1, y) − h (x, y) = h(x + 2, y) − 2h(x + 1, y) + h(x, y)
(16)
For the glass sample with ream defect, shown in Fig. 3(b), the automatic focal power detector is used to measure diopter. The accuracy of detector is ±0.01 D, 20 measurement values are
Positions in Fig. 15
The measuring value of focal power detector
The measuring value of this paper
Relative difference
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
−0.02 0.01 0.01 −0.01 −0.04 −0.09 0.24 0.05 −0.03 −0.07 −0.06 0.01 0.02 −0.01 0.01 0.01 −0.01 −0.01 0.01 0.01
−0.01 0.00 0.00 0.00 −0.07 −0.07 0.23 0.02 −0.02 −0.05 −0.10 0.01 0.02 0.00 0.00 0.00 0.00 −0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.03 0.02 0.01 0.03 0.01 0.02 0.04 0.00 0.00 0.01 0.01 0.01 0.01 0.00 0.00 0.00
recorded every 1 mm on the central line along x-direction. The comparison of the calculated D(x, y = 100) and measured results is presented in Fig. 15. As it can be seen in Fig. 15, there is no obvious difference in the distribution trends of the diopter measured by two methods. This confirms the effectiveness of the adopted method in this study. The 20 diopter values from both methods are summarized in Table 1. The maximum deviation (0.04 D) between the two measuring methods occurs at the edge of the defect, while the deviation in the defective and defect-free region is less than 0.02 D. Thus, the accuracy of this method reaches 0.04 D. 6. Conclusions In order to measure the optical distortion of float glass defect, this paper proposes an online measuring method based on digital grating projection. The transmissive grating projection method of online measurement using digital grating projector is designed, and the mathematical model between diopter and phase of fringe image is derived. The one-dimensional Fourier transform algorithm and spliced removal method are used to process the fringe image and the diopter map of fringe image is figured out. The experimental result shows that this method is able to realize the online measurement of optical distortion, and the measurement accuracy reaches 0.04 D. There are some advantages of online optical distortion measurement method based on the grating projection, i.e. simple structure, high accuracy and easy to maintain. It can also be predicted that the measurement accuracy will be further improved with the improvement of image processing algorithms and hardware performance. References
Fig. 15. The distribution of measurement value between two method.
[1] J. Neugebauer, Causes of optical distortions in glass, STAHLBAU 83 (2014) 277–288. [2] Q. Wang, J.-c. Yu, H. Chen, Y.-y. Wang, Measurement of lenses’ power using Moiré deflection technology, Opto-Electron. Eng. 35 (2008) 74–79 (in Chinese). [3] Y. Shulian, The measurement of the zebra angle of float glass with virtual Moire fringe method, Appl. Laser 25 (2005) 405–407 (in Chinese). [4] V. Meyer, Y. Surrel, Innovative technology for the glass quality control, in: Proc. International Conference Glass Performance Days, Tampere, Finland, 2007, pp. 782–785.
Y. Jin et al. / Optik 127 (2016) 2240–2245 [5] Y. Jin, Z. Wang, L. Zhu, J. Yang, B. Wei, Study on glass defect inspection based on projecting grating method, J. Test. Eval. 41 (2013) 332–339. [6] X.Y. Su, Q.C. Zhang, Dynamic 3-D shape measurement method: a review, Opt. Lasers Eng. 48 (2010) 191–204. [7] F.P. Da, S.Y. Gai, Flexible three-dimensional measurement technique based on a digital light processing projector, Appl. Opt. 47 (2008) 377–385.
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[8] L.-C. Chen, H.-W. Ho, X.-L. Nguyen, Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction, Opt. Lasers Eng. 48 (2010) 182–190. [9] E. Zappa, G. Busca, Comparison of eight unwrapping algorithms applied to Fourier transform profilometry, Opt. Lasers Eng. 24 (2007) 106–116.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
The online measurement of optical distortion for glass defect based on the grating projection method Yong Jin, Zhen Wang ∗ , Youxing Chen, Zhaoba Wang National Key Lab for Electronic Measurement Technology, North University of China, Taiyuan 030051, China
a r t i c l e
i n f o
Article history: Received 13 March 2015 Accepted 3 November 2015 Keywords: Optic distortion Online measurement One-dimensional Fourier transform Fringe image
a b s t r a c t In order to measure the optical distortion of float glass defect and obtain more accurate online measurement result, a novel online measuring method based on digital raster projection is proposed in the present paper. In this novel method, the deformation degree of optical distortion is represented as the diopter; thus a mathematical model is derived to describe the relationship between diopter and phase of fringe image. The image processing algorithm based on one-dimensional Fourier transform and elimination method of edge effect based on spliced removal method are used to process the fringe image. Finally, a diopter distribution of fringe image is obtained. The result shows that this novel method is able to fulfill the online measurement of the optical distortion with a higher accuracy of diopter which is up to 0.04 D. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The optical distortion is a main defect in glass products, which is also an important technical index in quality test of float glass [1]. The irregular surface and non-uniform refractive index of optical distortion causes the deformation when people look at scenery through the glass. The quality of glass products can be controlled by online measurement of optical distortion; higher accuracy of measurement will give rise to the final quality of glass. Therefore, in recent years, more and more attentions were paid on the optical distortion measurement. Concerning the optical distortion measurement, Qin Wang et al. [2] used Moiré deflection technology for realizing the measurement of lenses’ power. However, this method is not suitable for the measurement in production field due to the high mechanical stability standard required for measuring equipment. The Moiré fringe method is based on the interference of two precision grids at a defined distance. The glass is placed between the two precision grids, and an optical distortion between the grids changes the imaging of the grid on each other and produces intensity variations [3]. This method can calculate the intensity of optical distortion in glass ribbon with high sensitivity, but the dust and vibration in the environment have a great influence on the detection results, which can easily lead to false result.
∗ Corresponding author. Tel.: +86 15034158237. E-mail address: [email protected] (Z. Wang). http://dx.doi.org/10.1016/j.ijleo.2015.11.111 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Meyer and Surrel [4] used phase stepped deflectometry to measure the optical distortion, which was composed of imaging system including fringe display screen, glass holder and camera. The phase stepping method was implemented to calculate the map of optical distortion expressed in millidipoters reciprocal of focal length lens in meter, and the measurement of optical distortion in car windshield is realized. Unfortunately, multi fringe image with equally spaced phase is required to collect simultaneously. Therefore, it is unable to implement the online measurement with this method. As mentioned in [5], the linear Light Emitting Diode (LED) and grating are adopted to generate the fringe, and the online inspection of the glass defect is realized. But the inconsistent light intensity of the linear LED results in the inconsistent fringe intensity, which increases the complexity of fringe image processing. This method measures the optical deformation as a glass defect without quantifying the degree of optical distortion. In recent years, with the remarkable improvement of hardware performance and imaging precision, the digital grating projector has been widely applied in the optical precise measurement [6,7]. The digital grating is well-known for its uniform light distribution, high contrast, long operating life and variable grating pitch compared to grating. The theoretical model and processing algorithm of grating projection method have been improved by specialist in related field [8,9]. In this paper, the online measuring method is proposed based on digital grating projection. The imaging optical structure of online measurement is designed, and the mathematical model between diopter and phase of fringe image is derived. The one-dimensional
Y. Jin et al. / Optik 127 (2016) 2240–2245
2241
Fourier transform is adopted to process the collected fringe image. The first-order spectrum is extracted by band-pass filter in frequency domain. The phase map of fringe image is achieved by Inverse Fourier Transform. The elimination method of edge effect based on spliced removal method is developed. The diopter map of fringe image is figured out. The online measurement of optical distortion in glass ribbon is realized. The structure of the paper is organized as follows. Section 2 presents the measurement scheme. Section 3 explicates the processing and analyzing of fringe image. Section 4 presents the elimination of edge effect and phase unwrapping. Section 5 gives experiment and result analysis and Section 6 concludes the paper.
the defective modulation will cause the fringe moving from A to B, and the offset between A and B contains the information of defect density or surface curvature. The C is the symmetric point of C relative to the reference plane, H is the intersection point of BC and PC, H H is the vertical through H, the two triangles AHB and PHC are similar. According to the principle of similar triangles, Eq. (1) can be shown as:
2. The measurement scheme
h AB = d l−h
The transmissive grating projection method based on digital grating projector is designed to measure the optical distortion. The measurement principle is shown in Fig. 1. The linear CCD and digital grating projector are placed on the top and bottom sides of reference plane respectively, both of them locate on the same plane and focus on the reference plane. The glass ribbon parallels to the reference plane, and is located below the reference plane. Through the glass ribbon, the fringe from the digital grating projector forms a sinusoidal distribution on the reference plane. The glass ribbon moves at a certain constant speed, the linear CCD captures the fringe on the reference plane with uniform space, and transmits it to the computer, and thus the fringe image is captured. The optical distortion is a defect of glassy state compared with defect-free glass; therefore, the structure of optical path is simplified as the refraction in defect-free glass, as shown in Fig. 2. In which, P and C are optical center of grating projector and linear CCD respectively. The vertical dimension between P and reference plane is l, it is also the vertical dimension between C and the reference plane. The horizontal distance between P and C is d. If there is no defect, the fringe, which is generated from grating projector and captured by the linear CCD, will intersect at A with reference plane. In addition, the glass ribbon containing defect affected by
AB PC PC
=
HH
(1)
HH
The out of plane displacement is set as HH = h, HH = l − h, and = d can be seen in the figure, Eq. (1) can be converted to Eq. (2). (2)
Eq. (3) can be calculated by Eq. (2). h=
AB · l
(3)
d + AB
The point A is located at (x, y) in the fringe image captured by linear CCD. The image point (x, y) moves to B from A on reference plane because of the defective modulation, the corresponding phase of fringe image moves to B (x, y) from A (x, y), the distance of moving is: AB =
B (x, y) − A (x, y) (x, y) = 2f0 2f0
(4)
where f0 is the spatial frequency of the grating which is determined by the grid pitch of the grating. Eq. (5) can be calculated by substituting Eq. (4) to Eq. (3), h(x, y) =
l · (x, y) (x, y) + 2df0
(5)
According to Eq. (5), the relation between the out of plane displacement h(x, y) and phase difference (x, y) is derived. To x, taking the first and second order derivative to h(x, y) respectively, the h (x, y) and h (x, y) are achieved. According to the curvature formula, the curvature radius of image point (x, y) is given by: R(x, y) =
[1 + h (x, y)2 ] h (x, y)
3/2
(6)
According to the relation between diopter D(x, y) and curvature radius R(x, y), the diopter of the glass ribbon can be expressed as: D(x, y) =
(n − 1) = (n − 1) R(x, y)
h (x, y) [1 + h (x, y)2 ]
3/2
(7)
In which the refractive index of glass is n = 1.532. Therefore, the key of measuring optical distortion is to figure out the phase difference map (x, y) = B (x, y) − A (x, y) from the defect-free and defective fringe image. Fig. 1. The online measurement scheme of optical distortion.
3. The processing and analyzing of fringe image
Fig. 2. Optical path of measurement.
Because of the fast production and high measurement resolution, the multi uniformly spaced fringe images are difficult to be captured simultaneously in product line, and the phase shift method is not applicable for online processing of fringe image. For the single fringe image, the one-dimensional Fourier transform is used to solve the phase in this paper. One big fringe image in memory is divided into the multi-equally sized sub-blocks. The multi-core computer and multi-threading are employed to accelerate the processing for meeting the real-time measuring. Fig. 3 shows the sub-block of fringe image with a size of 200 × 200 pixels and resolution of 0.1 mm/pixel × 0.1 mm/pixel, Fig. 3(a) is the defect-free sub-block, and Fig. 3(b) is the ream defect.
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Fig. 3. The defective fringe sub-block: (a) defect-free; (b) ream.
Fig. 5. The position selected fringe curve in fringe image.
Fig. 6. The defect-free curve of fringe and phase.
Fig. 4. The spectrum of fringe image.
The grayscale distribution of the fringe image in Fig. 3 is given as: I(x, y) = a(x, y) + b(x, y)cos(2f0 x +
(x, y))
(8)
where I(x, y) is the fringe intensity, a(x, y) is the background intensity, b(x, y) is the fringe amplitude and (x, y) is the fringe phase; f0 is the spatial frequency of the grating, x and y are the spatial coordinates along the horizontal and vertical directions, respectively. According to the Euler formula, Eq. (9) can be expressed as: I(x, y) = a(x, y) + c(x, y)ej2f0 x + c ∗ (x, y)e−j2f0 x
(9)
Fig. 7. The fringe curve and phase curve of the defect in sub-block.
j (x,y)
where c(x, y) = b(x,y)e2 , c ∗ (x, y) = conjugate complex of c(x, y). To x, taking 1D Fourier transform on both sides of the Eq. (9), we have: F[I(x, y)] = A(f, y) + C(f − f0 , y) + C ∗ (f − f0 , y)
(10)
where A(f, y) denotes the zero-order and noise spectrum, C(f − f0 , y) and C*(f − f0 , y) denote the first-order spectrum. Fig. 4 shows the spectrum for a row of the fringe image. The band-pass filter H(f, y) can be designed to retain the first-order spectrum. According to the convolution theorem, the first-order spectrum is given by: 2C(f − f0 , y) = H(f, y) · F(I(x, y)) The phase map form for C(f, y):
(11)
(x, y) can be achieved by inverse Fourier trans-
Im(b(x, y)e Im{F −1 [C(f, y)]} = arctan (x, y) = arctan Re{F −1 [C(f, y)]} Re(b(x, y)e
(x,y) ) (x,y) )
(12)
Three rows in the sub-block of fringe image are selected: defectfree, defect in middle position, and defect in edge, which are shown in Fig. 5. Figs. 6–8 show the fringe curves at three positions and the corresponding phase curves obtained by inverse Fourier transforms.
Fig. 8. The fringe curve and phase curve of the defect on the edge of sub-block.
The fringe image is cut off in spatial domain for sub-block division. The truncation of fringe curve is processed by Fourier transform. It generates the Gibbs phenomenon in phase curve, which is also known as edge effect, and it is responsible for the both ends’ increase or decrease of phase values. As it is shown in Figs. 6–8, the defect-free phase suggests flat curve, but the phase value demonstrates dramatic changes at defective region. Apart from the changes caused by the defect, the edge effect, which occurs on both sides of the edge of the phase curve in the above three lines, leads to unreliable result occurring near the edge. Furthermore, the edge effect is in particular apparent when the defect is located at the edge of the sub-block. It is of great significance to eliminate
Y. Jin et al. / Optik 127 (2016) 2240–2245
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Fig. 9. The sketch of spliced removal method.
the edge effect for the benefits of guaranteeing the reliability of the edge value. Fig. 12. The wrapped phase map: (a) defect-free; (b) ream.
4. Elimination of edge effect and phase unwrapping In this work, spliced removal method is proposed to eliminate errors caused by edge effects. The realization of this method is described as follows: assuming row coordinate of any rows in subblock is y, the fringe curve with length 200 + 2 M pixels can be obtained by combining the both ends with M pixels at row coordinate y in the left adjacent sub-block, which is shown in Fig. 9. After the Fourier transform of the spliced fringe curve, the spliced M pixels at the left and right ends of the phase curve are removed, and the phase curve with effective length 200 pixels is obtained. The radix-2 FFT algorithm is used to speed up Fourier transform in the concrete implementation. The spliced length M is set as 28 pixels and the length of fringe curve which carries on radix-2 FFT is 256 pixels. Two fringe curves with length 200 pixels in Figs. 6 and 8 are spliced to length 256 pixels, respectively. The length of phase curve obtained by radix-2 FFT is 256 pixels, the effective phase curves with length 200 pixels are achieved by removing the spliced 28 pixels at the left and right ends. The results are shown in Figs. 10 and 11, respectively. Figs. 10 and 11 show that: (1) The edge effect of phase curve is eliminated in effective length, which ensures the reliability of phase value in fringe image subblock. (2) The jump occurs in phase curve, due to the phase value is calculated by the arctan function, and is wrapped in (− , ], which is known as wrapped phase are shown in Fig. 12.
In order to obtain the real phase, the wrapped phase should be unwrapped owing to the interrupted and jumped phase. A pointto-point unwrapping method with highest efficiency is adopted in this paper due to the small fringe deformation caused by optical distortion. It is a process to determine the integer n(x, y) in Eq. (13): (x, y) =
(x, y) + 2n(x, y)
(13)
where (x, y) is wrapped phase, (x, y) is unwrapped phase and n(x, y) is an integer. For each pixel in the wrapped phase map, the phase value between two adjacent pixels along the row direction is compared as follows: n(x, y) =
⎧ ⎨ 0 ⎩
−1 1
| (x, y + 1) − (x, y + 1) − (x, y + 1) −
(x, y)| < (x, y) ≥
(14)
(x, y) ≤ −
According to Eq. (14), the value of next pixel is added or subtracted 2 when the difference between two adjacent pixels exceeds threshold until all pixels in every row are completely unwrapped. The defect-free and defective unwrapping phase map A (x, y) and B (x, y) are shown in Fig. 13. 5. Experiment and result analysis The phase difference map (x, y) is obtained by (x, y) = B (x, y) − A (x, y). Where, A (x, y) is defect-free unwrapping phase map and B (x, y) is defective unwrapping phase map. The distance from
Fig. 10. The defect-free result of spliced removal method: (a) fringe curve; (b) spliced phase curve; (c) effective phase curve.
Fig. 11. The result of defect on the edge of image sub-block: (a) fringe curve; (b) spliced phase curve; (c) effective phase curve.
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Y. Jin et al. / Optik 127 (2016) 2240–2245 Table 1 The 20 diopter values from both methods.
Fig. 13. The unwrapped phase map: (a) defect-free; (b) ream.
Fig. 14. The map of out-of-plane displacement.
either linear CCD or grating projector to reference plane is set as l = 1000 mm. The horizontal distance between linear CCD and grating projector is d = 250 mm. The out-of-plane displacement h(x, y) is calculated by Eq. (5), as shown in Fig. 14. The diopter distribution map D(x, y) is calculated by Eq. (7). Where h (x, y) and h (x, y) are the first and second order difference of h(x, y) in x-direction. h (x, y) = h(x + 1, y) − h(x, y)
(15)
h (x, y) = h (x + 1, y) − h (x, y) = h(x + 2, y) − 2h(x + 1, y) + h(x, y)
(16)
For the glass sample with ream defect, shown in Fig. 3(b), the automatic focal power detector is used to measure diopter. The accuracy of detector is ±0.01 D, 20 measurement values are
Positions in Fig. 15
The measuring value of focal power detector
The measuring value of this paper
Relative difference
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
−0.02 0.01 0.01 −0.01 −0.04 −0.09 0.24 0.05 −0.03 −0.07 −0.06 0.01 0.02 −0.01 0.01 0.01 −0.01 −0.01 0.01 0.01
−0.01 0.00 0.00 0.00 −0.07 −0.07 0.23 0.02 −0.02 −0.05 −0.10 0.01 0.02 0.00 0.00 0.00 0.00 −0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.03 0.02 0.01 0.03 0.01 0.02 0.04 0.00 0.00 0.01 0.01 0.01 0.01 0.00 0.00 0.00
recorded every 1 mm on the central line along x-direction. The comparison of the calculated D(x, y = 100) and measured results is presented in Fig. 15. As it can be seen in Fig. 15, there is no obvious difference in the distribution trends of the diopter measured by two methods. This confirms the effectiveness of the adopted method in this study. The 20 diopter values from both methods are summarized in Table 1. The maximum deviation (0.04 D) between the two measuring methods occurs at the edge of the defect, while the deviation in the defective and defect-free region is less than 0.02 D. Thus, the accuracy of this method reaches 0.04 D. 6. Conclusions In order to measure the optical distortion of float glass defect, this paper proposes an online measuring method based on digital grating projection. The transmissive grating projection method of online measurement using digital grating projector is designed, and the mathematical model between diopter and phase of fringe image is derived. The one-dimensional Fourier transform algorithm and spliced removal method are used to process the fringe image and the diopter map of fringe image is figured out. The experimental result shows that this method is able to realize the online measurement of optical distortion, and the measurement accuracy reaches 0.04 D. There are some advantages of online optical distortion measurement method based on the grating projection, i.e. simple structure, high accuracy and easy to maintain. It can also be predicted that the measurement accuracy will be further improved with the improvement of image processing algorithms and hardware performance. References
Fig. 15. The distribution of measurement value between two method.
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