1. Introduction

As a powerful fringe analysis tool, phase-shifting algorithms have been widely used in various fields such as interferometry [1,2], holography [3], fringe projection profilometry (FPP) [4,5], and phase measuring deflectometry (PMD) [6,7], owing to their high precision, point-by-point calculation, and good flexibility. However, traditional equal-step phase-shifting algorithms [1] retrieve objective phases modulated in fringe patterns by strictly following the intensity distribution of sinusoidal or cosinoidal functions and ideal equal-step phase shifts between each frame. Unfortunately, the captured fringe patterns are often affected by various error factors in real measurements, which leads to the failure of the above assumptions. Consequently, phase demodulation errors occur with ripple-like artifacts being the most common one. Wherein noises caused by optoelectronic components [8], phase-shifting errors from imperfect calibrations of phase-shifters [9], fringe high-order harmonics due to non-linear response of optoelectronic devices [10], and temporal intensity variations of fringe patterns from fluctuations in light source or background [11] are the main error sources that lead to a decrease in the accuracy of wrapped phase demodulation for practical measurements. Therefore, how to simultaneously suppress or eliminate the hybrid effects of these major error factors has become one of the hot topics in the field of phase-shifting algorithm investigation, which has important research significance and practical application value.

In order to accurately extract the objective phase distribution from fringe patterns with foregoing error sources, varieties of phase-shifting algorithms have been proposed during the past few decades, which can be roughly classified into two categories: model-dependent [11–20] and model-independent ones [21–24]. In the field of model-dependent algorithms, typical methods that solely consider the influence of phase-shifting errors include advanced iterative algorithm (AIA) algorithm [12], principal component analysis (PCA) algorithm [13], Lissajous figure and ellipse fitting (LEF) algorithm [14] and universal phase-shifting algorithm (UPSA) based on Lissajous figures [15]. While for others only considering the influence of fringe high-order harmonics, there are Jiang et al.'s algorithm based on multi-frequency phase-shifts [16] and Lee et al.'s fast composite frequency phase extraction algorithm [17] for example. Although variable-frequency fringes are introduced by Jiang et al. [16], it is not able to handle the case of harmonics suppression for second order and above with only bi-frequency three-step phase-shifting fringe patterns. Once there are multiple error sources combined, the accuracy of these algorithms is severely limited which significantly constraints their generality.

Based on the algorithms mentioned above, researchers have conducted a combination analysis and suppression of several error sources. For instance, in order to suppress the comprehensive impact of high-order harmonics and phase-shifting errors, Xu et al. [18] proposed to directly incorporate the solution of high-order harmonic coefficients into the linear least-square framing of AIA for iterative calculation. While, Chen et al. proposed general iterative algorithm (GIA) [19] which comprehensively considers the influence of random phase-shifts, intensity harmonics and non-uniform phase-shifting distribution. However, GIA is constrained by the required number of phase-shifting steps (at least five steps) for high-order harmonics (second order and above) case and cannot suppress errors in three-step phase-shifting fringe patterns extremely susceptible to these harmonics. The reason is that the performance of the existing phase-shifting algorithms in suppressing high-order harmonics largely depends on the number of phase-shifts. When a corresponding algorithm with a single frequency fringe [19,20] is adopted, the solvable number (P) of harmonic coefficients is limited by the number (N) of phase-shifting steps, which must meet the requirement of N ≥ 2P + 1. In contrast, a three-step iteration algorithm with a grouped step-by-step trick was proposed by Tao et al. [11] to correct phase-shifting errors and temporal intensity fluctuations, but high-order harmonics suppression is still lacking.

Apart from the aforementioned model-dependent algorithms, many notable contributions have been also conducted in the field of model-independent ones [21–24]. For example, Wang et al. proposed a half-cycle phase histogram equalization algorithm (HPHE) [21] to mitigate the impact of phase-shifting errors. Meanwhile, shifted-phase histogram equalization (SHE) processing [22] has been introduced to suppress high-order harmonics errors as well. Xu et al. [23] similarly suppressed high-order harmonics errors with the 1/3 and the 1/6 period phase histogram equalizations. Furthermore, Yu et al. presented an algorithm using a phase-probability-equalization-based look-up table (PPE-LUT) [24] for both handling phase-shifting and high-order harmonics errors. These model-independent algorithms not only exhibit attractive performance in some specific error suppression, but also boast easily comprehensible principles and high computational efficiency. However, due to the fact that this category of algorithms is essentially error statistics one, it has relatively strict requirements on the selection of statistical areas and is highly susceptible to noise impact.

It is regrettable that existing phase-shifting algorithms [11–27] have not yet comprehensively considered the common influence of these three usual error sources and effectively suppressed them with a least number of fringe patterns while achieving absolute phase. However, this is of great significance for the research and application promotion of high-precision and excellent-performance general phase-shifting algorithms. Therefore, to the best of our knowledge, a general phase-shifting algorithm for hybrid errors suppression using variable-frequency fringes is proposed for the first time to overcome the issue in this paper. Theoretically, it can use several similar frequency combinations with a least number of multi-step phase-shifting fringe patterns to simultaneously suppress phase demodulation errors caused by the three error sources mentioned above. A new fringe model is deduced to represent real patterns more accurately under the hybrid influence of these error sources. Variable-frequency fringes are introduced to provide a least and sufficient system of equations, while a least-squares iterative method with a grouped step-by-step strategy is adopted for stable calculating a larger number of desired parameters in the constructed model. For the phase jump problem caused by non-full rank matrices at certain sampling points, a regularization combined with constraints between coefficients of high-order fringe harmonic components is further proposed for identification and processing. Effective and simultaneous suppression of the investigated three types of error sources has been eventually achieved with the least number of phase-shifting fringe patterns, using bi-frequency equal three-step as an example in this study. The main contributions of this paper are as follows:

The rest of this paper is organized as follows. Section 2 carries out the theoretical analysis of the proposed algorithm. Several numerical simulations considering these three types of error sources and corresponding discussions are presented in Section 3. Experimental results and related analysis are shown in Section 4. Finally, Section 5 summarizes the investigation. Detailed explanation of the proposed algorithm is in the Supplement 1.

2. Theoretical analysis

2.1 General phase-shifting fringe patterns model

The proposed algorithm aims to accurately retrieve phase from a least number of sinusoidal or cosinoidal phase-shifting fringe patterns, in which the hybrid influence of the investigated three usual error sources has been comprehensively considered. Therefore, it can be widely applied into various technical fields where variable-frequency phase-shifting fringe patterns are available. In order to facilitate the illustration of the principles, the subsequent parts of the paper use FPP as an example to explain the proposed algorithm, the flowchart of which is shown in Fig. 1. A classic monocular FPP system is mainly composed of a projector, a test object, a camera and a computer, as shown in the yellow box content of Fig. 1.

 

Fig. 1. Diagram of monocular FPP setup and flowchart of the proposed algorithm.

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When single-frequency phase-shifting fringe patterns are casted onto the test object through the projector, the distortional fringe patterns captured by the camera can be expressed as:

However, in real measurements, there may be a deviation in the nominal phase steps due to phase-shifting errors. In addition, the non-linear response of the used electronic devices, e.g., the projector and the camera in FPP, can cause Gamma distortion in the captured fringe patterns. Furthermore, if there is instability in the light source of the projection component or sudden changes in the external background light, temporal intensity fluctuations between phase-shifting fringe patterns will arise. Based on the analysis in Refs. [11] and [28], the distortional fringe patterns under the co-influence of phase-shifting errors, temporal intensity fluctuations and Gamma distortion can be written as below:

As pointed out in Ref. [28], the Gamma distortion of the fringe pattern can be equivalently represented as a high-order harmonic model. Taking into account the three types of error sources, Eq. (2) can be rewritten as:

2.2 Principle of the proposed algorithm

For classical single-frequency phase-shifting fringe algorithms, the solvable number of harmonic coefficients is constrained by the number of phase-shifting steps, i.e., N ≥ 2P + 1. Once the three types of error sources are included as shown in Eq. (3), it will be much more difficult to solve so many parameters using fewer single-frequency phase-shifting fringe patterns. Inspired by Ref. [16], variable-frequency phase-shifting fringes are introduced to construct a least and sufficient system of equations that satisfies estimation of the coefficients of these three types of error sources. Consequently, the corresponding universal fringe patterns model is expressed as:

2.2.1 Initial value estimation

After careful observation of Eq. (4), it is found that the fringe model will similarly degenerate into the one in Ref. [11] for single-frequency case if high-order harmonics, i.e., ${\textrm{B}_\textrm{j}}({\textrm{x},\textrm{y}} )$ with $\textrm{j} \ge 2$ are neglected. Therefore, the method of Ref. [11] named as Tao’s method for convenience, is a good choice to provide initial value estimation (IVE) for the other parameters of each single-frequency phase-shifting fringe patterns in Eq. (4). Once Tao’s method is adopted here, two threshold conditions are used as below:

2.2.2 Hybrid errors suppression process

In order to achieve more accurate results from the fringe model in Eq. (4) that contains numerous parameters, an iterative strategy of step-by-step parameterization is further proposed to suppress hybrid errors as shown in the red box content in Fig. 1. The unwrapped phase $\mathrm{\varphi }_1^0({\textrm{x},\textrm{y}} )$, the background intensity $\textrm{A}_0^0({\textrm{x},\textrm{y}} )$, the first harmonic coefficient $\textrm{B}_1^0({\textrm{x},\textrm{y}} )$, the phase-shifting error $\textrm{d}_{\textrm{mn}}^0$, and the coefficient of temporal intensity fluctuation $\textrm{a}_{\textrm{mi}}^0$ obtained by the above IVE are first used as initial values (setting the j-th harmonic coefficient as $\textrm{B}_\textrm{j}^0({\textrm{x},\textrm{y}} )= 0$ with $\textrm{j} \ge 2$) to form an iterative fringe pattern model as below:

Step 1. Absolute phase, background intensity and harmonic coefficients calculation

For the sake of simplicity, the spatial coordinates $({\textrm{x},\textrm{y}} )$ will be omitted in the subsequent equations. The first grouped parameters to be solved in Step 1 are ${\textrm{A}_0}$, ${\textrm{B}_\textrm{j}}$ and ${\mathrm{\varphi }_1}$, which have the characteristic of same values among variable-frequency phase-shifting fringe patterns in principle. A solving matrix that iteratively approximates ground-truth values of these parameters in real fringe patterns is constructed by using the deviation between the captured fringe patterns and the modeled ones, as well as the gradient of the parameters based on the proposed fringe model. The first-order Taylor expansion of Eq. (7) with respect to the parameters ${\textrm{A}_0}$, ${\textrm{B}_\textrm{j}}$ and ${\mathrm{\varphi }_1}$ can be expressed as follows:

Step 2. Phase-shifting errors calculation

The second grouped unknown phase-shifting errors ${\textrm{d}_{\textrm{mn}}}$ to be solved in Step 2 have the characteristic of unequal values among variable-frequency phase-shifting fringe patterns. The first-order Taylor expansion of Eq. (7) with respect to the parameters ${\textrm{d}_{\textrm{mn}}}$ can be expressed as below:

Similarly, Eq. (9) can be resolved using the same technique as Step 1. After the process in Step 2, the values of $\textrm{d}_{\textrm{mn}}^{\textrm{k} + 1}$ for the k-th iteration are achieved, which are served as the known ones for the next step in the current iteration cycle.

Step 3. Temporal intensity fluctuation coefficients calculation

The third grouped unknown parameters ${\textrm{a}_{\textrm{mi}}}$ have the characteristic of unequal values between variable-frequency fringe patterns. The first-order Taylor expansion of Eq. (7) with respect to the parameters ${\textrm{a}_{\textrm{mi}}}$ can be expressed as follows:

Equation (10) is also handled using the same way as Step 1. Therefore, the values of $\textrm{a}_{\textrm{mi}}^{\textrm{k} + 1}$ for the k-th iteration can be obtained through the process in Step 3.

After obtaining all the unknown parameters through the aforementioned three steps, a threshold judgment is conducted to determine whether the iteration is terminated. If the threshold conditions are not satisfied or the maximum number of iterations has not been reached, the three steps above are reiterated and vice versa. The desired fine results then can be finally achieved. The threshold judgment process is accordingly depicted as below:

3. Numerical simulations and discussion

Based on the principle and the derived formulas in Section 2, numerical simulations are carried out to analyze and demonstrate the effectiveness of the proposed method using the bi-frequency equal three-step phase-shifting case as an example. Because phase measurement accuracy mainly depends on results of the fringe patterns with a higher frequency, it is adopted to evaluate a series of simulations in the following study. To simulate non-linear distortion in real optoelectronic systems such as FPP, the Gamma distortion model is used instead of directly adding high-order harmonics. The temporal intensity fluctuations and the random phase-shifting errors as shown in Eq. (2) are also added to the simulated fringe patterns. It is noteworthy that the general formulas of the proposed method can be straightforwardly expanded to other multi-frequency and multi-step phase-shifting situations as long as the requirements of the variable-frequency and the phase-shifts are met. Owning to the negligible differences in final results of hybrid errors suppression, the comparison between utilizing Tao's method and UPSA as IVE has been omitted in the simulations for simplicity. Unless otherwise specified, the IVE of the proposed algorithm in this section will default to using Tao’s method.

3.1 Error-source combination analysis with a flat object

To verify the feasibility of the proposed method, the effects of discussed three types of error sources and their combinations are first analyzed in the absence of intensity noise. It is namely divided into seven cases as follows: (1) phase-shifting errors; (2) temporal intensity fluctuations; (3) Gamma distortion; (4) cases (1) and (2) combination; (5) cases (1) and (3) combination; (6) cases (2) and (3) combination; (7) cases (1), (2) and (3) combination. These simulated error influence cases are handled by the following four model-dependent methods: 1) the traditional three-step phase-shifting method; 2) Tao's method [11]; 3) Jiang's method [16]; 4) the proposed method in this section. The first simulated phase-object is a flat with a size of 256 × 256 pixels. The used two varied frequencies are set as f₁= 16/256 and f₂= 15/256 with equidistant three-step phase-shifts respectively, while the preset values of the error sources are shown in Fig. 2(h). To better demonstrate the comparisons, phase retrieval errors (in radians) of cross-sectional curves of the flat object’s central row by the four methods under the investigated three types of error sources and their seven combinations are accordingly plotted in Figs. 2(a) to 2(g). Root mean square error (RMSE) results (in radians) of the four model-dependent methods under these error-source combinations as shown in Table 1, where the statistical region of the data covers the full field of view (FOV). It is noted that corresponding to the error-combination case (3) in Table 1, the curves of the traditional and the Tao's methods have overlapped in Fig. 2(c), which is similar to those of the Jiang's and the proposed methods. It can be easily seen from the above simulations that the traditional three-step phase-shifting method cannot effectively suppress the influence of any one of these seven error-source combinations. While Tao's method can suppress the influence of phase-shifting errors, temporal intensity fluctuations or their combination, but its suppression effect significantly decreases once Gamma distortion is introduced. Although Jiang's method can suppress the influence of Gamma distortion, its suppression effect also significantly decreases once phase-shifting errors, temporal intensity fluctuations or their combination arises. In contrast, the proposed method not only shows the most significant suppression effect on any single error source, but also performs the best in suppressing any combination of the discussed three types of error sources.

 

Fig. 2. Phase retrieval errors comparisons (in radians) of cross-sectional curves of a flat object’s central row obtained by the four model-dependent methods under the investigated three types of error sources and their seven combinations in the absence of intensity noise: (a) to (g) corresponds to the error-source combination (1) to (7) case respectively. (h) denotes the preset values of the error sources.

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Table 1. RMSE (in radians) of the four model-dependent methods under seven error-source combinations

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3.2 Intensity noise-free simulations with a peak-like object

In order to further validate the effectiveness of the proposed method, additional numerical simulations are carried out with a more complex phase-object, which is a peak-like one with a size of 400 × 400 pixels as well. The simulated intensity noise-free bi-frequency phase-shifting fringe patterns, which have an equal phase-shift step of $2\pi /3$ with two similar frequencies (f₁= 16/400 and f₂= 15/400) under the same combined three types of error sources as those in Section 3.1, are shown in Figs. 3(a1) and (a2). Furthermore, non-uniformity distribution is conducted on both the background intensity and the fringe modulation to simulate spatial intensity fluctuation errors of the fringe patterns. Figure 3(b) illustrates a comparison of normalized fringe intensity along the red dashed line of the three-step phase-shifting fringe patterns in Fig. 3(a1). The retrieved absolute phase and the corresponding residual error maps (in radians) of the peak-like object by the four model-dependent methods in Section 3.1 under the investigated three types of error sources with intensity noise-free are shown in Figs. 3(c1) to 3(g2). To demonstrate the effectiveness of the regularization, the results of the proposed method without and with regularization are included in the comparison. The Max error and the RMSE results of the traditional method, the Tao's method, the Jiang's method, the proposed method without and with regularization accordingly shown in Figs. 3(c2), 3(d2), 3(e2), 3(f2) and 3(g2) are respectively 0.3563 and 0.2053, 0.3037 and 0.1822, 0.1861 and 0.0845, 0.5442 and 0.0156 rads, 0.0043 and 0.0013 rads, where the statistical region of the data also covers the whole FOV. Residual error comparison of central cross-sections along the red dashed line in Figs. 3(c2), 3(d2), 3(e2), 3(f2) and 3(g2) are plotted in Fig. 3(h) for a more detailed and intuitive comparison. From the simulations above, it can be easily seen that the proposed method is significantly better than the other three ones in suppressing the investigated three types of error sources for a more complex phase-object under the intensity noise-free condition, which is also similar to the case in Section 3.1. It is worth noting that the regularization plays an important role in significantly suppressing the occurrence of error jump points caused by matrix underdetermination, which are indicated by the yellow arrows in Fig. 3(f2) and well corrected in Fig. 3(g2). The detailed comparison can be found in Fig. 3(h), where those jump points on the purple curve have been effectively corrected on the red one. The Max error and the RMSE using the proposed method with regularization have been respectively decreased at least about 43 and 65 times here compared with the other three ones.

 

Fig. 3. Simulated results of intensity noise-free bi-frequency three-step phase-shifting fringe patterns under the combined three types of error sources for a peak-like object. (a1) and (a2) represent the high and the low frequency fringe patterns respectively. (b) represents a comparison of normalized fringe intensity along the red dashed line of the fringe patterns shown in Fig. 3(a1). The absolute phase and the corresponding residual error maps (in radians) are obtained by: the traditional method as (c1) and (c2); the Tao's method as (d1) and (d2); the Jiang's method as (e1) and (e2); the proposed method without and with regularization as (f1) and (f2), (g1) and (g2) respectively. (h) Residual error comparison of central cross-sections along the red dashed line in Figs. 3(c2), 3(d2), 3(e2), 3(f2) and 3(g2).

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3.3 Intensity noise-added simulations with a peak-like object

Due to the fact that there usually exist intensity noises in real experimental scenarios, Gaussian-type additive noises (a standard deviation ranging from 0% to 25% with a 5% each increment) are added to the same simulated bi-frequency equal three-step phase-shifting fringe patterns of a peak-like object as those in Section 3.2. The Max error and the RMSE results of the four methods aforementioned in Section 3.1 at different intensity noise levels are accordingly presented in Table 2. It can be readily seen that the Max error and the RMSE obtained by the proposed method for a peak-like object have been respectively decreased at least about 1.2 and 2.1 times at the level of 25% noise standard deviations (SDs) compared with the other three ones. The corresponding Max error and RMSE curves (in radians) varied with different noise SDs levels of Table 2 are plotted in Figs. 4(a) and 4(b) respectively. From the results shown in Fig. 4, it is found that as the noise level increases, the accuracy of the proposed algorithm linearly declines to a small extent. Compared to the other three ones, the overall accuracy of the retrieved results still shows a significant improvement. The reason why the three competitive algorithms have not been significantly affected by the intensity noises with different levels is that the phase retrieval error caused by the combined three types of error sources is too large, which makes the influence of intensity noise less obvious than that of the investigated three error ones. Since the noise is Gaussian-type additive, some simply denoise techniques can be applied to improve the phase retrieval results of the proposed algorithm.

 

Fig. 4. The corresponding (a) Max error and (b) RMSE curves (in radians) varied with different noise SDs levels of Table 2.

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Table 2. Max error and RMSE (in radians) of the four model-dependent methods with different noise SDs

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Generally speaking, the newly proposed method successfully uses the information of the bi-frequency phase-shifting patterns to achieve accurate numerical solution of numerous parameters of the proposed fringe model and finally extracts more satisfactory absolute phases, which can accomplish much higher precision than the other three model-dependent ones.

4. Experimental results and discussion

In this section, a flat surface and a complex plaster model object are adopted to ulteriorly confirm the effectiveness and performance of the proposed method compared with other competitive methods in real testing. To demonstrate the universality of the IVE module in the proposed algorithm, two different (i.e., Tao's and UPSA) methods are utilized as IVE here. Therefore, experimental comparisons of the complex plaster object are designed to two groups. The first group involves comparisons of the proposed algorithm (Tao's method IVE) with other three model-dependent ones (traditional three-step phase-shifting, Tao's and Jiang's methods). While, the second group comprises competition of the proposed method (UPSA IVE) with the model-dependent (UPSA) and the model-independent ones, the last of which are the PPE-LUT (Object-area based) and the PPE-LUT (Plane-area based) methods. The measurement system is consisted of a projector (TI DLP4500 with a resolution of 1140 × 912 pixels), a camera (MER-131-210U3M with a resolution of 1280 × 1024 pixels and 210 frame rate) equipped with a lens (M0814-MP2 with focal length as 8 mm) and a personal laptop, as shown in the upper right corner of Fig. 1. During the testing, a group of bi-frequency three-step phase-shifting fringe patterns are used and parallelly arranged along the horizontal direction with fringe pitches as 32/912 and 31/912 pixels respectively. Random phase-shifting errors and temporal intensity fluctuations are added to the projected fringe patterns. While, high-order fringe harmonics come from Gamma distortion of the used electronic equipment, i.e., the projector and the camera. Therefore, the investigated three types of error sources have been effectively combined in the measurement. It is worth noting that although the phase-shifting and the intensity values of the fringe patterns projected by DLP can be accurately set and controlled in advance, the final captured patterns are still prone to deviation from the ideal situation. Thus, the investigation of comprehensively considering these three types of error factors is really very meaningful for the research field of phase-shifting algorithms including FPP.

The flat surface used as a reference plane is first measured by the discussed four methods for comparison here. The corresponding retrieved absolute phase maps (in radians) are shown in Figs. 5(a) to 5(d) respectively. It should be noted that the ground true phase of reference plane is obtained by a standard six-step phase-shifting algorithm without the three error sources at the same high frequency. For simplicity, only one part result of the test flat surface is cropped and displayed, but it does not affect the correct performance demonstration of the related algorithms. Cross-sectional curves of the flat surface’s central row along the red dashed line in Figs. 5(a) to 5(d) obtained by the four methods are plotted in Fig. 5(e) for a more detailed and intuitive comparison. The RMSE results of the traditional three-step phase-shifting method, the Tao's method, the Jiang's method and the proposed method shown in Figs. 5(a) to 5(d) are 0.2074, 0.1405, 0.1071 and 0.0256 rads, respectively. It is obviously seen that, compared to the other three ones, the proposed algorithm has the highest accuracy in handling the fringe patterns of the flat surface containing the investigated three types of error sources, where the RMSE has been decreased at least about 4 times.

 

Fig. 5. The absolute phase maps (in radians) of a flat surface are obtained by: (a) the traditional method; (b) the Tao's method; (c) the Jiang's method; (d) the proposed method in the real experiment. (e) Cross-sectional curves of the flat surface’s central row along the red dashed line in Figs. 5(a) to 5(d).

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For the first group of comparative experiment for the complex plaster model, captured original bi-frequency equal three-step phase-shifting fringe patterns are shown in Figs. 6(a1) to 6(b3), where the influence of the three types of error sources can be clearly observed in these patterns. The corresponding absolute phase maps (in radians) obtained by the four model-dependent methods are shown in Figs. 6(c) to 6(f) respectively, where the ground true phase of the reference plane has been removed. Cross-sectional curves of the complex plaster model object’s certain row along the red dashed line in Figs. 6(c) to 6(f) obtained by the four methods are plotted in Fig. 6(g) for a more detailed and intuitive comparison as well. As show in Figs. 6(c) to 6(g), the proposed method is significantly better than the other three ones in suppressing the ripple-like phase errors caused by phase-shifting errors, temporal intensity fluctuations and high-order fringe harmonics (up to the third harmonic). Overall, as a universal one for phase-shifting fringe demodulation, the proposed method can be applied into various fields, e.g., interferometry, holography, FPP and PMD, to further improve their measurement accuracy, as long as fringe patterns to be processed are affected by aforementioned three types of error factors.

 

Fig. 6. Captured original bi-frequency three-step phase-shifting fringe patterns of a complex plaster model object in the real experiment: (a1) to (a3) and (b1) to (b3) are the high and the low frequency fringe ones respectively. The absolute phase maps (in radians) of the complex plaster model object are obtained by: (c) the traditional method; (d) the Tao's method; (e) the Jiang's method; (f) the proposed method. (g) Cross-sectional curves of the complex plaster model object’s certain row along the red dashed line in Figs. 6(c) to 6(f).

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The second group of comparative experiments for the complex plaster model is further carried out, where four experimental cases are established by combining three types of error sources and stochastically setting their values. To enhance the Gamma distortion of the used electronic equipment, the same Gamma value as 2.5 has been pre-encoded into the projected fringe patterns here. While the random phase-shifting errors of the projected fringe patterns and the varying LED current of DLP within each projection period simulating temporal intensity fluctuations for the experimental case 1 are set as in Fig. 7(a1). Captured original bi-frequency equal three-step phase-shifting fringe patterns are shown in Figs. 7(b11) to 7(c13), where the green and the blue boxes respectively denotes the statistical zone of the generated LUTs for the PPE-LUT (Object-area based) and the PPE-LUT (Plane-area based) methods used in the experimental case 1. Normalized fringe intensity curves of the test object’s certain row along the red dashed line in Fig. 7 (b11) are plotted in Fig. 7(d1). The absolute phase maps (in radians) obtained by the corresponding methods in the comparative experimental case 1 is shown in Figs. 7(e11) to 7(e14), where the ground true phase of the reference plane has been removed. Cross-sectional curves of the test object’s certain row along the same red dashed line in Figs. 7(e11) to 7(e14) are plotted in Fig. 7(f11). For a more detailed and intuitive comparison, the partial enlarged comparison of the results achieved by the PPE-LUT (Plane-area based) and the proposed (UPSA IVE) methods in the purple dashed box of Fig. 7(f11) is shown in Fig. 7(f12). The contents indicated in other subfigures of Fig. 7 corresponding to remaining three competitive experimental cases are similar to those of case 1. The Max error and the RMSE results of the second group comparative experiments are accordingly presented in Table 3, in which the statistical region of the data is the background plane. It is evident that the UPSA method exhibits the lowest accuracy, meanwhile the PPE-LUT (Object-area based) and the PPE-LUT (Plane-area based) methods achieve moderate and second-best ones respectively. In contrast, the proposed method based on UPSA IVE surpasses all the above competitors, the RMSE and Max error of which have been decreased at least about 1.5 and 1.2 times respectively. The reason for large deviations of the PPE-LUT method in two different zones is the fact that this type of algorithm is essentially error statistics one, which has relatively strict requirements on the selection of statistical areas and more easily affected by noises.

 

Fig. 7. Results of the competitive four methods in the second group comparative experiments. Figures 7(a1) to 7(f12), 7(a2) to 7(f22), 7(a3) to 7(f32) and 7(a4) to 7(f42) are corresponding to experimental cases 1, 2, 3 and 4 respectively.

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Table 3. Max error and RMSE results (in radians) of the second group comparative experiments

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5. Conclusion

To the best of our knowledge, a general phase-shifting algorithm for hybrid errors suppression using variable frequency fringes is proposed for the first time to reduce ripple-like phase artifacts in this paper. Simulations and experimental results have shown that compared with the existing techniques, the accuracy of the proposed algorithm has been significantly enhanced with the least number of phase-shifting fringe patterns. Besides the aforementioned ones, the proposed method also has the following advantages that are worth emphasizing. Firstly, although the harmonic model of the proposed algorithm is deduced from the Gamma model in FPP, it is also applicable to other fields as long as there is the influence of high-order harmonics in the testing scenario, while no particularly strict requirements for each harmonic coefficient. Secondly, it simultaneously retrieves absolute phases with the help of the used variable-frequency fringes, which means no pre-calibrations for the investigated three types of error sources are needed in advance. Thirdly, although a bi-frequency equal three-step phase-shifting realization is demonstrated as an example in the study, it can be easily extended to other multi-frequency and multi-step phase-shifting situations if the requirements of the variable-frequency and the phase-shifts are met.