Three-dimensional (3D) measurement has been widely used in the fields of medicine, chemical, engineering, and mechanical design^[1–3]. Among all 3D measurement methods, fringe projection profilometry (FPP) is one of the high-performance techniques, such as Fourier transform profilometry (FTP) and phase-shifting profilometry (PSP)^[4–6]. FTP uses only a single pattern for 3D shape measurement to promise the high speed. However, it suffers an overlapping spectrum, which limits its performance and makes it hard to calculate complex objects with large curved surfaces^[7–9]. Compared with FTP, PSP is popular among researchers because of its high stability and high efficiency^[9–11]. However, the retrieved phase information is wrapped in the range $(-\pi ,\pi )$ due to the arctangent function, leading to the need for more additional patterns to unwrap the phase^[6,12,13]. Therefore, many phase unwrapping methods have been proposed to unwrap the phase.

Generally, phase unwrapping is classified into spatial phase unwrapping (SPU) and temporal phase unwrapping (TPU) categories^[4]. SPU usually uses fringe patterns with a single frequency to obtain the wrapped phase^[14]. Then, the $2\pi$ phase ambiguity is solved by comparing it with the surrounding pixels. Although this method needs no extra patterns to recover the phase, it does not solve the phase ambiguity when measuring multiple isolated or abrupt depth changing objects, because the phase errors will propagate through the path on a discontinuous wrapped phase map^[14–16]. In order to overcome the deficiencies of the SPU algorithm, TPU methods have developed in recent years by projecting additional patterns to unwrap the phase^[15,17–19]. Compared with SPU, TPU retrieves the phase pixel-to-pixel without any other pixels’ information, which means it is suitable for any isolated object. Generally, TPU gets the wrapped phase (needs at least three patterns) and the fringe order (needs additional patterns to unwrap the phase) simultaneously. As two classic traditional TPU methods, Gray-code and multi-frequency phase unwrapping are commonly used. In Gray-code methods, several binary Gray-code patterns are sequentially projected after the phase-shifting patterns (to acquire wrapped phase) to obtain the corresponding fringe order. In this way, N periods wrapped phase theoretically requires ${\log }_{2}N$ Gray-code patterns for unwrapping^[20–22]. Consequently, many additional Gray-code patterns are employed to eliminate the $2\pi$ phase ambiguity of the high-frequency wrapped phase, which is not applicable in the field of high-speed 3D measurement. In multi-frequency methods, in order to obtain the absolute phase, it needs other wrapped phases with different periods (few periods, at least one) by phase shifting and then unwraps the phase with the aid of these wrapped phases based on the heterodyne principle^[23–25]. The traditional phase-shifting method requires at least three patterns to get a wrapped phase, and at least two different frequency wrapped phase maps are needed in multi-frequency methods. Thus, the minimum number of patterns required in multi-frequency methods is six. Obviously, it does not meet the requirements of fast measurement well.

To overcome the defects of these two traditional methods, numerous phase unwrapping algorithms have been developed. A series of phase-coding methods are proposed by embedding the information of the fringe order into fringe patterns to determine the corresponding fringe order^[26–30]. Wang et al. proposed a phase-coding method by projecting three-step phase-shifting patterns and three phase-coding patterns to acquire the absolute phase map^[31]. Zhou et al. employed a color phase-coding pattern to unwrap four-step phase-shifting patterns^[32]. Similarly, one complicated phase-coding pattern was used to get fringe order by Ma et al.^[33]. As shown above, this type of method requires at least four patterns, and it is difficult to unwrap the high-frequency fringe pattern. In recent years, the method of projecting a one-shot composite pattern, where multiple fringe patterns are embedded into one single pattern^[34], or a color pattern, can decrease the number of projected patterns a lot. However, using complex techniques to find the phase from a combined pattern or from a color pattern reduces measurement accuracy^[35]. Anyway, there are many drawbacks in the existing methods to achieve high-precision measurement results with high speed.

We propose a high-precision, high-speed 3D shape measurement method using only three inner shifting-phase patterns to acquire the 3D shape of measured objects. In this method, we encode the phase and the corresponding fringe order information into these three patterns. Thus, the wrapped phase and fringe order are obtained simultaneously without any additional patterns to unwrap the phase. Besides, it can suppress the influence of ambient light and noise well by encoding in the phase domain. Because only three patterns are projected, the proposed method can be applied in high-speed measurement fields. The experimental results demonstrate the stability and effectiveness of the proposed method. Furthermore, a high-speed measurement system is prepared to reconstruct moving objects at the camera’s full frame rate (1700 frames per second). The method of inner shifting-phase fringe projection was first proposed in our previous work [28,29], which required four patterns. In this paper, we reduced the number of patterns to three, which should achieve higher efficiency with fewer pictures.

We set up three patterns to achieve the wrapped phase and fringe order simultaneously by embedding the phase and fringe order information into the phase domain. These three projected patterns are shown in Fig. 1.

Three patterns are expressed as follows: $$I_1(x,y)=A+B[\sin (\varphi (x,y)+\alpha (x,y))-\cos (\varphi (x,y)+\alpha (x,y))],$$(1)$$I_2(x,y)=A+B[\sin (\varphi (x,y)-\alpha (x,y))-\cos (\varphi (x,y)-\alpha (x,y))],$$(2)$$I_3(x,y)=A+B[\sin (\varphi (x,y)+\alpha (x,y))-\cos (\varphi (x,y)-\alpha (x,y))],$$(3)where $I_n(x,y)$ represents the intensity of point $(x,y)$ in the $n$th pattern ($n=1,2,3$). $A$ and $B$ are the average intensity and modulated intensity, respectively. $\varphi (x,y)$ represents the absolute phase. $\alpha (x,y)$ is a single-frequency phase varying with vertical or horizontal pixels, so we name it shifting phase. If patterns are encoded with vertical fringe, then $\varphi (x,y)$and $\alpha (x,y)$ are written as follows: $$\varphi (x,y)=\frac{2\times \pi \times T\times x}{N},$$(4)$$\alpha (x,y)=\frac{\pi \times x}{N}-\frac{\pi }{2},$$(5)where $N$ represents the total number of pixels in one row of fringe pattern. $T$ is the number of fringe periods. The range of $\varphi$ is from 0 to $T\times 2\pi$, and the range of the shifting-phase $\alpha$ is from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$.

The wrapped phase and the shifting phase are calculated by Eqs. (1) to (3) as follows: $$\phi =\arctan \left(\frac{I_1-I_3}{I_3-I_2}\right),$$(6)$${\alpha }^{\prime }=\arctan \left(\frac{2I_3-I_2-I_1}{2A-I_2-I_1}\right)+\frac{\pi }{2},$$(7)$$2I_3-I_2-I_1=B\times \sin \alpha \times (\cos \phi -\sin \phi ),$$(8)$$2A-I_2-I_1=B\times \cos \alpha \times (\cos \phi -\sin \phi ).$$(9)

The range of the wrapped phase $\phi$ is from 0 to $2\pi$ in one period, and the range of the shifting-phase ${\alpha }^{\prime }$ is from 0 to $\pi$. $A$ is an unknown number, and we obtain its value with the same method in Ref. [36]. Equations (8) and (9) are the key steps to get Eq. (7). Since the wrapped phase is a curve with T periods, we use the single-frequency shifting phase to get the fringe order $k$ for each cycle of the wrapped phase: $$k=\mathrm{round}\left(\frac{{\alpha }^{\prime }\times 2T-\phi }{2\pi }\right),$$(10)where the function $\text{round}(·)$ is an integer function. In Eq. (10), the range of ${\alpha }^{\prime }\times 2T$ is from 0 to $T\times 2\pi$, which is the same as the absolute phase range that is encoded in Eq. (4). Thus, we get the fringe order by subtracting ${\alpha }^{\prime }\times 2T$ and the wrapped phase. The detail is shown in Fig. 2.

From the wrapped phase and fringe order, we get the absolute phase as follows: $$\varphi =\phi +2\pi \times k,$$(11)where $\varphi$ is the absolute phase. From Eq. (11), the accuracy of the absolute phase $\varphi$ depends on the unwrapped phase $\phi$ and the fringe order $k$. Since the fringe order is an integer, and the function “round” is used, $k$ is an accurate value. Moreover, the unwrapped phase $\phi$ is calculated by the light intensity value of three patterns ($I_1$, $I_2$, $I_3$) shown in the Eq. (6). Therefore, the accuracy of the unwrapped phase $\phi$ is only related to the light intensity. In general, the accuracy of absolute phase is only related to light intensity, because the fringe order $k$ is an accurate integer, and the unwrapped phase $\phi$ is only related to light intensity. Although ${\alpha }^{\prime }$ is a single-frequency phase, it does not affect the accuracy of absolute phase.

The procedure of our method is shown in Fig. 3.

The procedure of the proposed method can be listed as four steps.

Procedure 1: we achieve the wrapped phase by Eq. (6) and get the value of A with the method in Ref. [36] from the captured patterns.

Procedure 2: we obtain the shifting phase by Eq. (7) from the captured patterns and the value of A.

Procedure 3: we calculate the fringe order by Eq. (10) and calculate the absolute phase by Eq. (11) from the wrapped phase and fringe order.

Procedure 4: we retrieve the 3D reconstruction result by the absolute phase and calibration result.

In order to verify the performance of the proposed method, we set a high-speed FPP system, which includes a computer, an image grabbing card (Active Silicon CoaXPress-AS-FBD-4XCXP6-2PE8), a high-speed CMOS camera (Vision Research Phantom S210, $1280\times 1024$ resolution, full frame: 1700 frames per second), and a high-speed digital light processing (DLP) projection system. The high-speed DLP projection system consists of a DLP development kit (Texas Instruments DLP Discovery 4500) and a wide extended graphics array (WXGA) resolution ($912\times 1140$) digital micromirror device (DMD). On the basis of binary defocus technology, our projector can reach up to 4250 frames per second. The camera and the projector system are shown in the Fig. 4. In these experiments, we generated three high-frequency patterns with 70 fringe periods.

Section 1: we conducted an experiment to verify the accuracy of our method. In this experiment, we employed two methods to measure two standard ceramic spheres whose diameter is 50.805 mm, and distance between sphere centers (L) is 100.035 mm, based on manufacturing specification and precision. These two methods include the traditional method (three-frequency three-step algorithm)^[30,37] and the proposed method. The process and results of measuring standard ceramic spheres by the proposed method are shown in the Fig. 5. After obtaining 3D reconstruction results of these two methods, we compared the accuracy of two methods from five indices: diameter of the left sphere (DL), diameter of the right sphere (DR), L, standard deviation of the left sphere surface (SDL), and standard deviation of the right sphere surface (SDR). Figure 6 is the schematic diagram of measuring L and diameter of reconstruction results by MATLAB. In order to test the standard deviation, we reconstruct the measurement results in Geomagic Studio 2012 and fit them spherically. Moreover, for increasing the stability of results and reducing the randomness, standard ceramic spheres of six different angles have been measured to get an average value, as shown in Fig. 7. The average errors of center distance, standard deviation, and diameter are listed in Table 1. From Table 1, we see that the accuracy of the two methods is almost the same, confirming the high accuracy of the proposed method.

Section 2: we measured a cartoon mask to demonstrate the capability of absolute phase recovery of the proposed method. Figure 8(a) represents one of the three captured patterns of the cartoon mask. Figure 8(b) is the wrapped phase map of the cartoon mask. One cross section of the wrapped phase and its counterpart of the reference phase are plotted in Fig. 8(d). As shown in Fig. 8(c), a great 3D reconstruction result is achieved, proving the proposed method’s stability and robustness in measuring the complex object.

Section 3: the ability of calculating the wrapped phase and the fringe order pixel-to-pixel of our method can be proved by measuring isolated objects. We arranged an experiment to measure two sculptures (‘crab’ and ‘rabbit’) positioned on a black cloth background, as shown in Fig. 9(a). The black cloth completely separates the pixel points between the two sculptures. Figure 9(b) shows the wrapped phase of the two sculptures. One cross section of the wrapped phase and its counterpart of the reference phase are plotted in Fig. 9(d). The correct depth and good reconstruction results, as shown in Fig. 9(c), prove the feasibility of our method in measuring isolated objects.

Section 4: after confirming 3D measurement capability of the complex surface and the isolated objects, the proposed method can be widely used to measure colorful objects with varying textures in this subsection. Thus, we performed an experiment to measure a colorful ‘tortoise’ sculpture, as shown in Fig. 10(a). Figure 10(b) shows the wrapped phase map of this sculpture. One cross section of the wrapped phase and its counterpart of the reference phase are plotted in Fig. 10(c). Figure 10(d) shows the 3D reconstruction result of this colorful sculpture. Furthermore, we added the RGB information into the 3D reconstruction of this sculpture. Then, a colorful 3D reconstruction is obtained, as shown in Fig. 10(e). This experiment demonstrates that the proposed method reconstructs the colorful object well.

Section 5: a high-speed 3D measurement of the moving object (paper airplane with wings vibrating in the wind) is performed by our system. Since the proposed method uses three patterns to achieve the absolute phase map, we set the speed to the camera’s full frame rate (1700 frames per second) in this high-speed 3D measurement. The reconstruction speed of a moving object with the proposed method is 1700 frames per second, which is the camera’s full frame rate. For any frame of picture, the previous frame and the next frame are used to calculate the phase. In this way, we can get 3D information for each frame of the picture, as shown in Fig. 11. Figure 12(a) shows the paper airplane and the dynamics of the aircraft vibration. Figures 12(b) and 12(c) show one of the frames in the dynamic process (the collected picture and the reconstruction result). The wings of an airplane vibrate in the wind at a high frequency, and the amplitude of the vibration is within 20 mm, which shows that the small rapid vibration can be accurately captured and reconstructed by our method. We extract the corners (P1, P2, P3) shown in Fig. 12(c) of the paper airplane to show its vibration displacement (D) and frequency in the Z direction shown in Fig. 13. The whole 3D reconstruction process can be referred to in Visualization 1. In Visualization 1, the paper airplane is completely reconstructed with high quality, proving that the proposed method is capable of performing high-precision and high-speed 3D measurements.

Section 6: in the last experiment, we employed our system to measure the moving object (the fingers reciprocated in front of the camera) again. Pictures in the left of Figs. 14(a)–14(d) are representative camera images, and pictures in the right of Figs. 14(a)–14(d) are corresponding 3D reconstruction results at different frame points. We show the gestures of the fingers in different four frames (frame 44, frame 485, frame 1167, frame 1685). From the effect of the pictures below, we get good reconstruction results. A more detailed comparison of the original motion and reconstruction results is provided in Visualization 2. The success of this experiment proves that our method is well adapted to the high-speed 3D measurement field. The good reconstruction of fingers also shows that our method is applicable to measure multiple isolated objects.

A novel high-speed 3D shape measurement for moving objects has been presented. Since only three inner shifting-phase patterns are used to acquire the wrapped phase and the fringe order simultaneously, it meets the requirements of high-speed and full-field. The number of periods of projected patterns reaches 70, ensuring the accuracy. The experiments discussed above have confirmed the ability to measure complex, colorful, and isolated objects. Moreover, the high-speed 3D measurement experiment for moving objects at 1700 frames per second has also demonstrated its capability for high speed, high robustness, no geometry constraint, and full-field 3D shape acquisition.