## Abstract

In this paper, a fast rotary mechanical projector (RMP) is designed and manufactured for high-speed 3D shape measurement. Compared with the common high-speed projectors, RMP has a good performance in high-speed projection, which can obtain high quality projected fringes with shorter camera exposure time by using the error diffusion binary coding method and chrome plating technology. The magnitude, acceptability of systemic projection error is analyzed and quantified in detail. For the quantified error, the probability distribution function (PDF) algorithm is introduced to correct the error. Corrected projection error is reduced to more than one third of the original error. Subsequently, a monocular measurement system composed of the RMP and a single camera is constructed. The combination of the RMP device and PDF algorithm ensure the accuracy of a corresponding 3D shape measurement system. Experiments have demonstrated that the proposed solution has a good performance for the 3D measurement of high-speed scenes.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Structured light three-dimensional (3D) shape measurement technique is concerned with extracting the geometry information from the pattern of the measured object. Measured scene is sequentially illuminated by N-frame (N$\ge$1) encoded patterns, the distorted patterns modulated by the object height information are recorded by camera. The height information of a measured object can be recovered by demodulating the deformation patterns. Among them, fringe projection profilometry approaches have been proven to be the most promising technique and used for 3D sensing, machine vision, robot simulation, industrial monitoring, dressmaking, biomedicine, due to high precision and high flexibility [1–5]. Fourier transform profilometry (FTP) [6–8] and phase-shifting profilometry (PSP) [9–12] are the most widely used ones among the many proposed fringe projection profilometry. FTP method projects a single high-frequency fringe pattern to the measured object surface, and the phase can be extracted by applying a properly designed band-pass filter in the frequency domain. More technical details about FTP approaches can be found in the review articles [6,8]. Besides, the windowed Fourier transform (WFT) [13,14] and the wavelet transform (WT) [15,16] can also be used for the phase demodulation of single high-frequency fringe pattern. Meanwhile, PSP method generally requires at least three phase-shifting fringe patterns to demodulate the phase information of the measured object. It can provide higher measurement accuracy since it completely eliminates interferences from ambient light and surface reflectivity. Thus, PSP method is the first choice for high precision 3D measurement [12].

Recently, 3D shape measurement for a dynamic object or process has been a hot topic due to its wide field of application [2,3]. Compared with the static measurement, the difficulty of dynamic measurement is how to keep the object relatively static during the N-frame patterns projection to reduce or avoid the inter-frame error caused by motion. Among the research approaching in this field, three mainstream paradigms are observable. (1) Some scholars are committed to using the fringe projection method of one-shot, the representative method is FTP [17,18]. One-shot feature makes FTP method avoiding the inter-frame motion successfully, thus it highly suitable for 3D shape measurement of dynamic surfaces. The review article in Ref. [2] provides an overview of dynamic shape measurement based on FTP and its typical applications. However, due to the band-pass filtering, FTP method will loss precision when measuring an object with sharp edges, abrupt change or non-uniform reflectivity. (2) Some scholars developed methods to compensate for errors caused by inter-frame motion during measurement [19–21]. (3) In addition, some scholars are focusing on increasing availability of high-speed projection and imaging hardware to minimize inter-frame motion. With the increasing availability of high-speed projection and imaging hardware, multiple frames approaches will attain accuracy to levels unreached using one-shot approaches. High-speed imaging hardware with high sensitivity and high resolution have been commercially available for some time. However, the most commonly used high-speed projection device, such as liquid crystal display (LCD) or digital light processing (DLP) projector, can only operate properly within a limited spectral range and are restricted in terms of speed (especially in 8-bit greyscale mode).

To overcome the limitations of existing projection techniques, Patrick et al. developed a fast and low-cost structured light pattern sequence projection in 2011 [22], achieved a binary pattern projection at the rate of 200 Hz and the 3D reconstruction rate of 20 Hz. They adopted off-the-shelf components and ordinary machinery manufacturing for an economical perspective. Thus, the projection equipment made some concessions in accuracy and speed. Subsequently, Heist et al. proposed the GOBO (GOes Before Optics) projector [23,24]. The GOBO projector used a rotating slide structure to project aperiodic sinusoidal fringe patterns at high frame rate for 3D point clouds reconstruction. While, due to the nature of aperiodic sinusoidal fringe project method, an extra high-speed camera was required for 3D information reconstruction. At the same time, Hyun et al. [25] also developed a 3D measurement system in 2018, which consists of a metal-based pattern mechanical projector and two high-speed cameras. The encoded texture images and disparity map between two cameras were used to achieve high-speed 3D shape measurement. Despite both Heist’s and Hyun’s solutions having good performances, structured light measurement method with two cameras and one projector will bring more shadow-related 3D point cloud data holes since all three devices must cover the same point in order to recovery its height. Considering the system cost and result integrity, a monocular measuring system constituting of a single camera is still desirable and has its practical application requirements.

In our previous work [26,27], a monocular measuring system was presented by combining with a low-cost rotary mechanical projector (RMP) and single camera. This low-cost projector used squared binary defocusing (SBM) technique, Ronchi grating was printed on a film sheet as a projection disc, and sinusoidal fringe was obtained by defocusing the projection lens. Temporal Fourier transform profilometry (TFTP) method was used for phase extraction. Because the accuracy of TFTP method is not sensitive to precise phase-shifting, low-budget materials and manufacturing processes was adopted. The feature of TFTP makes its measuring accuracy to be affected when the sampling theorem in the temporal domain is not satisfied. Thus, for objects with complex dynamic distributions, this solution is likely to be limited because of the use of Fourier fringe analysis along the temporal axis.

Breaking the limitation of TFTP method and reaching a higher accuracy for 3D measurement of dynamic scenes are what we are doing. Compared to the TFTP method, PSP methods have advantages in accuracy and robustness. Moreover, PSP methods do not require filtering process, phase information can be restored from a minimum of three fringe patterns. Obviously, the device we proposed earlier does not apply to PSP methods. Therefore, obtaining precise phase-shifting fringe and improving projection performance, making RMP more flexibly applied to deal with the dynamic scene of complex changes, are the research targets to be done. To this end, the disc and structure of the RMP were redesigned in this paper.

Firstly, new RMP uses error diffusion binary coding grating and manufactured by chromium plating to substitute Ronchi grating that printed on film. Chrome plating technology provides more dots for binary grating coding, the error diffusion coded binary grating can produce high quality sinusoidal fringe by slightly defocusing, which will improve the projection quality of the system. Besides, better components and high-precision manufacturing processing are adopted to reduce the impact of system processing errors on projection accuracy. Secondly, the probability distribution function (PDF) is introduced for further phase-shifting error correction. The corrected system can obtain more precise fringe phase-shifting information, so the combination of new RMP device and PDF algorithm ensure the measurement accuracy of the proposed system. Several experimental results have proved the system’s ability in high-speed 3D measurement.

This paper is arranged as following: Section 2 illustrates the principle and system design exactly. Section 3 analyzes and corrects the system error, while evaluates projection performance of the new setup. Section 4 presents the experimental results to verify the performance of the proposed method. Section 5 discusses the strengths and weaknesses of the proposed method. Section 6 summarizes this work.

## 2. Principle

#### 2.1 Phase-shifting profilometry

In PSP method, multiple phase-shifting sinusoidal fringe patterns are projected sequentially onto an object surface. The distorted fringe distribution captured by the camera can be represented as:

where*A*(

*x*,

*y*) is the background intensity of the fringe image,

*B*(

*x*,

*y*) is the so-called intensity modulation,

*ϕ*(

*x*,

*y*) is the corresponding phase to be measured,

*δ*is the certain phase-shift, and

_{n}*n*represents the phase-shift time index

*n =*0, 1, 2, 3….

*A*(

*x*,

*y*),

*B*(

*x*,

*y*),

*ϕ*(

*x*,

*y*) are three unknowns, it is obvious that at least three independent equations are required to solve$\textrm{\; }\phi $(

*x*,

*y*). For simplicity, the variable’s pixel coordinates in the formula below are omitted. For example,

*ϕ*(

*x*,

*y*) is replaced by

*ϕ*.

Taking the five-step phase-shifting as an example, the captured fringes are:

The least squares estimate to this series of *N* equations is given by the solution of the following matrix equation:

After finding the matrix *X*, the wanted phase information $\phi $ can be obtained as:

Since the arctangent function only ranges from −π to π, the obtained phase value provided from Eq. (5) will have 2π phase discontinuities. The corresponding natural and continuous unwrapped phase can be obtained by the phase unwrapping algorithm. In this paper, the phase unwrapping algorithm guided by the reliability map is adopted [28]. Intensity modulation is selected as the parameter to identify the reliability of the phase data and the direction of phase unwrapping. The path of phase unwrapping is guided according to the parameter map and will always along the direction from the pixel with higher reliability value to the pixel with low reliability value to enhance noise immunity.

In order to obtain 3D surface information of the object, the system needs to be calibrated using the phase-to-height algorithm [29]. Equation (6) can be used to reconstruct the height of a measured object.

where Δ*ϕ*is the phase value of the measured object, relative to the reference plane.

_{r}*h*is the measured object’s height. Four planes with known height distributions should be measured for 3 unknown parameters’

*u*,

*v*and

*w*calculation, and the 3D shape result of the measured object can be restored from Eq. (6).

The camera calibration technique proposed by Zhang [30] is implemented to calibrate the camera in our developed system for X-axis and Y-axis calibration.

#### 2.2 Projection principle and system design

Projection principle of rotary mechanical projector (RMP) is shown in Fig. 1. The RMP contains a light source, beam shaping optics, a custom-made grating disc, an actuating motor and projection lens. The light from the source is collimated and homogenized by beam shaping optics and then the custom-made grating disc is illuminated. Servo motor drives the disc to rotate, fringe patterns are transported and imaged onto an object to be measured through projection lens.

Light source adopts a 50-watt white LED, enough radiant flux can make the camera to work with a short exposure time, which is conducive to high-speed measurement. Designed grating disc and magnified part are shown in Fig. 2. The diameter of the disc is 100 mm. Inner ring of the disc is radial binary-encoded grating, which is encoded by the binary error diffusion method for sinusoidal structure illumination [31,32].

Design method of error diffusion radial grating is as follows. The standard sinusoidal transparency can be encoded:

*P _{p}* is the period of the illumination structure at the grating plane.

As an active Halfton technique, error diffusion algorithm was first proposed in 1994 [33]. The idea is that the quantization error of the processed elements is weightedly diffused to the untreated elements for reducing the quantization error of the whole coding field. The revised sinusoidal pattern by error diffusion *f*(*x _{P}*,

*y*) can be expressed as:

_{P}*d*$\textrm{(i,j)}$ is the distribution coefficient of error diffusion controls the direction of error diffusion. $\textrm{d(i,j)} \ne \textrm{0}$ is for the unprocessed pixel, and $\textrm{d(i,j)}\,\textrm{ = }\,\textrm{0}$ for the processed pixel. $\textrm{e(}{\textrm{m}_\textrm{p}}\textrm{,}{\textrm{n}_\textrm{p}}\textrm{)}$ is the diffusion error of pixel $\textrm{(}{\textrm{m}_\textrm{p}}\textrm{,}{\textrm{n}_\textrm{p}}\textrm{)}$, $\textrm{t(}{\textrm{m}_\textrm{p}}\textrm{,}{\textrm{n}_\textrm{p}}\textrm{)}$ is the threshold, generally set as 0.5 for binarization.

The error diffusion grating binary-encoded utilizes two gray scales to approach the sinusoidal transparency function in the spatial domain. Though the slightly defocused projection and the inherent effect of lowpass filter by optical systems, a high-quality sinusoidal pattern is obtained in the image plate.

Outer ring of the disc is the signals part, detected by an optical synchronization unit to track and feedback phase-shifting information. The implementation process is shown in Fig. 3. Optical synchronization unit adopts an optocoupler sensor. It will collect the phase-shifting information and generate the square wave to trigger the high-speed camera strictly. When the optical transmission of optocoupler sensor is blocked by the signal, high level signal is output and camera is triggered to capture deformed fringe.

As shown in Fig. 3, the motor drives the disc to rotate continuously, each signal will trigger a record of the currently projected fringe and provide the amount of phase-shifting between the two signals. Signals part used in this paper will feedback the five-step phase-shifting signal and produce 35 signals within one round for the realization of 3D measurement profilometry.

This disc is manufactured by using etching chromium plates technique. Chrome plate uses quartz glass as the base material, and the surface is coated with chrome film. For every unit, using polar coordinate laser direct writing system, the chromium plate is heated, oxidized and chemically etched according to designed transparency function, which finally produces a high-resolution optical mask. Chromium plating technology can achieve the location of micrometer unit. The size of a single micrometer unit of grating template is 4 μm. For about 20 mm×20 mm illuminated areas, 5000×5000 dots can be used for binary encoded.

Compared with the Ronchi grating printed on film in the previous work, the error-diffused chrome-plate grating needs less defocusing to obtain sinusoidal fringe, which can guarantee the fringe contrast and obtain higher phase precision. In addition, chrome plate etching technology provides more coding dots in a smaller area, which can make the equipment have both better precision and integration.

The speed of the rotational motion driven by the actuating motor can be adjusted from 0 to a maximum of 3000 rpm (round per minute). Due to grating disc producing 35 signals within one round, the projection frequency of our RMP can reach up to 1.75kHz. Moreover, the light source of RMP is interchangeable, RMP can also be used in a broad spectrum.

## 3. Projection error analysis and correction strategy

During the RMP manufacturing process, such as the manufacture of the disc, the assembly of the motor and disc, there are inevitably some manufacturing errors in the device. For RMP, the possible manufacturing errors are mainly from eccentricity error, axial error and uneven disc-making. Eccentricity error will occur when concentric runout exists in the motor shaft and center disc hole. Axial error will occur when the disc assembly is non-perpendicular to the rotation axis. The existence of these errors may reduce the phase-shifting fringe accuracy which is directly related to the quality of 3D reconstruction.

In our equipment, the eccentricity error is controlled within 0.01 mm and axial machining error is controlled within 0.03 mm during the high precision machining process, and all of them are further corrected and compensated during the measuring process. To evaluate the projection phase accuracy obtained by RMP system, a plate covering the entire measurement field is repeatedly measured and the results are analyzed.

The measured plate remains stationary, four groups of phase-shifting fringes are obtained from four areas (corresponding to the 0, 1/4, 1/2, 3/4 areas of disc, as shown in Fig. 4) within one rotation of the disc.

One of five phase-shifting deformation fringe patterns at four different projected areas are shown in Figs. 5(a)–5(d). Their corresponding wrapped phase and unwrapped phase are shown in Figs. 5(e)–5(h) and 5(i)–5(l) respectively. As can be seen from the results, the measured phase has been major influenced by periodic fluctuations related to double frequency of the projected fringe.

Schwider et al. [34,35] demonstrated that the phase-shifting error will inevitably influence the accurate phase reconstruction if a phase-shifting technique is used, and the corresponding phase error $\mathrm{\Delta }\phi \textrm{(x,y)}$ calculated by standard *N*-step phase-shifting algorithm is:

*ϕ*(

*x*,

*y*) is the measured phase, the coefficients ${\textrm{S}_\textrm{1}}$, ${\textrm{S}_\textrm{2}}$, and ${\textrm{S}_\textrm{3}}$ are approximately proportional to the phase-shifting error. The phase error is related to 2$\phi $, and the frequency of phase error is twice that of the fringe frequency. The phase recovery results indicate that manufacturing error of the system affects the precise feedback of phase-shifting.

In an ideal state, four exactly same phase results will be obtained. And there will be deviation between the four phase results when systemic projection error exists. Thus, taking the result in area 0 as a benchmark, the differences of unwrapped phase obtained from area 0 and other areas are compared and analyzed. These three differences are labeled as group 1 (area 0 and area 1/4), group 2 (area 0 and area 1/2), group 3 (area 0 and area 3/4) respectively, which can reflect the systematic projection error clearly. Three sets phase differences are shown in the Figs. 6(a)–6(c), and Figs. 6(d)–6(f) give their cross lines on 400th row.

The phase differences measured at different areas around the disc also mainly come from phase-shifting error*.* The means of error are 0.0004, -0.0057, -0.0128 respectively, and the Standard Deviation (STD) is 0.0111, 0.0136 and 0.0090. For higher precision phase acquisition, probability distribution function (PDF) is introduced for further phase-shifting error correction.

#### 3.1 Probability distribution function

Probability distribution function (PDF) is a common statistical method [36,37]. Using PDF as the objective function of wrapped phase, which can be represented as:

*M*is the number of sampling points in the range of [-π, π],

*m*= 0, 1, 2, …,

*M*-1. For example,

*M*is chosen to be 64, it means that the wrapped phase is divided into 64 regions for evaluation and correction. The number of pixels in each wrapped phase region is counted, and its ratio to the total number of pixels is the PDF value of the sampling point. When the number of sampling point is sufficient, the larger

*M*is, the more accurate the results will be.

If there is no phase-shifting error in the measurement system, then the probability of each phase value is theoretically the same, and the PDF curve is a uniform curve. An existing phase-shifting error will change the probability of the original phase value and make the PDF curve change. After statistically obtaining the measured PDF curve in the actual fringe, the precise phase-shifting value can be recovered with its minimum STD value.

Taking the five-step phase-shifting fringes as an example, we conduct simulation experiments about the influence of phase-shifting error on phase accuracy and PDF curve. Figure 7(a) shows the phase errors under several increasing phase-shifting errors. Legends 0∼ pi/10 indicate phase-shifting error of 0∼π/10 for each phase-shifting step. Figure 7(b) shows the PDF curves under the corresponding phase-shifting error. Obviously, the amplitude of the PDF curve is changing with the phase-shifting error. PDF curve is a straight line when the phase-shifting error does not exist, and the probability of each sample point is 1/*M*, i. e. *F*(*M*) = 0.0156 when *M* is chosen as 64. Thus, the STD of the PDF curves is used as the criterion to judge the existence of phase-shifting error. For multiple phases obtained under the same conditions, the phase with the smallest phase-shifting error will be found when STD of the PDF curve is minimum.

#### 3.2 Phase correction

The phase correction process shown in Fig. 8 can be described in the following four steps.

Step 1: For deformed fringes with phase-shifting error, multiple sets of phase-shifting errors are generated with interval *U* in the possible phase-shifting error range.

Step 2: Calculating phase distributions and PDF curves of each set of phase-shifting error.

Step 3: Calculating the STD of PDF curves.

Step 4: Determining the corresponding phase-shifting error with the minimum STD value, then the precise phase-shifting value and accurate phase can be obtained.

From the simulation results in section 3.1, it can be seen that the phase-shifting error of π/300 will result in a small phase error $\mathrm{\Delta }\phi \textrm{(x,y)\; }$with the STD of 0.0039 and the mean of 0, which is very tiny within the acceptable range. Thus, the interval *U* is set to π/300 for phase error correction of RMP in this paper.

According to Eq. (2), for the first set of five-step phase-shifting fringe patterns *I*_{0}∼*I*_{4}, there are four uncertain phase-shifting values *δ*_{1}∼*δ*_{4}, which will be retrieved after step1-4 performing. Since the projection pattern of RMP is continuously producing a series of phase-shifting fringes during measurement, the *n*-frame multiple phase-shifting fringes *I*_{0}∼*I _{n}* should also be subsequently corrected. For next group phase-shifting fringes

*I*

_{1}∼

*I*

_{5}, the phase-shifting values of

*I*

_{1}∼

*I*

_{4}have been confirmed, and only last one uncertain phase-shifting of

*δ*

_{5}should be determined. Similarly, steps 1-4 are performed again on

*I*

_{1}∼

*I*

_{5}to obtain the precise phase-shifting value of

*δ*

_{5}. Repeat this process and the rest uncertain phase-shifting in

*I*

_{0}∼

*I*can be calculated in this same manner for the entire measurement process.

_{n}For the measured plate, the corrected phases shown in Fig. 9 are almost not affected by the phase-shifting error. The original and corrected PDF curves are shown in the Fig. 10. It can be seen that the original PDF curve appears with periodic fluctuations suffering from the phase-shifting error, and the corrected one tends to be even with minor fluctuations caused by noise in the system.

Differences of the plate phase after phase-shifting correction on the plate are shown in the Figs. 11(a)–11(c). Same as Fig. 6, Figs. 11(d)–11(f) show the cross lines on 400th row. The error amplitude after correction is about 1/4∼1/3 of that before correction. The STD and mean of the corrected phase differences and original phase are shown in Table 1. As can be seen from this result, the projection errors associated with the phase-shifting error are well suppressed.

#### 3.3 System repeatability

In this section, the repeatability of the RMP system is tested. For the plate covering the measured field, multiple measurements were made and the measuring results in different rotation of disc were analyzed. Taking one measuring result as reference, the difference between three other random samples and the reference are shown in Fig. 12, and their STD is 0.0012, 0.0021, and 0.0017 radian respectively. Experimental results show that RMP has a favorable repeatability.

#### 3.4 Projection performance

In this section, the RMP and common high-speed projection device DLP4500 were compared with a Baumer HXC40NIR camera to illustrate their performance. The used camera was symmetrically placed on the central axis of two projection system, the working distances and the angles between the RMP/DLP projector and the camera were kept the same. Thus, the phase reconstruction accuracy of each monocular 3D measuring system is adopted to evaluate the projection accuracy.

In comparative experiment, a precisely made standard ceramic plate was placed at their respective measurement volumes and randomly transformed into four different positions during the measurement process. The DLP4500 projected the five-step phase-shifting binary error diffusion coded radial fringes with the same period as that of RMP. The camera’s exposure time was set as 300 μs for RMP and 2200 μs for DLP respectively to ensure their recorded images have same intensity.

In Table 2, the third column shows the deformed fringes in four different positions captured by RMP and DLP 4500, the fourth column shows the fringe’s profile in same row. The phase retrieve results are shown in fifth column. Eighty percent of the area of the ceramic plates are used for accuracy evaluation. Their error distributions relative to its own fitted plane are shown in sixth column. In this experiment, the measuring accuracy is evaluated by the root-mean-square (RMS) of non-outlier points from the fit surface. And the RMS values of RMP and DLP projector are labeled in each subgraph respectively.

Experimental results demonstrate that the RMS values of recovered phase have very small difference between RMP and DLP4500. With the greater light flux of the rotating grating, RMP can take a shorter exposure time, only one-seventh of DLP projector, to obtain the appropriate fringe contrast. While the DLP projector works, the high-speed camera needs longer exposure time which limits the measuring speed. However, compared with DLP4500, RMP cannot change projection mode flexibly.

## 4. Experiments and results

The prototype implementing the described concept is depicted in Fig. 13(a), which consists of a Baumer HXC40NIR high-speed camera, an optical synchronization unit and the RMP. Figure 13(b) shows the servo motor used to drive the disc rotation. One measurement process is shown in Visualization 1.

#### 4.1 Accuracy analysis

For a quantitative evaluation 3D reconstruction performance of this proposed method, we evaluate the measuring accuracy from three aspects: probing error, spacing error and flatness error. First, a standard ceramic ball was measured to evaluate the probing error, corresponding to the volume accuracy of measurement system. As shown in Fig. 14(a), the radius of the ceramic ball is 25.3996 mm measured by the coordinate measurement machine. Figure 14(b) shows its wrapped phase. And Fig. 14(c) shows the overlays the ideal sphere and the measured data. The measured radius is 25.3483 mm and its RMS is 0.0904 mm. The error distribution is shown in Fig. 14(d).

Then, a step-shaped workpiece shown in Fig. 15(a) was measured to evaluate the spacing error and flatness error. The step height is designed to be 30 mm, and the machining error is less than 30 microns. Figures 15(b) and 15(c) give the wrapped phase and its 3D reconstruction result. The flatness error of the measured three planes is derived from the deviation of their fitted planes. The error distribution of each step is shown in Fig. 15(d), and their RMS errors are 0.0809 mm (Plane 1), 0.0333 mm (Plane 2) and 0.0534 mm (Plane 3).

Because the optimal defocusing level is in the middle region of the calibrated volume, Plane 2 exhibits its best flatness. Figure 15(e) shows the lateral view of the 3D reconstruction result. The mean value of height on three planes are 0.0020 mm (Plane 1), 29.9503 mm (Plane 2) and 59.9308 mm (Plane 3).

#### 4.2 Measurement on dynamic scenes

Lastly, two group experiments were conducted to demonstrate the capability of our proposed method for high-speed 3D shape measurement. The first measured scene was a swing hand. The rotation speed of the mechanical projector was adjusted to 515 rpm (to match the camera's fastest shooting speed). The RMP sends out 35 pulses per rotation to feedback the phase-shifting information. In the whole process, the Baumer HXC40NIR camera resolution was set as 1024 × 1024 pixels, the exposure time was set as 300 µs to capture 300 frames image in one second. Representative texture map (average intensity of the five phase-shifting fringe images) and one of the corresponding deformation fringe patterns are respectively shown in Figs. 16(a) and 16(b). The 3D reconstruction results with the rate of 296fps are shown in Fig. 16(c).

In second experiment, the process of a sculptural model moving forward in space was measured. Different from the first dynamic experiment, the changes in this experiment mainly focus on the direction parallel to the camera’s optical axis. Rotation rate of RMP was also 515 rpm. Camera resolution and exposure time was set as 1024 × 1024 pixels and 300 µs. The high-speed camera captured 300 frames per second of distorted fringe and reconstructed speed is 296 frames. Figure 17(a) shows a few representative 3D frames. One of the corresponding deformed fringes and the reconstructed results are shown in Figs. 17(b) and 17(c) separately.

The results of two experiments confirmed that the proposed solution has good performance for high-speed scenarios measurement. It should be noted that RMP was set at a low rotation rate and does not give full play to its performance in both experiments, which is mainly limited by the shooting rate of the camera.

## 5. Discussion

The major differences between our presented work and that developed by Heist in Refs. [23] and [24] are listed for better distinguish: 1) our projection disc is designed by error diffusion method and made by chromium technology for producing high quality sinusoidal fringes, instead of a metal disc encoded by aperiodic fringe; 2) the fringe phase-shifting information is feedback through the signal part of the disc and the camera is triggered synchronously to get the phase-shifting fringe; 3) PDF method is presented for phase-shifting error correction to reach the more precise result; 4) a monocular system composed of RMP and a single high-speed camera is used for phase extraction and height recovery in this work.

The major advantages between this proposed method and our previous work as follow:

- 1.
*The projection disc is designed by error diffusion binary fringe coding method and manufactured by chromium plating technique.*Compared to Ronchi grating, which is printed on film, chrome plating allows coding of micron-sized units. Although the size of the disc is reduced by nearly half, the number of dots available for coding on the disc increases to 1.9 billion. In addition, the error diffusion coded binary grating can obtain high quality sinusoidal fringe with less defocusing, which will also improve the projection quality of the system. - 2.
*Better components are selected and high-precision manufactured to reduce the system errors.*In RMP provided in this paper, components of the projector are optimized. For instance, servo motor provides better stability instead of low-cost motor, a better uniform collimation equipment is adopted and so on. Meanwhile, high precision machining process reduces the mechanical error on projection accuracy to a great extent. Moreover, the magnitude and acceptability of residual system mechanical errors are further evaluated, analyzed and corrected. - 3.
*Probability distribution function is introduced for further error correction.*By simulating the PDF curve clusters with different phase-shifting error and searching for the minimum STD position, the phase-shifting closest to the real value can be found. However, PDF method is a process of circular search. This means that there is a time consumption compared to the regular PSP method. Time consumption is mainly related to the magnitude of phase-shifting error and the selection of interval*U*. The larger the phase-shifting error is, the larger the possible error range will be. When the interval*U*is determined, more possible phase-shifting error values will be generated, increasing the number of cycle search and time consumption. Similarly, within the range of possible phase-shifting error, the smaller the interval*U*is, the higher the phase accuracy will be obtained. But it will also bring more circular search and time consumption. The phase-shifting error caused by the RMP will not exceed π/300 or less, depending on the tradeoff between a higher precision and time consumption. - 4.
*RMP performs better at high-speed fringe projection.*The projection performance of RMP was assessed by comparison with the common DLP4500 projector. High-speed camera with RMP can take a shorter exposure time (around 1/7 of DLP4500) when obtaining the same quality of deformed fringes. This means that RMP has obvious advantages in high-speed fringe projection measurement. And the features of RMP make it suitable for a wider spectrum. But at the same time, more flexible mode transformation and almost nonexistent phase-shifting error of DLP4500 can be not ignorable.

## 6. Conclusions

This paper has proposed a solution for high-speed 3D shape measurement. A monocular measurement system using rotary mechanical projector (RMP) is constructed. RMP adopts error diffusion binary coding method and chrome plating technology to encoded high accuracy sinusoidal fringe pattern. Compared with common high-speed projectors, RMP can obtain almost same quality projected fringes with a shorter exposure time, which is a clear advantage in high-speed projection. Moreover, RMP can be applied to a broader light spectrum.

High-reliable components and high-precision machining process are adopted to greatly reduce the possible systemic errors. The residual error is analyzed quantitatively and the probability distribution function (PDF) method is introduced for phase correction.

The integration of RMP device and PDF algorithm ensures the projection accuracy of phase-shifting. Using the PSP method, standard parts and two dynamic scenarios were measured. Experimental results have demonstrated that the proposed method can achieve 3D measurement of high-speed dynamic scenes at a rate of 296 Hz, which is mainly limited by the captured speed of the camera.

## Funding

National Natural Science Foundation of China (62075143, 61675141).

## Disclosures

The authors declare no conflicts of interest.

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