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High-speed height measurement is required in industrial fields for analyzing the behavior of a breaking object, a vibrating object or a rotating object. A shape measurement performed using a phase-shifting method can measure the shape with high spatial resolution because the coordinates can be obtained pixel by pixel. A light-source-stepping method (LSSM) that uses a linear LED array by means of a whole-space tabulation method (WSTM) has been proposed. Accurate shape measurement can be performed using this method. The response speed of the LED array is greater than 12 kHz. In this paper, height measurement is performed using WSTM and LSSM with a linear LED array and a high-speed camera. It was verified that the response speed of the linear LED is greater than 200 kHz. The phase shifting was performed at 12 kHz, and the height measurement of the vibrating woofer was performed at 4 kHz using a 3-step phase-shifting method.
© 2013 Optical Society of America
1. Introduction
Accurate 3D shape measurement is required in industrial fields. High-speed shape measurement is also required to analyze the behavior of a breaking object or a vibrating object. The maximum deformed point on the breaking or vibrating specimen surface can be obtained by a high-speed shape measurement. A shape measurement performed using a phase-shifting method can measure the shape with high spatial resolution because the coordinates can be obtained pixel by pixel.
This method requires a phase-shifting mechanism in the projector as a primary function. High stability, high brightness and high-speed response are required for a projector to be used for shape measurement. In the conventional mechanical grating panel-movement method [1, 2], a liquid crystal display (LCD) projector [3, 4], a digital light processing (DLP) projector using a digital micro-mirror device (DMD) [5–7], a microelectromechanical system (MEMS) scanner grating projector [8, 9], a projector with free-form mirror [10], or a rotating wobbling mirror [11] are employed to project phase-shifted grating patterns. High-speed shape measurement has been performed using a DLP projector [6,7] and the rotating wobbling mirror [11].
In the case of the DLP projector, the limitation of speed depends on the device and the driver chip. This method has a potential for achieving 3-D shape measurement with a frequency of MHz [6]. A special driver is necessary to control each mirror with the frequency. It is not easy to produce the special driver. In the case of a rotating wobbling mirror, the speed depends on the mechanical limitation of rotating speed of the mirror. It is also not easy to produce a compact system and to produce a measurement system with high accuracy because the rotate motor makes a vibration.
Previous studies proposed a light-source-stepping method (LSSM) using a linear LED array [12,13]. The basic idea was also mentioned in a patent description approximately 10 years ago [14]. However, the possibility of performing shape measurements using this idea was apparently not realized at that time. The disadvantage of LSSM is that the shifted phase varies according to the z position. However, because the z coordinate and the phase obtained using the phase-shifting method have a one-to-one relationship, accurate shape measurement can be realized using the whole-space tabulation method (WSTM) proposed previously [15,16]. Using this method, the shape measurement equipment can be produced inexpensively. Furthermore, the equipment is compact, and no phase-shifting mechanical systems are required for this method. Additionally, the method enables quick 3D shape measurement by rapidly switching the light sources. These are the advantage points against the conventional phase-shifting methods mentioned above. The phase of the projected grating is calculated pixel by pixel with a phase-shifting method. The height is also obtained pixel by pixel with the WSTM. The result of obtained height is therefore unaffected by the heterogeneity of the surface of the object in the proposed method at least in principle.
In this paper, the response speed of a linear LED array produced by the authors is verified and high-speed height measurement with 4 kHz is realized. An LSSM with the linear LED array and the WSTM are applied to high-speed height measurement with a high-speed camera. An experimental trial to measure the height distribution of a woofer, which is a speaker for low-pitched tone, vibrating at 4 kHz with phase shifting at 12 kHz is performed.
2. Light-source-stepping method (LSSM)
The light-source-stepping method makes it possible to shift the projected grating pattern with high-speed. Figure 1 presents a schematic illustration of a projected grating pattern with a point light source and a grating plate, such as a Ronchi ruling. The grating plate is placed between the light source and the object. When the light source is assumed to be a point source, the shadow of the grating plate is projected on the object. As shown in this Fig., the projected phase ϕ(xp, zp) at point P(xp, zp) on the object is equal to the phase at point G(xg, zg) on the grating plate. Point G is the intersection of a line that connects light source S(xs, zs) and point P with the grating plate.
The phase ϕ on the grating plate is expressed in Eq. (1),
where p is the pitch of the grating and ϕ 0 is the intercept value at x = 0. The x coordinate at point G is calculated from the coordinates of point S and point P using Eq. (2),The projected phase ϕ (xp, zp) at point P is obtained using Eq. (3),Figure 2 shows a schematic illustration of phase-shifted projected grating patterns using a light-source-stepping method. The projected grating pattern is changed by stepping the light source position, as shown in Fig. 2. The projected phase at point P on an object is also changed according to the position of the light source. When the position of the light source is stepped from S to S', which are separated by a distance Δx, the intersection of the line that connects the light source and point P on the grating plate is changed from G to G' and moves a distance Δxg, which is given by Eq. (4),The projected phase at point P is shifted by Δϕ, which is given in Eq. (5),This equation implies that the projected phase is shifted in proportion to the light source stepping distance Δx, and the proportionality coefficient is determined by the grating pitch of the grating plate and the z positions of the light sources, grating plate and point P on the object. Because the grating pitch of the grating plate, the z position of the light sources and the z position of the grating plate are fixed, the proportionality coefficient corresponds to the z position of the object surface.In reality, the light source is not a point but rather is spatially spread. The light diffracts at the grating plate. Thus, the exact projected phase differs from the right side of Eq. (3). However, when the grating pitch is approximately 1 mm, the influence of the diffraction is sufficiently small that the difference from Eq. (3) is small. For example, in the case that the grating pitch is 0.5 mm, the wavelength is 630 nm and the distance between a light source and the gratingplate is 15 mm, the diffraction angle becomes 0.00126 radians. This angle is around 4% of the angle of projected grating.
In the case of using a linear light source with the normal direction to the xz-plane instead of a point light source, the same principle is available. The intensity of the projected grating can be increased using the linear light source. Figure 3 shows the schematic illustration of a projected grating using a linear light source. The direction of a linear light source is the y-direction. When the direction of the linear light source is aligned with the direction of the grating slits, the projected grating patterns obtained by each point light source on the linear light source are overlapped. The grating alignment and the phase obtained by each point light source become conformable as shown in Fig. 3. As a result, the brightness of the projected grating pattern becomes higher.
3. Whole-space tabulation method (WSTM)
In the LSSM, the shifted phase varies according to the z position, as described above. The z coordinate and the phase obtained using the phase-shifting method have a one-to-one relationship.
A calibration method for an accurate and high-speed shape measurement using multiple reference planes were proposed. Figure 4 presents the principle of the calibration method. A reference plane oriented vertically to the z-direction is translated in the z-direction by small amount. A camera and a projector are fixed above the reference plane. The grating is projected from the projector onto the reference planes. The phase of the projected grating can be easily obtained using the phase-shifting method. A pixel of the camera obtains an imagealong the ray line L in Fig. 4(a). The pixel contains images of the points P0, P1, P2…PN on the reference planes R0, R1, R2…RN, respectively. At each point, the grating phases θ0, θ1, θ2…θN can be calculated using the phase-shifting method. Therefore, the correspondence between the heights z0, z1, z2…zN and the phases θ0, θ1, θ2…θN, respectively, is obtained. From these phase-shifted images, calibration tables are formed to obtain the z coordinate from the phase θ at each pixel, as shown in Fig. 4(b). This table can be constructed in the range that the phase is changed within 2π. The range has some limitation. There are no disambiguation of phases in this method.
Fig. 4 Principle of the whole-space tabulation method. (a) Correspondence between the height and phase of the projected grating at a pixel line. (b) Calibration tables to obtain the x coordinate from the phase θ.
This method theoretically excludes the effects of lens distortion and intensity warping of the projected grating on the measurement results. Although the imaging lens and the projected lens have some distortion or the wave profile of the projected grating is warped from the sinusoidal wave, the position of projected grating and each imaging ray line are fixed. Therefore the correspondence relationship between the phase and the height along a ray line is obtained accurately. Tabulation makes short-time measurement possible because the z coordinate can be obtained from the phase at each pixel using the calibration lookup tables, and this operation does not require any time-consuming complex calculation.
4. Linear LED device and response speed
A prototype of a linear LED device is produced for this experiment. Figure 5(a) shows a photograph of the developed linear LED array device. There are 9 parallel lines with 0.5-mm pitch. Each line has 30 LED chips connected in series. The size of the chip is 350 μm by 350 μm. The wavelength of the LED is approximately 630 nm. In Fig. 5(a), the leftmost line is switched on and the other lines are switched off. Figure 5(b) shows the driver circuit for the linear LED array. Each line has an NPN transistor (Toshiba, 2SC2240) as a switching device.
Verification of the response speed of the linear LED array is performed. A pulse generator produces a square wave, and the wave is applied to the base electrode of a transistor that is used as a switching device. The voltage of power supply is 117 V, and the lighting current is 113 mA. The response speed of the photosensor and the amplifier is 100 MHz. A change of lighting power of the linear LED array is detected by the photosensor. The amplitudes of the input square wave and output wave are simultaneously recorded by an oscilloscope as a time series. The frequency of the input square wave is changed from 1 kHz to 3,000 kHz.
Figures 6(a)-6(c) show waveforms output by the photosensor and input by the pulse generator with frequencies of 1 kHz, 12 kHz and 100 kHz, respectively. The result for the response speed of the linear LED array is shown in Fig. 7. The output amplitudes of the photo sensor are almost the same from 1 kHz to 200 kHz. The output amplitude of the photo sensor at 1,000 kHz is half of the output amplitude at 200 kHz. This result demonstrates that the linear LED array can be applied for high-speed shape measurement.
Fig. 6 Waveforms output by the photosensor and input by the pulse generator with frequencies of 1 kHz, 12 kHz and 100 kHz.
Fig. 7 Characteristic amplitude versus frequency for a rectangular wave for frequency ranging from 1 kHz to 3,000 kHz.
5. Experiment for high-speed height measurement
5.1 Experimental setup
Figure 8(a) shows a block diagram of the experimental setup. A high-speed camera (Redlake, Motion Xtra HG-100K) is used as an imaging device. The camera is set in an external trigger mode. The grabbed image size is set in 192 x 192 pixels. The maximum frame rate is 20,000 fps in this setting.
The orientation angle between the camera and the projector composed of the linear LED array device is approximately 30 degrees. The distance between the linear LED array device and the grating plate is 15 mm. The pitch of the grating plate is 1.02 mm. The distance between the grating plate and the lens is 30 mm. The projected grating pitch on the object can be reduced to 4 mm by the lens [12]. It is necessary to increase the brightness to capture the image at high speed.
Figure 8(b) shows a photograph of the experimental setup. In this photograph, a white woofer is placed in the measurement area. In this setup, the measurement area is approximately 20 mm x 20 mm and the measurement height is approximately 5 mm.
A flat white plate is used as a reference plane. The maximum difference between the ideal flat plane was 22 μm and the standard deviation from the ideal flat plane was 7 μm in the region of 40 mm x 40 mm in the reference plane. It is set on a linear stage such that it can be shifted in the normal direction. The repetitive accuracy and the positioning accuracy are 3 μm and 5 μm, respectively. The normal direction is defined as the z direction. Before measuring an object, the reference plane set on the linear stage is placed on the measurement area to produce phase-height tables for the WSTM. The reference plane is translated from 0 to 6.0 mm in 0.4 mm increments. The relationship between the phases of the grating projected onto the reference plane and the z position is recorded. The calibration tables are produced at every 2π/1000 of phase in a pixel-by-pixel manner. Figure 9 shows the phase-height table at the center pixel of the measurement area.
In this experiment, 3-step phase shifting method is employed. The phase θ at a pixel is calculated using the relationship between phase and phase shifted intensities as shown in Eq. (6),
where I0, I1 and I2 are intensities obtained at a pixel with 3-step phase shifting.5.2 Confirmation of accuracy using the reference plane
The reference plane is a flat plane object that is measured to confirm the accuracy of the system. The reference plane is moved from 0.5 to 3.5 mm in every 0.5 mm. The interval is different from the interval used when producing the phase-height tables, as mentioned in the preceding section.
Figure 10 shows a grating image, a phase map and a height map of the reference plane placed at 1.0 mm. Figure 11 shows cross sections taken along the horizontal center line of the height measurement results for the reference plane. Table 1 presents the average height, error and standard deviation along the cross section shown in Fig. 11. This result demonstrates that the error is less than 10 μm and the standard deviation of the height measurement is approximately 40 μm. No distortions are obtained.
Fig. 11 Cross sections along the horizontal center line of the height measurement results for the reference plane.
Table 1. Measurement results of planes on a horizontal center line
5.3 High-speed height measurement for a vibrating woofer
Height measurement was performed for a vibrating woofer. A sinusoidal signal with frequency of 50 Hz was applied to the woofer. The phase-shifting speed was 12 kHz. The grating images were obtained by a high-speed camera, as shown in Fig. 8. The camera was synchronized with the phase shifting using a trigger signal. Figure 12(a) shows a grating image projected onto the woofer. Phase shifting with three steps was performed in this experiment.
Figures 12(a)-12(c) show phase-shifted grating images projected onto the vibrating woofer. Figure 13(a) shows one of the height maps obtained in this experiment. Phase shifting with three steps was performed in this experiment. The phase map was obtained with the 3-step phase shifting method at 1/4000-second intervals. The height distribution was also obtained at 1/4000-second interval. Figure 13(b) shows a height map obtained from the phase map shown in Fig. 13(a) using the WSTM.
Figure 14 shows cross sections of the time series at 1/1000-second intervals along the horizontal broken line labeled A in Fig. 13(b). The height distribution of the vibrating woofer was measured at 1/4000-second intervals.
Fig. 14 Cross sections in time series at 1/1000-second intervals along the horizontal broken line A shown in Fig. 13(b).
Figure 15 shows the height change at point B on the woofer shown in Fig. 13(b). The results of a displacement performed using a laser displacement meter are plotted on the same graph. The laser displacement meter captured data at 1/1000-second intervals. The amplitudes obtained by the proposed method and the laser displacement meter were 2.01 mm and 2.12 mm, respectively. The difference was 0.11 mm.
These results demonstrate that the proposed method has the ability to measure the height of a vibrating object with a high-speed camera.
6. Conclusions
High-speed height measurement with 4 kHz with a phase-shift grating projection method using a high-speed camera was performed in this study. The light-source-stepping method (LSSM) using a linear LED array was employed for the grating projector. One of the advantage points of the LSSM is that high-speed phase-shift can be performed with changing the electrical signal. The whole-space tabulation method (WSTM) was employed to analyze the height distribution from the phase distribution obtained. The WSTM is suitable for the LSSM. It was experimentally verified that the response speed of the linear LED array produced by the authors was greater than 200 kHz. This response speed means the total response with the linear LED array and the transistor as a switching device. If a high-speed switching device such as an FET is used, the improvement of response speed can be expected. This response speed is sufficient for commonly used high-speed cameras.
In this experiment, the phase-shifting projection was performed at 12 kHz and the height measurement of a woofer vibrating at 50 Hz was performed at 4 kHz with a 3-steps phase-shifting method. The limit on the phase-shifting speed in this experiment depends on the brightness of projected grating.
Acknowledgments
This work was financially supported by the Hyogo COE Program Promotion Project (2010-2011) from Hyogo Prefecture, Japan. The authors would like to thank Asort Co., Ltd., Moire Institute Inc. and USHIO LIGHTING, INC.
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