Three-dimensional (3D) shape reconstruction has been widely used in the practical applications of modern information technology in machine design, industrial inspection, robotics, and game industry due to its high speed and high flexibility^[1–6]. The fringe projection profilometry (FPP) technique is considered to be one of the most reliable methods for restoring the 3D shape of objects due to its efficiency^[7–9]. The FPP technique is principally used to project the fringe patterns onto the target object and to capture the distorted fringe pattern(s) by an angled camera. Then, the corresponding wrapped phase can be recovered from the captured images by using different fringe analysis algorithms, which derived phase-shifting measurement profilometry (PMP)^[10–12], Fourier transform profilometry (FTP)^[13,14], etc. These methods extract the phase information of the measured object’s height by using arctangent calculation. Therefore, the phase distribution is wrapped within the value range from $-\pi$ to $\pi$; sequentially, the phase unwrapping must be implemented to acquire a continuous phase map for accurate 3D shape reconstruction.

The phase unwrapping algorithms can be commonly classified into spatial phase unwrapping^[15–17] and temporal phase unwrapping^[18–21]. For spatial phase unwrapping, each single wrapped phase map of a measured object can be unwrapped in the spatial domain, which usually uses the phase relationship between adjacent pixels to obtain the unwrapped phase value. However, such spatial phase unwrapping algorithms have one common limitation: they tend to fail if the abrupt surface changes introduce more than $2\pi$ phase variation from one sampling point to the next. In contrast, the temporal phase unwrapping algorithm fundamentally tries to eliminate the limitation of spatial phase unwrapping by capturing more patterns. They can be used to analyze the highly discontinuous object because the unwrapping process is independent for each spatial point.

Among temporal phase unwrapping algorithms, one popular method is to use the dual-frequency (or wavelength) phase-shifting technique^[22–24]. The projected fringe patterns with different frequencies are used to generate an equivalent phase map, ${\phi }_{\text{eq}}$, whose frequency can be determined with the difference of $f_1$ and $f_2$, i.e., $f_{\text{eq}}=f_1-f_2$. Since the synthetic frequency is lower than both frequencies of the projected fringes, the measurement range depth can be extended. However, the measurement accuracy is reduced with the increasing noise and lower equivalent frequency phase.

The phase sum with high sensitivity in two-wavelength phase-metrology (TWPM) was proposed for the first time, to the best of our knowledge, by Mustafin and Seleznev in 1970^[25], and then Di et al. applied phase sum in digital holography for improving the measurement accuracy^[26,27]. Inspired by their study, combined with the superiority of the higher- and lower-frequency fringes by synthesizing the projected dual-frequency fringes, a method of 3D shape reconstruction based on dual-frequency fringes is proposed, in which large measurable depth and high measurement accuracy can be achieved simultaneously. The phase difference of the projected dual-frequency fringes is used to produce a wrapped phase with synthetic lower-frequency information, which enlarges the range of the measurable depth. With the help of the phase difference, the unwrapped phase sum is obtained as the final measurement results. The unwrapped phase of synthetic higher-frequency information, which is extracted from the phase sum, can improve the measurement accuracy of the system. The simulation and experimental results verify the availability of this proposed method.

In 3D shape measurement using fringe structured light, the intensity of deformed fringes recorded by a digital camera can be given by $$I_n(x,y)=R(x,y)\{A(x,y)+B(x,y)\cos [\varphi (x,y)-2\pi n/N]\},$$(1)where $(x,y)$ is the pixel coordinate, $R(x,y)$ represents the uneven reflectivity of the tested object surface, $A(x,y)$ is the background intensity, $B(x,y)$ is the modulation intensity, $N$ is the phase-shift index, $n=0,1,2,\dots ,N-1$, and the wrapped phase $\varphi (x,y)$ reflects the height of the object information, which can be extracted by the following equation: $$\varphi (x,y)=\arctan \frac{{\sum }_{n=0}^{N-1}{I}_n(x,y)\sin (2\pi n/N)}{{\sum }_{n=0}^{N-1}{I}_n(x,y)\cos (2\pi n/N)}.$$(2)

Since there are three unknowns, $A(x,y)$, $B(x,y)$, and $\varphi (x,y)$, in Eq. (1), at least three images with different phase shifting should be used to calculate the wrapped phase $\varphi (x,y)$.

The phase value provided from Eq. (2) contains $2\pi$ phase discontinuities. Its corresponding continuous phase distribution $\mathrm{\Phi }(x,y)$ can be unwrapped as Eq. (3): $$\mathrm{\Phi }(x,y)=\varphi (x,y)+2k\pi ,$$(3)where $k$ is the integer number to indicate the fringe orders.

As shown in Fig. 1, in 3D shape measurement based on dual-frequency fringes, two projected fringes with different frequencies, $f_h$ and $f_l$, are projected to obtain their corresponding wrapped phase ${\varphi }_h$ and ${\varphi }_l$ by Eq. (2). The basic process of this dual-frequency method is to subtract the wrapped phase ${\varphi }_l$ from ${\varphi }_h$, sum the wrapped phase ${\varphi }_h$ and ${\varphi }_l$, and then adjust the results to be within the range of $[0,2\pi ]$: $$\begin{gathered} {\varphi }_d(x,y)=\{\begin{array}{cc}{\varphi }_h(x,y)-{\varphi }_l(x,y),& \text{if}{\varphi }_h\ge {\varphi }_l,\\ {\varphi }_h(x,y)-{\varphi }_l(x,y)+2\pi ,& \text{if}{\varphi }_h<{\varphi }_l,\end{array} \\ {\varphi }_s(x,y)=\{\begin{array}{cc}{\varphi }_h(x,y)+{\varphi }_l(x,y),& \text{if}{\varphi }_h+{\varphi }_l\ge 0,\\ {\varphi }_h(x,y)+{\varphi }_l(x,y)+2\pi ,& \text{if}{\varphi }_h+{\varphi }_l<0,\end{array} \end{gathered}$$(4)where ${\varphi }_d$ and ${\varphi }_s$ represent the phase difference and phase sum. Their corresponding lower ($f_d$) and higher ($f_s$) equivalent frequencies are defined as $$\begin{gathered} f_d=f_h-f_l, \\ {f}_s=f_h+f_l. \end{gathered}$$(5)

If the phase-difference map ranging from 0 to $2\pi$ covers the whole range of the surface, phase unwrapping is not necessary because ${\varphi }_d$ can be regarded as the unwrapped phase, i.e., ${\mathrm{\Phi }}_d={\varphi }_d$. ${\mathrm{\Phi }}_d$ can be used to obtain the fringe order for each point on the low-frequency $f_l$ wrapped phase for phase unwrapping. We choose the low frequency $f_l$ rather than the high frequency $f_h$, which is to reduce the phase unwrapped order errors caused by noise amplification.

The fringe order $k_l$ of ${\varphi }_l$ is $$k_l=\text{round}\left(\frac{{\mathrm{\Phi }}_d·f_l/f_d-{\varphi }_l}{2\pi }\right),$$(6)where round ($·$) is the operator that gives the nearest integer number.

This results in the unwrapped phase of $f_l$, which is given by $${\mathrm{\Phi }}_l(x,y)={\varphi }_l(x,y)+2\pi ·k_l(x,y).$$(7)

The fringe order $k_s$ of ${\varphi }_s$ is $$k_s=\text{round}\left(\frac{{\mathrm{\Phi }}_l·f_s/f_l-{\varphi }_s}{2\pi }\right).$$(8)

This results in the unwrapped phase of $f_s$, which is given by $${\mathrm{\Phi }}_s(x,y)={\varphi }_s(x,y)+2\pi ·k_s(x,y).$$(9)

The measured object’s absolute phase map can be obtained by subtracting the phase of the reference plane from the unwrapped phase in the phase-sum information.

The synthetic frequency information and their phases depend on the frequencies of the projected fringes. By selecting the appropriate fringe’s frequency, the synthetic frequency will change accordingly. The phase difference extends the range of measurable depth variation by the synthetic lower frequency. In addition, the synthetic phase sum has an averaging effect in the process of calculating the synthetic frequency and thus smoothing the phase noise.

In order to illustrate the relationship of phase difference with phase sum, the sensitivity gain $G$ is defined as $$G=\frac{f_h+f_l}{f_h-f_l}.$$(10)

That is, the phase sum is $G$ times more sensitive than the phase difference. The purpose of using phase difference is to synthesize a lower frequency $f_d$. In other words, if the phase difference is meaningful, the phase-difference frequency must be lower than the original low frequency of $f_l$, so it is necessary that $$f_d=f_h-f_l<f_l\Rightarrow f_h<2f_l\Rightarrow G=\frac{f_h+f_l}{f_h-f_l}>3.$$(11)

However, the synthesis fringes corresponding to the phase difference should cover the whole measurement area, which means that the two projected frequencies are close to each other, and $G$ is much larger than three. The measurement accuracy increases with the increasing sensitivity gain $G$. Therefore, the value of $G$ can be used to guide the selection of frequency.

We simulated 3D shape measurement by using FPP for a tested object whose shape distribution acts like the peaks function. The diagram of the computer simulation process of this work is shown in Fig. 2. The process of phase unwrapping uses the phase difference ${\varphi }_d$ to help the phase-sum ${\varphi }_s$ unwrapping.

The continuous phase is $500\times 500$ pixels. The frequency of a projected fringe pattern is actually defined as $1/T$ in an FFP system, where $T$ is the number of pixels in each period. According to Eq. (5), both of the original fringe frequencies $f_l$ and $f_h$ ($T_l=170\mathrm{pixels}$, $T_h=150\mathrm{pixels}$) are lower than the synthetic frequency $f_s$ ($T_s=79.7\mathrm{pixels}$). Taking the data along 250th row as an example, the wrapped phase, the unwrapped phase, the object’s height, and the height error are plotted in Fig. 3. The phase differences are shown in Fig. 3(a), and the phase sum is shown in Fig. 3(b). Figures 3(c) and 3(d) show the process of the phase unwrapping.

In order to compare the performance of the proposed method, a random Gaussian noise with a signal-to-noise ratio (SNR) of 27 dB is designed to be added in the simulated fringes, and all of the phase distributions have been transformed to the height distribution. Figure 3(e) shows the cross section along one row of the restored object’s height. Related to the simulated object’s truth shape, the corresponding height errors restored by two original projected fringes and the synthetic fringes were analyzed, and their cross sections along one row are shown in Fig. 3(f). Due to the higher synthetic frequency and denser wrapped phase fringes, we can see that the noise of the synthetic frequency’s restored height (magenta line) is obviously smaller than that of the other two. The standard deviation (STD) of the synthetic frequency’s height error distribution is 0.325, while those of $f_l$ and $f_h$ are 0.490 and 0.433, respectively. In effect, the noise is reduced by the superimposing operations in the process of calculating the synthetic wrapped phase. Therefore, the measurement accuracy can be increased with this proposed dual-frequency technique.

In this simulation, the phase sensitivity gain $G$ between the phase sum and phase difference is given by $$G=\frac{f_h+f_l}{f_h-f_l}=\frac{1/150+1/170}{1/150-1/170}=16.$$(12)

To confirm the relationship between the sensitivity gain $G$ and the measurement accuracy, we carried out simulation by selecting a fixed high frequency of 1/30 and changed the low frequency from 1/31 to 1/125. Figure 4 shows the trend of measurement accuracy changing with the low frequency.

It is clear that the accuracy is the best when the dual-frequency fringes are close to each other. We conclude that the effect of phase sum for improving accuracy relies on the ratio of $f_s$ to $f_d$. As can be seen from the figure, the phase sum is not meaningful after the intersection of two lines. Therefore, we find $G$ is approximately 2.4 at the intersection. From the above, it is meaningful to use dual-frequency fringe for $G>3$. At the point of $G=3$, the accuracy of the proposed method is higher than that of the conventional method. Therefore, the phase sum in the dual-frequency technique can obviously improve the measurement accuracy.

To further verify the performance of the proposed method, we developed an FFP system including a projection system [digital projector with a digital light processing (DLP) chip and the resolution of $800\times 1280$ pixels], imaging equipment (UI-3250CP-M-GL camera with the resolution of $1200\times 1600$ pixels), and a computer. The distance between the projector and the camera was about 200 mm. The measured objects were placed at around 500 mm in front of the FFP system, where a flat board was used as the reference plane. Fringe patterns were generated and then projected onto the tested surface. Afterward, the deformed fringe patterns were captured by the camera.

In our experiments, the absolute phase calculated by a 24-step phase-shifting algorithm is used as the ground truth. It should be noted that all of the images have been cropped according to the tested object in the following illustrations. In these experiments, we compared the 3D reconstruction accuracy of the proposed method with that of the conventional dual-frequency method reported in literature [24]. In order to verify the proposed method, the reference plane was first measured, and the reconstruction phase error is shown in Fig. 5. The measurement accuracy of the conventional dual-frequency method is mainly determined by the high frequency of two projected fringes, so it will not be improved even though the low-frequency fringe is changing. The STD of the conventional dual-frequency method is 0.0165, while the STD of the proposed dual-frequency method is, respectively, 0.0130, 0.0143, 0.0154, 0.0164, and 0.0170 with the increasing $T_l$.

Then, one simple object with a smooth surface and one complex object were measured. The unwrapped phase was obtained to reconstruct the 3D shape of the measured object. Figures 6(a) and 6(b) show one of the captured fringe patterns with very close spatial frequencies having $T_h=80$ pixels and $T_l=85$ pixels. Figures 6(c) and 6(d) show the wrapped phase of the projected dual-frequency fringes. Figures 6(e) and 6(f), respectively, present the information of phase difference and phase sum. Figure 6(g) presents the fringe order $k_l$. After applying Eq. (7), the unwrapped phase ${\mathrm{\Phi }}_l$ is retrieved, as shown in Fig. 6(h). Figure 6(i) presents the fringe order $k_s$, and the unwrapped phase ${\mathrm{\Phi }}_s$ is retrieved by Eq. (9) and shown in Fig. 6(j). The phase sensitivity gain $G$ between the phase sum and phase difference is given by $$G=\frac{f_h+f_l}{f_h-f_l}=\frac{1/80+1/85}{1/80-1/85}=33.$$(13)

In order to research the relationship between the sensitivity gain $G$ and the measurement accuracy, a series of experiments were carried out. The second experiment was carried out on a complex object with the projected fringes of $T_h=45$ pixels and $T_l=50$, 70, 90, 120, and 140 pixels. Figure 7 illustrates the 3D reconstruction results, where the first one gives the 3D shape reconstructed by the conventional method, and the others are the 3D shapes reconstructed by our proposed method with different lower-frequency fringes. Meanwhile, to show their difference more clearly, the corresponding errors related to the absolute phase, which were calculated by the 24-step phase-shifting algorithm, have also been calculated and shown in Fig. 8. The STD of the conventional dual-frequency method is 0.0234, while the STD of the proposed dual-frequency method is, respectively, 0.0187, 0.0202, 0.0217, 0.0235, and 0.0241 with the increasing $T_l$.

Figure 9 is a trend chart of measurement accuracy of actual experiments with a fixed high frequency and a varied low frequency. The measured object in Fig. 9(a) is the foam heart in the first experiment scene, and the measured object in Fig. 9(b) is the petal in the second experiment scene. It is obvious that the phase error becomes larger as the frequency difference between the two original fringes increases. It can also be concluded that the higher the sensitivity gain $G$, the higher the measuring accuracy. However, there will be many interference factors in the actual experiment, such as the nonlinear error of the projector, uneven reflectivity of the object surface, camera resolution, shadow, environmental light, etc. It can be seen from the phase error distribution that it is obviously periodic, which is mainly caused by the nonlinear error of the projector. The experimental results are not as accurate as the simulation ones, but the trend is basically the same. At the point of $G=3$, the measuring accuracy of the proposed method is higher than that of the conventional method, which verifies that the phase sum in dual-frequency method can improve the measuring accuracy.

In this study, a dual-frequency method in 3D measurement is certified to enhance the measurement accuracy. Compared with the conventional dual-frequency method, the proposed dual-frequency method employs the phase sum of two different frequency projected fringes to improve the measuring accuracy. A synthetic fringe is acquired, which is higher in frequency than either of the original projected fringes. The measurement results with the synthetic fringes are superior to that of a single frequency fringe, and higher measurement accuracy is also obtained. In addition, the relationship between the sensitivity gain $G$ and the measurement accuracy is also studied to guide the frequency selection in actual measurements. We found that the larger the sensitivity gain $G$ is, the higher the measurement accuracy will be. Especially, when $G=3$, the measurement accuracy of the proposed method is also higher than that of the conventional method. Both the computer simulation and actual experimental results demonstrate the viability of the proposed method.