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Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement
Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement Mingteng Lu a, Xianyu Su a,n, Yiping Cao a, Zhisheng You b, Min Zhong a a b
Opto-Electronics Department, Sichuan University, Chengdu 610064, China College of Computer Science, Sichuan University, Chengdu 610064, China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 June 2015 Received in revised form 23 November 2015 Accepted 17 December 2015
In order to determine Dynamic 3-D shape with vertical measurement mode, a fast modulation measuring profilometry (MMP) with a cross grating projection and single shot is proposed. Unlike the previous methods, in our current projection system, one cross grating is projected by a special projection lens consisting of a common projection lens and a cylindrical lens. Due to the characteristics of cylindrical lens, the image of the vertical component and the horizontal component of the cross grating is separated in the image space, and the measuring range is just the space between the two image planes. Through a beam splitter, the CCD camera can coaxially capture the fringe pattern of the cross grating modulated by the testing object's shape. In one fringe pattern, by applying Fourier transform, filtering and inverse Fourier transform, the modulation corresponding to the vertical and horizontal components of the cross grating can be obtained respectively. Then the 3-D shape of the object can be reconstructed according to the mapping relationship between modulation and height, which was established by calibration process in advance. So the 3-D shape information can be recorded at the same speed of the frame rate of the CCD camera. This paper gives the principle of the proposed method and the set-up for measuring experiment and system calibration. The 3-D shape of a still object and a dynamic process of liquid vortex were measured and reconstructed in the experiments, and the results proved the method’s feasibility. The advantage of the proposed method is that only one fringe pattern is needed to extract the modulation distribution and to reconstruct the 3-D shape of the object. Therefore, the proposed method can achieve high speed measurement and vertical measurement without shadow and occlusion. It can be used in the dynamic 3-D shape measurement and vibration analysis. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Modulation measuring profilometry Vertical measurement Dynamic 3-D shape measurement Vibration analysis Fringe analysis
1. Introduction Optical 3-D shape measurement techniques based on the structured light illumination have been applied in numerous areas such as 3-D sensing, machine vision, robot simulation, industry monitoring, automated manufacturing, and quality control, etc. Among them, the techniques based on the structured light triangulation principle, such as phase measuring profilometry (PMP) [1], Fourier transform profilometry (FTP) [2], and moiré profilometry, etc, have made considerable achievements. However, in the triangulation techniques, the optical axis of the projector and that of the CCD camera need to form an offset angle to modulate the height information of the object into the fringe's phase. Thus, any area on the testing object, which is unseen by camera or projector, cannot be reconstructed successfully. n Correspondence to: Opto-Electronic Department, College of Electronics and Information Engineering, Sichuan University, Chengdu, Sichuan 610064, China. E-mail address: [email protected] (X. Su).
Therefore, these methods cannot be used to measure surface shapes with shutoff or steep slope. To overcome the problems of shadow and occlusion, some vertical measurement techniques have been proposed [3–8], such as modulation measurement profilometry (MMP) based on phase shifting and vertical scanning [3], MMP based on Fourier transform and vertical scanning [4], absolute three-dimensional shape measurement technique using coaxial and coimage plane optical systems combining Fourier fringe analysis [5], and the method using liquid crystal grating and liquid varifocus lens [6], etc. In MMP, it is the modulation distribution instead of the phases of sinusoidal fringe that are used to reconstruct the 3D surface of the testing object. In the previous MMP [3], M frames of modulation patterns would be obtained by shifting the projection device along the optical axis step by step, and then the height of the tested object was reconstructed by searching for the maximum modulation for each pixel. This method achieved the purpose of vertical measurement and high precision, but it needed an intermittent scanning process, which
http://dx.doi.org/10.1016/j.optlaseng.2015.12.011 0143-8166/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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means “go-stop-go”, therefore the measurement speed was limited. To improve the measurement speed, several fast MMP methods based on Fourier analysis were proposed. Among them, Yoichiro Kasai, Mitsuo Takeda, etc. proposed a method that two sinusoidal gratings with orthogonal direction were located axially separated along the optical axis [7]. The method improved the measurement speed because of only single fringe pattern was needed. In another fast MMP methods proposed by Dou et al. [8], a beam splitter was used to separate the two gratings, so the probability of frequency overlapping was reduced greatly compared to the two sinusoidal gratings axially separated. However the use of beam splitter in
illumination and projection made the system complex for collimation of the light path. Another recent uniaxial 3-D shape measurement by analyzing phase error was introduced into the 3-D shape measurement [9], and this method was based on skillful application of the relationship between the phase error caused by improperly defocused binary structured patterns and the depth z. The extraction of depth information was based on the defocusing degree instead of triangulation relationship formed by the optical axis of projector and that of the image device, so it can also overcame the problems of shadow and occlusion. However, this technique had lower spatial
Fig. 1. (a) The designed cross grating pattern. (b) Details of highlighted area of (a).
Fig. 2. The principle diagram of the proposed 3-D surface measurement method.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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resolution comparing with the other uniaxial 3-D shape measurement techniques. Among the studied fringe projection profilometry, FTP introduced first by Takeda [2], is particularly suitable for high-speed measurement because of only one fringe needed [10,11]. In the past years, the FTP method for dynamic 3-D shape measurement has been extensively studied [12]. But FTP based on the structured light triangulation principle is still facing with the problem of shadow and occlusion. So a simpler MMP is expected, which not only has the characteristics of vertical measurement, but also is capable for dynamic measurement. In this paper, a fast MMP method is proposed. A cross sinusoidal grating is projected onto the testing object through a special lens which consists of a common projection lens and a cylindrical lens. The CCD camera captures the deformed modulation fringes pattern through a beam splitter. By applying Fourier transform,
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filtering the fundamental frequency of the vertical fringes and horizontal fringes respectively, and inverse Fourier transform, only one captured cross grating pattern can provide the modulation distribution of the vertical fringes and the horizontal fringes on the testing object. The mapping relationship between the height (or depth) in the measurement space and the modulation distribution of vertical and horizontal fringes is provided by the system calibration. Then based on the relationship, the 3-D shape of the testing object can be reconstructed successfully. No scanning (or moving) device is needed in the system, which simplifies the measuring system greatly. And the 3-D shape reconstruction can be achieved with only one captured deformed grating pattern. Therefore this method has capability for both high speed measurement and vertical measurement without shadow and occlusion. The basic principle is introduced in Section 2; In Section 3, system calibration process is detailed; In Section 4, experiments
Fig. 3. Fringe patterns and their spatial frequency spectra.
Fig. 4. (a) The modulation cross curve. (b) The variation curve of depth parameter MR.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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verifying the proposed method and further a dynamic 3-D shape measurement experiment are also carried out. The discussion and conclusion involving our method applied in 3-D shape measurement are respectively in Sections 5 and 6.
2. Principle The cross grating in our method is binary dithered patterns, also called halftoning pattern, which is designed by error diffusion [13], and manufactured on a glass plate by E-beam lithography. The dot size of the grating is 2 μm. The transmittance function of the plate is designed as the addition of two sinusoidal gratings orthogonally, and can be written as: !) ( 1 1 x 1 1 y þ cos 2π þ cos 2π þ tðx; yÞ ¼ ð1Þ 4 4 px 4 4 py where the px and py are the period length of the vertical and horizontal grating fringes respectively. The designed cross grating is shown in Fig. 1. The principle diagram of the proposed vertical measurement setup is shown in Fig. 2. In this system, a cross sinusoidal grating is projected by a special projection lens, which consists of a common projection lens L1 and a cylindrical lens L2. The Z direction is set along the optical axis. The cylindrical lens changes the focus of the projection system in one direction (vertical or horizontal, and vertical direction is taken as an example in this paper), and this direction aligns with the cylindrical lens' generatrix. According to the imaging theory of the cylindrical lens, a point at the object space will “image” two lines separately in the image space, whose direction are parallel to and perpendicular to the cylindrical lens' generatrix respectively, and the parallel one is nearer to the projection lens. Then we align one component fringe of the cross grating (and take vertical fringes as an example) with the cylindrical lens' generatirx. According to the analysis above, the vertical fringe will form its “image” nearer and the horizontal fringe farther because the point on the vertical fringe will defocus at the horizontal fringe's “image” plane. So, by adding the cylindrical lens in the projection system, we can separate the “image” of the vertical fringe and horizontal fringe when projecting a single cross grating. And they will be imaged separately at P1 and P3 respectively. The CCD camera captures the deformed fringe pattern through the beam splitter.
From the analysis above, it is obvious that to perfectly align the grating with the cylindrical lens is vital. In the projection device, the projection lens and the cylindrical lens are required to be coaxial, then the cross grating is set perpendicular to the optical axis, and either the vertical component fringes or the horizontal fringes will be aligned with the cylindrical lens' generatrix. In the actual experiments, the standard to judge whether the grating is aligned with the cylindrical lens is to check whether the vertical fringes and the horizontal fringes “image” clearly at each image plane. Due to the different defocusing distance at the tow directions caused by the cylindrical lens, the vertical and horizontal component fringes of the cross grating recorded by the CCD have different fringe contrasts cv[Z(x, y)] and ch[Z(x, y)] at a certain image plane, therefore, the captured fringe pattern can be written as: rðx; yÞ 1 1 þ cv ½Zðx; yÞ∙ cos ð2π f v xÞ gðx; yÞ ¼ I þ I 0 b 2 4 M 2T 1 ð2Þ þ ch ½Zðx; yÞ∙ cos ð2π f h yÞ 4 where rðx; yÞ is the nonuniform reflectivity of the object surface, MT is the transverse magnification of the CCD lens, Ib is the environment background light, I0 is the uniform projection background light, fv and fh are the spatial carrier frequency of the vertical and horizontal fringes, respectively. When the image plane moves along the optical axis: at the starting position where Z¼ 1025 mm (distance from image plane to the cylindrical lens), high-contrast fringe is observed mostly for the vertical grating fringe with spatial frequency component fV, as shown in Fig. 3(a); and then the image plane passes through the intermediate position, where Z¼1075 mm, the fringe components with spatial frequency components fV and fH have similar contrasts, as shown in Fig. 3(b); in the end, at position where Z¼1125 mm, the fringe component with spatial frequency component fH has maximum contrast, as shown in Fig. 3(c). The spectra of the corresponding fringes patterns are shown below in Fig. 3(d), (e) and (f), respectively. To detect the fringe modulations of the spatial frequency components fV and fH on one image plane, we compute the Fourier transform of the captured two-dimensional cross grating pattern (as expressed in Eq. (2)): Rðf x ; f y Þ Rðf x ; f y Þ I 0 Gðf x ; f y Þ ¼ FT gðx; yÞ ¼ Ib þ M 2T M 2T 2
Fig. 5. Schematic diagram for the system calibration and testing plane measurement.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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þ þ
Rðf x ; f y Þ M 2T Rðf x ; f y Þ M 2T
I0 I0 C V ðf x f V ; f y ÞÞ þ C V ð ðf x þ f V Þ; f y Þ 8 8
I0 I0 C H ðf x ; f y f H ÞÞ þ C H ðf x ; ðf y þ f H ÞÞ : 8 8
optical axis, and it is defined by: MR ¼ ð3Þ
In Eq. (3), is the convolution operator, C V and C H are the conjugate items of C V and C H respectively. And the capital letters are the frequency spectra of the corresponding frequency components corresponding to the terms expressed as lowercase letters in Eq. (2). By filtering the corresponding spectra and computing the inverse Fourier transform, the modulation information corresponding to the vertical component and the horizontal component fringes can be obtained:
" #
Rðf x ;x f y Þ Rðx; yÞ
I 0 cv ½Zðx; yÞ ¼ IFT C V ðf X f V ; f Y Þ ð4Þ M V ðx; yÞ ¼ 2 2
M M T
M H ðx; yÞ ¼
Rðx; yÞ M 2T
5
T
" #
Rðf x ;x f y Þ
I 0 ch ½Zðx; yÞ ¼ IFT C ðf ; f f Þ
H X Y H
M2 T
ð5Þ Fig. 4(a) shows the variation of fringe modulation as the object plane moves in the Z-direction along the optical axis. Within the range Z¼1025 –1125 mm, the modulation variation trends of the vertical fringe and the horizontal fringe are monotonous but opposite. In order to enhance the depth resolution and to reduce the influence of the nonuniform reflectivity of the object surface, we combine the two modulations Mv and Mh to a new depth parameter, i.e. Modulation ratio (MR) to detect the depth along the
M V ½Zðx; yÞ M H ½Zðx; yÞ cv ½Zðx; yÞ ch ½Zðx; yÞ ¼ M V ½Zðx; yÞ þ M H ½Zðx; yÞ cv ½Zðx; yÞ þch ½Zðx; yÞ
ð6Þ
The distribution of the parameter is shown in Fig. 4(b). It shows that, this depth parameter exhibits relatively good linearity within the range Z¼ 1030 –1120 mm, which can be used as a look-up table for converting the detected fringe modulations into the actual physical depth value. Another merit of using this look-up table is that it will not be influenced by the reflectivity of the object surface because the depth parameter is based on the ratio of the fringe modulations.
3. System calibration The schematic diagram of the calibration set-up used for verification of our method is shown in Fig. 5. The projection system consists of one LED illumination light source of 10 W, one 40 40 mm2 cross sinusoidal grating with 4 lines/mm in vertical and horizontal direction, one projection lens with focal length of 135 mm and one cylindrical lens with focal length of 10 m. The projected grating pattern is observed through a beam splitter by a fast CCD camera (Baumer sxc10) with the image size of 1024 1024 pixels. Due to the optical aberration and the field curvature of the projection system, modulation distribution is different for every pixel, so system calibration is necessary. In system calibration the CCD captures N frames grating patterns (N ¼50 in our experiment) of a moving plane along the projection optical axis in the image space with interval of 5 mm, and then compute the modulation
Fig. 6. The measured plots at several pixels: (a) modulation cross curves and (b) MR curves inside the purple square area of (a).
Fig. 7. The measurement results of the testing plane: (a) fringe pattern; (b) reconstructed 3D shape.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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distribution by the algorithm mentioned in Section 2. The modulation curves at 4 pixels are shown in Fig. 6(a), and the position is the distance from the image plane to the cylindrical lens. In Fig. 6(a), the full lines represent the modulation curves of the vertical fringes, and the dash dot lines represent the modulation curves of the horizontal fringes. The different colors distinguish different pixel positions, and they agree with the legend shown in Fig. 6(b). From Fig. 6(a), the field curvature is clearly observed for the clear imaging position (maximum modulation value position) of the pixel at the center region (pixel (550, 550) e.g.) is different from the pixels at the edge region(pixel (50, 50) e.g.). After checking plenty of pixels, a proper range (1035 –1100 mm) is chosen as the actual measuring range, which includes the monotonous down slope of the vertical fringe's modulation curve (full line) and the monotonous upslope of the horizontal fringe's modulation curve (dash dot line) for most pixels (which are at the main field of the CCD and will include the testing object totally), as is shown in the square in Fig. 6(a). Fig. 6(b) shows the MR curve calculated from the modulation distribution in the actual measuring range. Within this range, the image plane farthest from the cylindrical lens is set as the reference plane and the position in Fig. 6(b) is the distance from the image plane to the cylindrical lens. From Fig. 6(b) we can see that the measuring range is 65 mm. After calibration, the look-up
table (the MR curve) for every pixel on CCD camera is constructed with which we can convert the detected fringe modulations into the actual physical depth value.
4. Experiments 4.1. Testing plane measurement To estimate the precision of the method, we measured a flat plane which was set 47.50 mm from the reference plane. The captured grating fringes pattern is shown in Fig. 7(a), and the reconstructed 3D shape is shown in Fig. 7(b). The measured mean height is 47.40 mm, the RMS value is 0.03 mm and the maximum absolute error is 0.76 mm. These results are found to be reasonably in agreement with the nominal height value of 47.50 mm. 4.2. Static surface measurement A Buddha statue was measured to illustrate that our method can achieve vertical measurement and reconstruct 3-D shape with sharp height distribution. Fig. 8(a) shows the fringe pattern of the Buddha statue which was detected by the CCD camera with 1024 1024 pixels. As seen in the figure, no shading effect is
Fig. 8. (a) The fringe pattern of the Buddha statue. (b) The reconstructed 3-D shape of the Buddha statue.
Fig. 9. Schematic diagram of the dynamic vortex experiment.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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observed even at the nose top area with steep change of height, and the contrast of the fringe in this part of object surface is high for the vertical component fringe, while the fringe contrast on the reference plane is high for the horizontal component fringe. We computed the fringe modulations by the Fourier transform method as described above, and obtained the height distribution using the calibration look-up table. Fig. 8(b) shows the height distribution of the object. As seen in the figure, steep changing height distribution was reconstructed from the single fringe pattern shown in Fig. 8(a). Some errors are observed at the boundary of the edges, and it is also found that the shape tends to be measured as a smooth distribution. These are caused by the use of the Fourier transform method that loses the high spatial frequency component in the filtering process. 4.3. Dynamic vortex measurement Another experiment is to measure a vortex shape in dynamic process and to prove that our method can implement dynamic
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vertical measurement. The schematic diagram is shown in Fig. 9. The projection system was the same as last experiment and had been calibrated. The cross grating was projected onto the liquid surface in a container through a mirror. An electromagnetic stirrer revolved the liquid to form the vortex. The liquid was mixed with white poster paint to increase the liquid surface’s reflectivity. The deformed fringe pattern was observed by the fast CCD camera. The captured image size was 1024 1024 pixels at the speed of 25 frames per second. In the experiment, we started to capture the deformed fringe pattern as soon as we started the stirrer. From the still to stable rotation, 149 frame images of the vortex surface were captured, and the whole process lasted for about 6 s. One of the deformed fringe pattern is shown in Fig. 10(a), and the associated Media 1 at top left corner. In Fig. 10(b) it shows the profiles of the reconstructed vortex when t¼0.84 s, 1.68 s, 2.52 s, 3.6 s, 4.2 s, and 5.32 s whose corresponding frame number is 21, 42, 63, 84, 105 and 133.
Fig. 10. (a) The vortex shape for measurement (the area inside the square is to be reconstructed and the video is shown in Media 1 at top left corner.) (b): the profile chart of reconstructed vortices (the numbers in the chart indicate the time point of the curve and the video is shown in Media 1 at bottom right corner.).
Fig. 11. The reconstructed shape of vortices (drawing inverse shape for observation convenience). And the video Media 1 will show the original reconstructed 3-D shape of the vortex without inverse shape.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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The reconstructed 3-D shape grid charts of the corresponding frames are given in Fig. 11. The inverse shape is given for better observation. And the dynamic 3-D shape of the reconstructed vortex is shown at the top right corner in Media 1. The video Media1.mov is given for better illustration of the dynamic experiment result. The video shows the vortex fringe pattern, the reconstructed 3-D shape of the vortex, top view of the reconstructed vortex and the cross section of the vortex with frame synchronization. And the video speed is slowed down to 9 fps to show the result of the total 149 frames (which were originally captured at 25 fps by the fast CCD) more clearly. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.optlaseng.2015.12.011. Compared with the previous proposed vortex measurement method [14] based on structured light triangulation, our method realizes vertical measurement, avoids the shadow and occlusion, and can measure vortexes with bigger depth-width ratio.
5. Discussion The theoretical analysis and the experiment results indicate that this modulation measuring profilometry with cross grating projection has the following merits: (1) The use of an additional cylindrical lens can produce the different defocusing degrees for the vertical and the horizontal component fringes of the projected cross grating pattern in the image space, which achieves the cross curve of the fringes' modulation with simple optical device. (2) One cross grating with binary dithered patterns is designed and manufactured, and its transmittance function is designed as two additive orthogonal sinusoidal gratings. The spectra of the corresponding grating fringes are also linear added. So the convolution terms are avoided, which reduce the probability of spectrum overlapping. (3) With the uni-axial measurement system and the advantages described above, our method can use only one fringe pattern to extract the modulation distribution and reconstruct the 3-D shape of the object. The measurement speed only depends on the hard ware, i.e. the frame rate of the CCD camera. (4) The measurement range in our experiment is 65 mm, but the range can be conveniently changed according to the demand. By properly changing the parameters (such as: focal length and aperture size of the projecting lens, the grating's pitch and the focal length of the cylindrical lens), we can design the proper measurement range we need. Despite its many advantages, the proposed method has some limitations. In order to get the modulation from only one fringe pattern, Fourier transform and filtering are applied in the method. Some higher frequency information is lost in the filtering process. So the measurement precision for the surface shape with considerable variation in height is somewhat less than the MMP method with phase shifting and vertical scanning [15]. To some extent, we achieve the high measuring speed at the cost of sacrificing some measurement accuracy. However, there do exist solutions to improve the measurement accuracy, by increasing the number of fringes and increasing the CCD camera's resolution for instance. Another method for accuracy improvement is to use the phase shifting algorithm. If we use electrical projector (DPL or LCD
projector) with an additional cylindrical lens as before, six phase shifting fringe patterns can be captured severally for example, among which three are for the vertical fringes and three are for the horizontal fringes. Phase shifting algorithm can achieve point– point calculation, which could measure complex surface with higher accuracy. Of course in this case, the measurement speed will drop to one sixth of the proposed method in this paper.
6. Conclusion In this paper, a dynamic vertical measurement method based on MMP is proposed and demonstrated by experiments. It has obvious advantages in dynamic measurement and avoids the difficulties of shadow and occlusion which traditional methods based on structured illumination triangulation can hardly overcome. And the most innovative point is the use of cylindrical lens and one cross grating projection which not only makes the measuring system achieve the vertical measurement and dynamic measurement, but also greatly simplifies the measuring system. With the development of high-resolution and high speed CCD cameras, the proposed method should be a promising dynamic and vertical measurement method.
Acknowledgment The authors acknowledge the support by National Key Scientific Apparatus Development Project (2013YQ49087901), the National Natural Science Foundation of China (NSFC) (61177010).
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Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement Mingteng Lu a, Xianyu Su a,n, Yiping Cao a, Zhisheng You b, Min Zhong a a b
Opto-Electronics Department, Sichuan University, Chengdu 610064, China College of Computer Science, Sichuan University, Chengdu 610064, China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 June 2015 Received in revised form 23 November 2015 Accepted 17 December 2015
In order to determine Dynamic 3-D shape with vertical measurement mode, a fast modulation measuring profilometry (MMP) with a cross grating projection and single shot is proposed. Unlike the previous methods, in our current projection system, one cross grating is projected by a special projection lens consisting of a common projection lens and a cylindrical lens. Due to the characteristics of cylindrical lens, the image of the vertical component and the horizontal component of the cross grating is separated in the image space, and the measuring range is just the space between the two image planes. Through a beam splitter, the CCD camera can coaxially capture the fringe pattern of the cross grating modulated by the testing object's shape. In one fringe pattern, by applying Fourier transform, filtering and inverse Fourier transform, the modulation corresponding to the vertical and horizontal components of the cross grating can be obtained respectively. Then the 3-D shape of the object can be reconstructed according to the mapping relationship between modulation and height, which was established by calibration process in advance. So the 3-D shape information can be recorded at the same speed of the frame rate of the CCD camera. This paper gives the principle of the proposed method and the set-up for measuring experiment and system calibration. The 3-D shape of a still object and a dynamic process of liquid vortex were measured and reconstructed in the experiments, and the results proved the method’s feasibility. The advantage of the proposed method is that only one fringe pattern is needed to extract the modulation distribution and to reconstruct the 3-D shape of the object. Therefore, the proposed method can achieve high speed measurement and vertical measurement without shadow and occlusion. It can be used in the dynamic 3-D shape measurement and vibration analysis. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Modulation measuring profilometry Vertical measurement Dynamic 3-D shape measurement Vibration analysis Fringe analysis
1. Introduction Optical 3-D shape measurement techniques based on the structured light illumination have been applied in numerous areas such as 3-D sensing, machine vision, robot simulation, industry monitoring, automated manufacturing, and quality control, etc. Among them, the techniques based on the structured light triangulation principle, such as phase measuring profilometry (PMP) [1], Fourier transform profilometry (FTP) [2], and moiré profilometry, etc, have made considerable achievements. However, in the triangulation techniques, the optical axis of the projector and that of the CCD camera need to form an offset angle to modulate the height information of the object into the fringe's phase. Thus, any area on the testing object, which is unseen by camera or projector, cannot be reconstructed successfully. n Correspondence to: Opto-Electronic Department, College of Electronics and Information Engineering, Sichuan University, Chengdu, Sichuan 610064, China. E-mail address: [email protected] (X. Su).
Therefore, these methods cannot be used to measure surface shapes with shutoff or steep slope. To overcome the problems of shadow and occlusion, some vertical measurement techniques have been proposed [3–8], such as modulation measurement profilometry (MMP) based on phase shifting and vertical scanning [3], MMP based on Fourier transform and vertical scanning [4], absolute three-dimensional shape measurement technique using coaxial and coimage plane optical systems combining Fourier fringe analysis [5], and the method using liquid crystal grating and liquid varifocus lens [6], etc. In MMP, it is the modulation distribution instead of the phases of sinusoidal fringe that are used to reconstruct the 3D surface of the testing object. In the previous MMP [3], M frames of modulation patterns would be obtained by shifting the projection device along the optical axis step by step, and then the height of the tested object was reconstructed by searching for the maximum modulation for each pixel. This method achieved the purpose of vertical measurement and high precision, but it needed an intermittent scanning process, which
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Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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means “go-stop-go”, therefore the measurement speed was limited. To improve the measurement speed, several fast MMP methods based on Fourier analysis were proposed. Among them, Yoichiro Kasai, Mitsuo Takeda, etc. proposed a method that two sinusoidal gratings with orthogonal direction were located axially separated along the optical axis [7]. The method improved the measurement speed because of only single fringe pattern was needed. In another fast MMP methods proposed by Dou et al. [8], a beam splitter was used to separate the two gratings, so the probability of frequency overlapping was reduced greatly compared to the two sinusoidal gratings axially separated. However the use of beam splitter in
illumination and projection made the system complex for collimation of the light path. Another recent uniaxial 3-D shape measurement by analyzing phase error was introduced into the 3-D shape measurement [9], and this method was based on skillful application of the relationship between the phase error caused by improperly defocused binary structured patterns and the depth z. The extraction of depth information was based on the defocusing degree instead of triangulation relationship formed by the optical axis of projector and that of the image device, so it can also overcame the problems of shadow and occlusion. However, this technique had lower spatial
Fig. 1. (a) The designed cross grating pattern. (b) Details of highlighted area of (a).
Fig. 2. The principle diagram of the proposed 3-D surface measurement method.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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resolution comparing with the other uniaxial 3-D shape measurement techniques. Among the studied fringe projection profilometry, FTP introduced first by Takeda [2], is particularly suitable for high-speed measurement because of only one fringe needed [10,11]. In the past years, the FTP method for dynamic 3-D shape measurement has been extensively studied [12]. But FTP based on the structured light triangulation principle is still facing with the problem of shadow and occlusion. So a simpler MMP is expected, which not only has the characteristics of vertical measurement, but also is capable for dynamic measurement. In this paper, a fast MMP method is proposed. A cross sinusoidal grating is projected onto the testing object through a special lens which consists of a common projection lens and a cylindrical lens. The CCD camera captures the deformed modulation fringes pattern through a beam splitter. By applying Fourier transform,
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filtering the fundamental frequency of the vertical fringes and horizontal fringes respectively, and inverse Fourier transform, only one captured cross grating pattern can provide the modulation distribution of the vertical fringes and the horizontal fringes on the testing object. The mapping relationship between the height (or depth) in the measurement space and the modulation distribution of vertical and horizontal fringes is provided by the system calibration. Then based on the relationship, the 3-D shape of the testing object can be reconstructed successfully. No scanning (or moving) device is needed in the system, which simplifies the measuring system greatly. And the 3-D shape reconstruction can be achieved with only one captured deformed grating pattern. Therefore this method has capability for both high speed measurement and vertical measurement without shadow and occlusion. The basic principle is introduced in Section 2; In Section 3, system calibration process is detailed; In Section 4, experiments
Fig. 3. Fringe patterns and their spatial frequency spectra.
Fig. 4. (a) The modulation cross curve. (b) The variation curve of depth parameter MR.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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verifying the proposed method and further a dynamic 3-D shape measurement experiment are also carried out. The discussion and conclusion involving our method applied in 3-D shape measurement are respectively in Sections 5 and 6.
2. Principle The cross grating in our method is binary dithered patterns, also called halftoning pattern, which is designed by error diffusion [13], and manufactured on a glass plate by E-beam lithography. The dot size of the grating is 2 μm. The transmittance function of the plate is designed as the addition of two sinusoidal gratings orthogonally, and can be written as: !) ( 1 1 x 1 1 y þ cos 2π þ cos 2π þ tðx; yÞ ¼ ð1Þ 4 4 px 4 4 py where the px and py are the period length of the vertical and horizontal grating fringes respectively. The designed cross grating is shown in Fig. 1. The principle diagram of the proposed vertical measurement setup is shown in Fig. 2. In this system, a cross sinusoidal grating is projected by a special projection lens, which consists of a common projection lens L1 and a cylindrical lens L2. The Z direction is set along the optical axis. The cylindrical lens changes the focus of the projection system in one direction (vertical or horizontal, and vertical direction is taken as an example in this paper), and this direction aligns with the cylindrical lens' generatrix. According to the imaging theory of the cylindrical lens, a point at the object space will “image” two lines separately in the image space, whose direction are parallel to and perpendicular to the cylindrical lens' generatrix respectively, and the parallel one is nearer to the projection lens. Then we align one component fringe of the cross grating (and take vertical fringes as an example) with the cylindrical lens' generatirx. According to the analysis above, the vertical fringe will form its “image” nearer and the horizontal fringe farther because the point on the vertical fringe will defocus at the horizontal fringe's “image” plane. So, by adding the cylindrical lens in the projection system, we can separate the “image” of the vertical fringe and horizontal fringe when projecting a single cross grating. And they will be imaged separately at P1 and P3 respectively. The CCD camera captures the deformed fringe pattern through the beam splitter.
From the analysis above, it is obvious that to perfectly align the grating with the cylindrical lens is vital. In the projection device, the projection lens and the cylindrical lens are required to be coaxial, then the cross grating is set perpendicular to the optical axis, and either the vertical component fringes or the horizontal fringes will be aligned with the cylindrical lens' generatrix. In the actual experiments, the standard to judge whether the grating is aligned with the cylindrical lens is to check whether the vertical fringes and the horizontal fringes “image” clearly at each image plane. Due to the different defocusing distance at the tow directions caused by the cylindrical lens, the vertical and horizontal component fringes of the cross grating recorded by the CCD have different fringe contrasts cv[Z(x, y)] and ch[Z(x, y)] at a certain image plane, therefore, the captured fringe pattern can be written as: rðx; yÞ 1 1 þ cv ½Zðx; yÞ∙ cos ð2π f v xÞ gðx; yÞ ¼ I þ I 0 b 2 4 M 2T 1 ð2Þ þ ch ½Zðx; yÞ∙ cos ð2π f h yÞ 4 where rðx; yÞ is the nonuniform reflectivity of the object surface, MT is the transverse magnification of the CCD lens, Ib is the environment background light, I0 is the uniform projection background light, fv and fh are the spatial carrier frequency of the vertical and horizontal fringes, respectively. When the image plane moves along the optical axis: at the starting position where Z¼ 1025 mm (distance from image plane to the cylindrical lens), high-contrast fringe is observed mostly for the vertical grating fringe with spatial frequency component fV, as shown in Fig. 3(a); and then the image plane passes through the intermediate position, where Z¼1075 mm, the fringe components with spatial frequency components fV and fH have similar contrasts, as shown in Fig. 3(b); in the end, at position where Z¼1125 mm, the fringe component with spatial frequency component fH has maximum contrast, as shown in Fig. 3(c). The spectra of the corresponding fringes patterns are shown below in Fig. 3(d), (e) and (f), respectively. To detect the fringe modulations of the spatial frequency components fV and fH on one image plane, we compute the Fourier transform of the captured two-dimensional cross grating pattern (as expressed in Eq. (2)): Rðf x ; f y Þ Rðf x ; f y Þ I 0 Gðf x ; f y Þ ¼ FT gðx; yÞ ¼ Ib þ M 2T M 2T 2
Fig. 5. Schematic diagram for the system calibration and testing plane measurement.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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þ þ
Rðf x ; f y Þ M 2T Rðf x ; f y Þ M 2T
I0 I0 C V ðf x f V ; f y ÞÞ þ C V ð ðf x þ f V Þ; f y Þ 8 8
I0 I0 C H ðf x ; f y f H ÞÞ þ C H ðf x ; ðf y þ f H ÞÞ : 8 8
optical axis, and it is defined by: MR ¼ ð3Þ
In Eq. (3), is the convolution operator, C V and C H are the conjugate items of C V and C H respectively. And the capital letters are the frequency spectra of the corresponding frequency components corresponding to the terms expressed as lowercase letters in Eq. (2). By filtering the corresponding spectra and computing the inverse Fourier transform, the modulation information corresponding to the vertical component and the horizontal component fringes can be obtained:
" #
Rðf x ;x f y Þ Rðx; yÞ
I 0 cv ½Zðx; yÞ ¼ IFT C V ðf X f V ; f Y Þ ð4Þ M V ðx; yÞ ¼ 2 2
M M T
M H ðx; yÞ ¼
Rðx; yÞ M 2T
5
T
" #
Rðf x ;x f y Þ
I 0 ch ½Zðx; yÞ ¼ IFT C ðf ; f f Þ
H X Y H
M2 T
ð5Þ Fig. 4(a) shows the variation of fringe modulation as the object plane moves in the Z-direction along the optical axis. Within the range Z¼1025 –1125 mm, the modulation variation trends of the vertical fringe and the horizontal fringe are monotonous but opposite. In order to enhance the depth resolution and to reduce the influence of the nonuniform reflectivity of the object surface, we combine the two modulations Mv and Mh to a new depth parameter, i.e. Modulation ratio (MR) to detect the depth along the
M V ½Zðx; yÞ M H ½Zðx; yÞ cv ½Zðx; yÞ ch ½Zðx; yÞ ¼ M V ½Zðx; yÞ þ M H ½Zðx; yÞ cv ½Zðx; yÞ þch ½Zðx; yÞ
ð6Þ
The distribution of the parameter is shown in Fig. 4(b). It shows that, this depth parameter exhibits relatively good linearity within the range Z¼ 1030 –1120 mm, which can be used as a look-up table for converting the detected fringe modulations into the actual physical depth value. Another merit of using this look-up table is that it will not be influenced by the reflectivity of the object surface because the depth parameter is based on the ratio of the fringe modulations.
3. System calibration The schematic diagram of the calibration set-up used for verification of our method is shown in Fig. 5. The projection system consists of one LED illumination light source of 10 W, one 40 40 mm2 cross sinusoidal grating with 4 lines/mm in vertical and horizontal direction, one projection lens with focal length of 135 mm and one cylindrical lens with focal length of 10 m. The projected grating pattern is observed through a beam splitter by a fast CCD camera (Baumer sxc10) with the image size of 1024 1024 pixels. Due to the optical aberration and the field curvature of the projection system, modulation distribution is different for every pixel, so system calibration is necessary. In system calibration the CCD captures N frames grating patterns (N ¼50 in our experiment) of a moving plane along the projection optical axis in the image space with interval of 5 mm, and then compute the modulation
Fig. 6. The measured plots at several pixels: (a) modulation cross curves and (b) MR curves inside the purple square area of (a).
Fig. 7. The measurement results of the testing plane: (a) fringe pattern; (b) reconstructed 3D shape.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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distribution by the algorithm mentioned in Section 2. The modulation curves at 4 pixels are shown in Fig. 6(a), and the position is the distance from the image plane to the cylindrical lens. In Fig. 6(a), the full lines represent the modulation curves of the vertical fringes, and the dash dot lines represent the modulation curves of the horizontal fringes. The different colors distinguish different pixel positions, and they agree with the legend shown in Fig. 6(b). From Fig. 6(a), the field curvature is clearly observed for the clear imaging position (maximum modulation value position) of the pixel at the center region (pixel (550, 550) e.g.) is different from the pixels at the edge region(pixel (50, 50) e.g.). After checking plenty of pixels, a proper range (1035 –1100 mm) is chosen as the actual measuring range, which includes the monotonous down slope of the vertical fringe's modulation curve (full line) and the monotonous upslope of the horizontal fringe's modulation curve (dash dot line) for most pixels (which are at the main field of the CCD and will include the testing object totally), as is shown in the square in Fig. 6(a). Fig. 6(b) shows the MR curve calculated from the modulation distribution in the actual measuring range. Within this range, the image plane farthest from the cylindrical lens is set as the reference plane and the position in Fig. 6(b) is the distance from the image plane to the cylindrical lens. From Fig. 6(b) we can see that the measuring range is 65 mm. After calibration, the look-up
table (the MR curve) for every pixel on CCD camera is constructed with which we can convert the detected fringe modulations into the actual physical depth value.
4. Experiments 4.1. Testing plane measurement To estimate the precision of the method, we measured a flat plane which was set 47.50 mm from the reference plane. The captured grating fringes pattern is shown in Fig. 7(a), and the reconstructed 3D shape is shown in Fig. 7(b). The measured mean height is 47.40 mm, the RMS value is 0.03 mm and the maximum absolute error is 0.76 mm. These results are found to be reasonably in agreement with the nominal height value of 47.50 mm. 4.2. Static surface measurement A Buddha statue was measured to illustrate that our method can achieve vertical measurement and reconstruct 3-D shape with sharp height distribution. Fig. 8(a) shows the fringe pattern of the Buddha statue which was detected by the CCD camera with 1024 1024 pixels. As seen in the figure, no shading effect is
Fig. 8. (a) The fringe pattern of the Buddha statue. (b) The reconstructed 3-D shape of the Buddha statue.
Fig. 9. Schematic diagram of the dynamic vortex experiment.
Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i
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observed even at the nose top area with steep change of height, and the contrast of the fringe in this part of object surface is high for the vertical component fringe, while the fringe contrast on the reference plane is high for the horizontal component fringe. We computed the fringe modulations by the Fourier transform method as described above, and obtained the height distribution using the calibration look-up table. Fig. 8(b) shows the height distribution of the object. As seen in the figure, steep changing height distribution was reconstructed from the single fringe pattern shown in Fig. 8(a). Some errors are observed at the boundary of the edges, and it is also found that the shape tends to be measured as a smooth distribution. These are caused by the use of the Fourier transform method that loses the high spatial frequency component in the filtering process. 4.3. Dynamic vortex measurement Another experiment is to measure a vortex shape in dynamic process and to prove that our method can implement dynamic
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vertical measurement. The schematic diagram is shown in Fig. 9. The projection system was the same as last experiment and had been calibrated. The cross grating was projected onto the liquid surface in a container through a mirror. An electromagnetic stirrer revolved the liquid to form the vortex. The liquid was mixed with white poster paint to increase the liquid surface’s reflectivity. The deformed fringe pattern was observed by the fast CCD camera. The captured image size was 1024 1024 pixels at the speed of 25 frames per second. In the experiment, we started to capture the deformed fringe pattern as soon as we started the stirrer. From the still to stable rotation, 149 frame images of the vortex surface were captured, and the whole process lasted for about 6 s. One of the deformed fringe pattern is shown in Fig. 10(a), and the associated Media 1 at top left corner. In Fig. 10(b) it shows the profiles of the reconstructed vortex when t¼0.84 s, 1.68 s, 2.52 s, 3.6 s, 4.2 s, and 5.32 s whose corresponding frame number is 21, 42, 63, 84, 105 and 133.
Fig. 10. (a) The vortex shape for measurement (the area inside the square is to be reconstructed and the video is shown in Media 1 at top left corner.) (b): the profile chart of reconstructed vortices (the numbers in the chart indicate the time point of the curve and the video is shown in Media 1 at bottom right corner.).
Fig. 11. The reconstructed shape of vortices (drawing inverse shape for observation convenience). And the video Media 1 will show the original reconstructed 3-D shape of the vortex without inverse shape.
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The reconstructed 3-D shape grid charts of the corresponding frames are given in Fig. 11. The inverse shape is given for better observation. And the dynamic 3-D shape of the reconstructed vortex is shown at the top right corner in Media 1. The video Media1.mov is given for better illustration of the dynamic experiment result. The video shows the vortex fringe pattern, the reconstructed 3-D shape of the vortex, top view of the reconstructed vortex and the cross section of the vortex with frame synchronization. And the video speed is slowed down to 9 fps to show the result of the total 149 frames (which were originally captured at 25 fps by the fast CCD) more clearly. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.optlaseng.2015.12.011. Compared with the previous proposed vortex measurement method [14] based on structured light triangulation, our method realizes vertical measurement, avoids the shadow and occlusion, and can measure vortexes with bigger depth-width ratio.
5. Discussion The theoretical analysis and the experiment results indicate that this modulation measuring profilometry with cross grating projection has the following merits: (1) The use of an additional cylindrical lens can produce the different defocusing degrees for the vertical and the horizontal component fringes of the projected cross grating pattern in the image space, which achieves the cross curve of the fringes' modulation with simple optical device. (2) One cross grating with binary dithered patterns is designed and manufactured, and its transmittance function is designed as two additive orthogonal sinusoidal gratings. The spectra of the corresponding grating fringes are also linear added. So the convolution terms are avoided, which reduce the probability of spectrum overlapping. (3) With the uni-axial measurement system and the advantages described above, our method can use only one fringe pattern to extract the modulation distribution and reconstruct the 3-D shape of the object. The measurement speed only depends on the hard ware, i.e. the frame rate of the CCD camera. (4) The measurement range in our experiment is 65 mm, but the range can be conveniently changed according to the demand. By properly changing the parameters (such as: focal length and aperture size of the projecting lens, the grating's pitch and the focal length of the cylindrical lens), we can design the proper measurement range we need. Despite its many advantages, the proposed method has some limitations. In order to get the modulation from only one fringe pattern, Fourier transform and filtering are applied in the method. Some higher frequency information is lost in the filtering process. So the measurement precision for the surface shape with considerable variation in height is somewhat less than the MMP method with phase shifting and vertical scanning [15]. To some extent, we achieve the high measuring speed at the cost of sacrificing some measurement accuracy. However, there do exist solutions to improve the measurement accuracy, by increasing the number of fringes and increasing the CCD camera's resolution for instance. Another method for accuracy improvement is to use the phase shifting algorithm. If we use electrical projector (DPL or LCD
projector) with an additional cylindrical lens as before, six phase shifting fringe patterns can be captured severally for example, among which three are for the vertical fringes and three are for the horizontal fringes. Phase shifting algorithm can achieve point– point calculation, which could measure complex surface with higher accuracy. Of course in this case, the measurement speed will drop to one sixth of the proposed method in this paper.
6. Conclusion In this paper, a dynamic vertical measurement method based on MMP is proposed and demonstrated by experiments. It has obvious advantages in dynamic measurement and avoids the difficulties of shadow and occlusion which traditional methods based on structured illumination triangulation can hardly overcome. And the most innovative point is the use of cylindrical lens and one cross grating projection which not only makes the measuring system achieve the vertical measurement and dynamic measurement, but also greatly simplifies the measuring system. With the development of high-resolution and high speed CCD cameras, the proposed method should be a promising dynamic and vertical measurement method.
Acknowledgment The authors acknowledge the support by National Key Scientific Apparatus Development Project (2013YQ49087901), the National Natural Science Foundation of China (NSFC) (61177010).
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Please cite this article as: Lu M, et al. Modulation measuring profilometry with cross grating projection and single shot for dynamic 3D shape measurement. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2015.12.011i