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Enhanced Fourier-Hilbert-transform suppression for saturation-induced phase error in phase-shifting profilometry

Yingying Wan, Yiping Cao, Min Xu, and Tao Tang

Open Access Open Access

Abstract

Intensity saturation tends to induce severe errors in high dynamic range three-dimensional measurements using structured-light techniques. This paper presents an enhanced Fourier-Hilbert-transform (EFHT) method to suppress the saturation-induced phase error in phase-shifting profilometry, by considering three types of residual errors: nonuniform-reflectivity error, phase-shift error, and fringe-edge error. Background normalization is first applied to the saturated fringe patterns to suppress the effect of the nonuniform reflectivity. A self-correction method is proposed to correct the large phase-shift error in the compensated phase. The self-corrected phase error is detected to assist in locating the fringe-edge area, within which the true phase is computed based on the sub-period phase error model. Experimental results demonstrated the effectiveness of the proposed method in suppressing the saturation-induced phase error and other three types of residual errors with fewer images.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Three-dimensional (3D) information has been widely used in many aspects such as defect detection, online inspection, face recognition, historical relics protection, and 3D display [1–4]. Structured-light (SL) based 3D measurement technology has proven to be one of the most promising techniques because of its non-contact and high accuracy [5–8]. However, due to the limited dynamic range of the camera sensors, it is a challenge to measure objects with high dynamic range (HDR) surface. High reflection from the object surface tends to cause intensity saturation, and result in measurement errors.

Researchers have deeply studied a variety of HDR 3D measurement methods based on structured light, especially in fringe projection profilometry (FPP) [9]. A group of the HDR 3D measurement methods are based on equipment techniques, which adjust the hardware parameters or use additional devices to avoid the intensity saturation. These equipment-based techniques can be classified into three major categories: adaptive projection methods [10,11], multi-exposure methods [12,13] and additional equipment methods [14,15]. The solution of the adaptive projection methods is adjusting the encoded intensity of the projected images pixel-by-pixel according to the reflected images, so that the final captured images are unsaturated [16]. However, this kind of methods require to calibrate the pixels mapping between the camera and projector in advance, and the pixel mapping relationship needs recalibration once the object moves. Compared with adaptive projection methods, the multi-exposure methods capture a set of images under different exposure time and fuse them into a single HDR image. Zhang et al. [17] captured a sequence of phase-shift fringe patterns with different exposures, and the fringe patterns were fused by choosing the brightest but unsaturated point pixel-by-pixel for phase retrieval. This method is easy to implement, but it is difficult to choose the exposure time so that a large number of images are needed. To reduce the human interference, the same group proposed an automatic optimal exposure control method, which determined the global optimal exposure time using only one exposure image [18]. However, this technique still requires multiple images, which would reduce the measurement efficiency and limit its application in dynamic scenarios. To speed up the measurement, binary defocus method [19] and color encoded method [20] etc. were used, but their measurement accuracy was sacrificed to some extent. Some scholars suppressed the intensity saturation by introducing additional equipment in a conventional projector-camera FPP system, such as hyperspectral camera [21], digital micromirror device (DMD) [22], transparent screen [23], linear polarizer [24], etc. However, these methods have some shortcomings, such as reduced signal-to-noise ratio (SNR) or high hardware cost.

Another group of HDR 3D measurement methods are algorithm-based techniques, which use algorithms to obtain the high accurate phase from the saturated fringe patterns without additional equipment or special hardware settings. Jiang et al. [25] proposed an inverted-fringe method to complement original fringe patterns for phase-shifting profilometry (PSP). If the original fringe patterns are saturated, the inverted fringe patterns are used in lieu of the original saturated fringes for phase calculation. Wang et al. [26] combined this method with three-frequency temporal phase unwrapping (TPU) using fewer fringe patterns for 3D measurement of isolated objects. However, the phase calculation equations were completely different depending on the saturation conditions, which would increase the complexity of the algorithm. Qi et al. [27] developed a saturation-induced phase error theory for standard N-step phase-shifting algorithm and proved that phase error would decrease with a large phase-shift number N. Multi-step phase-shifting algorithms could be used for reducing the phase errors and seven-step was suggested to be used in practical measurement. Wan et al. [28] developed a complementary phase method for compensating the phase errors caused by intensity saturation using three-step phase-shifting algorithm, but additional images were required. To improve the measurement efficiency, deep learning was applied to address the problem of fringe saturation [29,30]. As a data-driven method, the deep-learning-based methods have challenges on collecting the training data and generalization ability for diverse samples [31]. Recently, Tan et al. [32] proposed a saturation-induced error correction method by applying joint Fourier transform and Hilbert transform (FHT) to the fringe patterns for generating a compensated phase map. This method was efficient and effective for reducing the saturation-induced phase error without requiring pixel mapping calibration, additional fringe patterns, or high hardware cost. However, errors might occur on the nonuniform reflectivity surface and at fringe edges in compensation methods based on Hilbert transform [33,34]. In addition, we find that it is prone to introducing phase-shift errors in the compensated phase map.

To address the saturation-induced phase error, this paper presents an enhanced Fourier-Hilbert-transform (EFHT) method considering three types of residual errors: nonuniform-reflectivity error, phase-shift error, and fringe-edge error. Firstly, the nonuniform-reflectivity error is suppressed by employing background normalized (BN) to the captured saturated fringe patterns. Afterward, the BN fringe patterns are applied with Fourier transform (FT) and Hilbert transform (HT) sequentially to obtain an FHT-compensated phase. Next, a self-correction method is proposed to correct erroneous points with phase-shift error of the FHT-compensated phase. The phase error excluding the phase-shift errors is detected. With the statistics analysis of the detected self-corrected phase error, the fringe-edge area (FEA) is located. The true phase within the FEA is computed using an iteration algorithm based on the sub-period phase error model. The final phase is then a fusion of the original phase, the self-corrected FHT-compensated phase and the true phase in the FEA. The experimental results demonstrated the effectiveness of the proposed method in significantly reducing the saturation-induced phase error in HDR scenarios, as well as the three residual errors without requiring additional images.

2. Principles

2.1 Saturation-induced phase error compensation in N-step phase shifting profilometry

Phase-shifting profilometry is one of the most representative techniques in fringe projection profilometry due to its high speed and high accuracy. The fringe intensity of the phase-shifting fringe patterns can be expressed as:

$${I_n}(x,y) = A(x,y) + B(x,y)\cos [\varphi (x,y) + 2\pi n/N],\textrm{ }n = 1,2,\ldots ,N,$$
where $(x,y)$ are the image coordinates, n is the image index, N is the phase-shift step, $A(x,y) = R(x,y)a$ is the background intensity, $B(x,y) = R(x,y)b$ is the intensity modulation, $R(x,y)$ is the reflectivity, and a, b are the defined constants. $\varphi (x,y)$ is the phase modulated by depth information, and can be calculated using a standard N-step phase-shifting algorithm ($N \ge 3$):
$$\varphi (x,y) = \arctan \frac{{\sum\limits_{n = 1}^N {{I_n}(x,y)\sin (2\pi n/N)} }}{{\sum\limits_{n = 1}^N {{I_n}(x,y)\cos (2\pi n/N)} }}.$$
The phase $\varphi (x,y)$ is wrapped phase with 2π discontinuities due to the arctangent function, and it needs to be unwrapped into a continuous phase using a phase unwrapping algorithm [35] for 3D shape reconstruction.

When the intensity is saturated, the fringe can be considered as the aliased signal with fundamental harmonic and high-order harmonics:

$$I_n^s = A + \sum\limits_{j = 1}^p {{B_j}} \cos [j(\varphi + 2\pi n/N)],$$
where ${B_j}$ is the modulation of the jth harmonic, and p is the number of the harmonics. Note that (x, y) are omitted for brevity in the remainder of this paper. The high-order harmonics can be eliminated by the low-pass filtering operation after applying Fourier transform to the fringe pattens (details please refer to the Appendix A), and the FT-filtered fringe patterns can be described as:
$$I_n^F = \textrm{real\{ IFT\{ filter} \cdot \textrm{[FT(}I_n^s\textrm{)]\} \} ,}$$
where $\textrm{filter[]}$ is the low-pass filter, FT() and IFT{} denote the Fourier transform and the inverse Fourier transform operations, and real{} means to take the real part of a number. The saturation-induced phase error can be obtained by taking the difference of the true phase and the FT-filtered phase ${\varphi ^F}$ calculated using the FT-filtered fringe patterns, and it can be approximately modeled as [32]:
$$\begin{aligned} \Delta {\varphi ^F}&\textrm{ = }{\varphi ^F} - \varphi \\ &\textrm{ } \approx \arctan \frac{{ - {C_{N - 1}}\sin (N\varphi )}}{{1 + {C_{N - 1}}\cos (N\varphi )}},\\ &\textrm{ } \approx{-} {C_{N - 1}}\sin (N\varphi ) \end{aligned}$$
where constant ${C_k} = {{{B_k}} / {{B_1}}}$. It demonstrates that the frequency of the saturation-induced phase error is N-folder of the original fringe, and the phase error can be modeled as sinusoidal distribution.

To correct the saturation-induced phase error, the Hilbert transform is applied to the FT-filtered fringe patterns to generate the HT-based fringe patterns $I_n^H$:

$$I_n^H = \textrm{HT(}I_n^F\textrm{) = }A^{\prime} + \sum\limits_{j = 1}^p {{B_j}} \sin [j(\varphi + 2\pi n/N)],$$
where HT() denotes the Hilbert transform operation (details please refer to the Appendix B). In the similar way, the phase error using HT-based fringe patterns can be expressed as:
$$\begin{aligned} \Delta {\varphi ^H}&\textrm{ = }{\varphi ^H} - \varphi \\ &\textrm{ } \approx \arctan \frac{{{C_{N - 1}}\sin (N\varphi )}}{{1 + {C_{N - 1}}\cos (N\varphi )}},\\ &\textrm{ } \approx {C_{N - 1}}\sin (N\varphi ) \end{aligned}$$
where ${\varphi ^H}$ is the HT-based phase calculated using the HT-based fringe patterns. Based on Eq. (5) and Eq. (7), it can be concluded that $\Delta {\varphi ^F}$ and $\Delta {\varphi ^H}$ has the opposite distribution. Consequently, the saturation-induced phase error can be canceled out by taking the average of FT-filtered phase ${\varphi ^F}$ and HT-based phase ${\varphi ^H}$, and an FHT-compensated phase can be obtained:
$${\varphi ^{FHT}} = ({\varphi ^F} + {\varphi ^H})/2.$$
However, there are some residual phase errors in the FHT-compensated phase when employing the above FHT method, and we classify them into three categories:

Nonuniform-reflectivity errors which vary greatly with the nonuniform reflectivity of the surface.

Phase-shift errors which occur at the wrapped phase edges due to the phase shift between the FT-filtered phase and HT-based phase.

Fringe-edge errors which occur at both ends of the fringe due to the HT operation.

2.2 Suppression of nonuniform-reflectivity errors

Object surfaces usually have nonuniform reflectivity in practice, especially in HDR scenarios. The measurement accuracy would be affected by the nonuniform reflectivity as shown in Fig. 1. The intensity fluctuates steeply on the boundary of two different reflectivity (Fig. 1(a)) so that the corresponding phase error changes greatly while employing the FHT method as the blue line in Fig. 1(b) (the simulation parameters are presented in Sec. 3).

 figure: Fig. 1.

Fig. 1. Effect of the nonuniform reflectivity: (a) intensity on the surface with two different reflectivity, and (b) phase error by employing FHT method before and after background normalization.

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To suppress the effect of the nonuniform reflectivity, the captured fringes are first background normalized (BN):

$$I_n^{BN} = \frac{{I_n^s - {I_{bg}}}}{{{I_{bg}}}},$$
where $I_n^{BN}$ is the fringe patterns after background normalization, and ${I_{bg}}$ is the background intensity and can be obtained by:
$${I_{bg}} = \sum\limits_{n = 1}^N {I_n^s} /N.$$
Due to the intensity saturation, the ${I_{bg}}$ deviates from the true background intensity, and can be further expressed as:
$${I_{bg}} = A + \sum\limits_{i = 1}^q {{B_{iN}}\cos (iN\varphi )} ,$$
where ${B_{iN}}$ is the modulation of the (iN)th harmonic, and q is the number of harmonics. The BN fringe patterns in Eq. (9) can be rewritten as:
$$I_n^{BN} = \frac{{\sum\limits_{j = 1}^p {{B_j}} \cos [j(\varphi + 2\pi n/N)] - \sum\limits_{i = 1}^q {{B_{iN}}\cos [iN(\varphi + 2\pi n/N)]} }}{{A + \sum\limits_{i = 1}^q {{B_{iN}}\cos (iN\varphi )} }}.$$
Considering A is much larger than the modulations ${B_{iN}}$, the above equation can be approximately as:
$$I_n^{BN} = \sum\limits_{j = 1}^p {{D_j}} \cos [j(\varphi + 2\pi n/N)] - \sum\limits_{i = 1}^q {{D_{iN}}\cos [iN(\varphi + 2\pi n/N)]} ,$$
where ${D_j} = {{{B_j}} / A}$, ${D_{iN}} = {{{B_{iN}}} / A}$. The background intensity of the BN fringe patterns is removed, and the reflectivity is cancelled out. It should be noted that the iN-folder harmonics are also removed, which will not affect the phase error model in Sec. 2.1. Then FT and HT are sequentially applied to the BN fringe patterns to generate the FT-filtered phase ${\varphi ^F}$ and HT-based phase ${\varphi ^H}$, respectively. The FHT-compensated phase map ${\varphi ^{FHT}}$ is obtained by averaging the two phase maps as in Eq. (8). The nonuniform-reflectivity phase error can be suppressed after background normalization, which is illustrated as the red line in Fig. 1(b). It should be noted that some residual errors still occur at the fringe edges, which will be further eliminated in Sec. 2.4.

2.3 Self-correction for phase-shift errors

In the above FHT-compensated phase, large errors tend to occur at some period edges of the phase, since the HT-based phase is prone to have a phase-shift to the FT-filtered phase as shown in Fig. 2(a) (the simulation parameters are presented in Sec. 3). The phase in the green dashed region is zoomed in for better illustration. After averaging the two phase maps to obtain the FHT-compensated phase, the phase has a large jump at some period edges of the wrapped phase as shown in the circles in Fig. 2(b). To correct the phase-shift error in the FHT-compensated phase, a self-correction method is proposed.

 figure: Fig. 2.

Fig. 2. Self-correction for phase-shift error: (a) FT-filtered phase and HT-based phase, (b) FHT-compensated phase, (c) phase error, and (d) FHT-compensated phase before and after self-correction.

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According to the phase error model in Sec. 2.1, the saturation-induced phase error can be obtained using the FT-filtered phase ${\varphi ^F}$ and the HT-based phase ${\varphi ^H}$ as:

$$\Delta \varphi = ({\varphi ^H} - {\varphi ^F})/2.$$
It can be found that the phase-shift error fluctuates largely at the wrapped phase edges, which is much larger than that in the other regions, as shown in Fig. 2(c). Thus, FHT-compensated phase can be corrected by identifying the erroneous points using a threshold:
$${\varphi ^{FHTC}} = \left\{ \begin{array}{lc} {\varphi^{FHT}} - \pi ,&\textrm{ if }\Delta \varphi > Th1\\ {\varphi^{FHT}} + \pi ,&\textrm{ if }\Delta \varphi < - Th1\\ {\varphi^{FHT}},&\textrm{ others} \end{array} \right.,$$
where ${\varphi ^{FHTC}}$ is the self-corrected FHT-compensated phase, Th1 is the threshold for identifying the erroneous points with phase-shift error. The FT-filtered phase and HT-based phase generally have a phase-shift of one or two pixels, which will result in the distribution of the phase-shift errors around ${\pm} \pi$. Thus, Th1 can be set to be ${\pi / 2}$ empirically in the following implementation. The FHT-compensated phase before and after self-correction are shown in Fig. 2(d), and the phase in the orange dashed region is zoomed in for better illustration. It should be noted that the self-correction method can correct the phase-shift error by making use of the FT-filtered phase and the HT-based phase without additional images.

In addition, the phase error excluding the phase-shift error can be detected:

$$\Delta {\varphi ^C} = \left\{ \begin{array}{lc} \Delta \varphi - \pi ,&\textrm{ if }\Delta \varphi > Th1\\ \Delta \varphi + \pi ,&\textrm{ if }\Delta \varphi < - Th1\\ \Delta \varphi ,&\textrm{ others} \end{array} \right..$$
The self-corrected phase error $\Delta {\varphi ^C}$ will assist the fringe-edge area detection, which will be explained in the following section. It should be noted that if Eq. (14) is employed to the unwrapped phase, there will be no phase-shift errors. However, when the FT-filtered phase and HT-based phase are unwrapped first, the saturation-induced errors will lead to severe unwrapping errors. Thus, the FT-filtered phase and HT-based phase need to be averaged first to cancel out the saturation-induced error, then phase-shift errors are compensated on the wrapped phase of FHT-compensated phase.

2.4 True phase extraction within fringe-edge area (FEA)

The last residual phase error due to the conventional FHT method is the fringe-edge error. When HT is applied to the non-continuous fringe with fractional period, errors would occur at both ends of the fringe edge due to the spectrum leakage, as shown in Fig. 3. The FHT-compensated phase obtained by averaging the FT-filtered phase and the HT-based phase at the fringe edges deviates from the true phase value. To extract the true phase, we first detect the FEA and then compute the true phase value within the FEA.

 figure: Fig. 3.

Fig. 3. phase error before and after fringe-edge error correction.

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The distribution of the self-corrected phase error in Sec. 2.3 is found to follow a normal distribution: $\Delta {\varphi ^C} \sim N(0,{\sigma ^2})$, where 0 and ${\sigma ^2}$ are the mean and variance, respectively. The fringe-edge errors at the very end of the fringes are large, and typically much larger than phase error induced by intensity saturation. The 3-sigma rule in statistics indicates that nearly all data are within three standard deviations of the mean. Thus, $3\sigma$ is calculated as a criterion for identifying the large fringe-edge error points. If the $\Delta {\varphi ^C}$ of a pixel is larger than $3\sigma$, the pixel is considered as a large fringe-edge error point. Since the phase within the fractional period is incorrect, the fringe periods where the large fringe-edge error points located are detected as the FEA.

Then, the true phase within FEA is computed based on the FT-filtered phase and the phase error model. As mentioned in Sec. 2.1, the saturation-induced phase error of FT-filtered phase can be modeled by Eq. (5), thus iterative algorithm can be applied to obtain the true phase. The iteration formula can be expressed as:

$${\varphi ^{k + 1}}\textrm{ = }{\varphi ^F} + K\sin (N{\varphi ^k}),$$
where ${\varphi ^{k + 1}},\textrm{ }{\varphi ^k}$ are the solution of (k + 1)th and kth round iterations, and coefficient K represents the amplitude of the phase error. The initial guess of the iteration can use ${\varphi ^0}\textrm{ = }{\varphi ^F}$, and the convergence condition can be set as $\textrm{abs[}{\varphi ^{k + 1}} - {\varphi ^k}\textrm{]} < 0.001\textrm{ rad}$, where abs[] means to take the absolute value of a number. In practice, the saturation degree might vary on the object surface, thus sub-period phase error model needs to be established, that is, the sub-K in each fractional period of the FEA needs to be estimated. Since the phase error fluctuates around the ideal phase periodically, and the phase in the fractional period with high frequency can be considered to change smoothly, the linear fitting is applied to the FT-filtered phase in the sub period. The 3-sigma of the difference between the fitted phase and the FT-filtered phase can be approximated as a sub-K value. Once the iterative algorithm converges, the true phase within the FEA can be extracted. The phase error before and after fringe-edge error correction are shown in Fig. 3.

2.5 Summary of method

The workflow of the proposed EFHT method for saturation-induced phase error compensation is presented in Fig. 4, and the proposed method is summarized as follows:

  • 1) Perform background normalization to the captured saturated fringe patterns (Sec. 2.2).
  • 2) Apply FT and HT sequentially to the BN fringe patterns, compute the FT-filtered phase and HT-based phase, and extract the FHT-compensated phase by averaging (Sec. 2.1).
  • 3) Detect phase error distribution, and perform self-correction (Sec. 2.3).
  • 4) Detect the fringe-edge area, and compute the true phase (Sec. 2.4).

 figure: Fig. 4.

Fig. 4. Workflow of proposed EFHT method.

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The final phase in the region of interest is a fused phase map by fusing the original phase in the unsaturated area, the true phase in the FEA, and the self-corrected FHT-compensated phase in the rest area.

3. Numerical simulations

Some numerical simulations were conducted to verify the performance of the proposed method. The intensities of the phase-shifting fringes are encoded as:

$${I_n} = 127.5 + 127.5\cos [2\pi x/T + 2\pi n/N],$$
where T is the period of the fringe, T = W/F, where W is the horizontal resolution, and F is fringe number. The captured intensities of the fringe patterns are expressed as:
$$I_n^r = R\{{127.5 + 127.5\cos [2\pi x/T + 2\pi n/N]} \},$$
where R is the reflectivity of the surface. To simulate the nonuniform reflectivity, we set R = 1 in the left half of the fringe pattern and R = 1/3 in the right half. An example of the fringe intensity with two different reflectivity is shown in Fig. 1(a). If the fringe patterns are saturated, the intensities can be rewritten as:
$$I_n^s = \left\{ \begin{array}{ll} I_n^r\textrm{, }&I_n^r \le 255\\ \textrm{255, }&I_n^r > 255 \end{array} \right.,$$
where 255 is the maximum dynamic range (eight-bit).

The simulated results are shown in Fig. 5, where the simulation parameters are N = 3, W = 300 pixels, F = 6.5, R = R0 = 1.5, if $x \in [1,150]$ and R = R0 /3, if $x \in [151,300]$. One of the saturated fringe patterns is shown in Fig. 5(a), and its intensity cross section is shown in Fig. 5(b), where the left half of the fringe is saturated. The saturation level in this paper is defined as the ratio between the number of saturated pixels and all the pixels in saturated region, and the saturation level in this case is 42.67%. The wrapped phase without compensation, with the conventional FHT method [32] and the proposed EFHT method are shown in Fig. 5(c), and the corresponding phase errors are shown in Fig. 5(d). Some errors occurred at the nonuniform reflectivity surface, at some wrapped phase edges and at fringe edges in the phase with the conventional FHT method, which could be effectively suppressed by background normalization (Sec. 2.2), self-correction method (Fig. 2 in Sec. 2.3), and true phase extraction within FEA (Sec. 2.4) using the proposed EFHT method. It should be noted that the final phase is a fusion of the original phase in the unsaturated area, the true phase in the FEA, and the self-corrected FHT-compensated phase in the rest area.

 figure: Fig. 5.

Fig. 5. Simulation results: (a) one phase-shifting fringe pattern, (b) intensity cross section, (c) upper to lower: wrapped phase without compensation, with conventional FHT method, and with proposed method, and (d) upper to lower: phase error without compensation, with conventional FHT method, and with proposed method.

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To verify the robustness of the proposed method to noise, a random noise with Gaussian distribution is added to the fringe patterns (R0 = 1.5). The simulation is carried out in the case of added random Gaussian noise of 0 mean and 0 to 5 variances. The root-mean-square (RMS) errors without compensation, with the conventional FHT and with the proposed EFHT were obtained as shown in Fig. 6(a). The results demonstrated that the proposed method was tolerant to the noise, probably due to the low-pass filtering operation in FT process. Furthermore, to select a proper filtering window size, the RMS errors were computed by applying low-pass Gaussian filters with different cutoff frequencies, as shown in Fig. 6(b). The phase error of both methods decreases dramatically at a particular cutoff frequency, and then increases slightly with the increasing cutoff frequency. The window size of the low-pass filter can be selected with the frequency at the valley of the curve meanwhile considering the fringe frequency in actual experiments.

 figure: Fig. 6.

Fig. 6. Phase errors: (a) with different noise, and (b) with different cutoff frequencies of the low-pass filter.

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To further evaluate the performance of the proposed method under different saturation levels, the RMS errors were computed with different R0 values, and the results are shown in Fig. 7. The phase error without compensation increases largely with the increase of the saturation level, and that using the conventional FHT method keeps a stable level around 0.2 rad with some slight fluctuations. In contrast, the phase error using the proposed EFHT method is reduced to a low level below 0.1 rad when the saturation level is lower than 60%, and then increases to be close to the result of the conventional FHT method with increasing saturation level.

 figure: Fig. 7.

Fig. 7. Phase errors comparison under different saturation levels.

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4. Experiments and results

To verify the performance of the proposed EFHT method on reduction of saturation-induced phase error, a measurement system was set up, which consisted of a digital-light-processing (DLP) projector (Wintech PRO4500) with 912 × 1140 resolution, one mono camera (Basler acA2040-120um) with 2048 × 1536 resolution and a 16 mm focal length lens. The 3-step phase-shifting algorithm was adopted, and a three-frequency heterodyne temporal phase unwrapping algorithm [34] was used to obtain the continuous phase in the following experiments. The algorithm is implemented with MATLAB 2018a on a computer equipped with an Intel Core i7-7700 processor (3.6 Ghz main frequency) and 16 GB RAM. The iterative algorithm usually required 1 to 4 iterations to converge, and the mean execution time of one round iteration was about 49.14 ms, which can be further reduced by improving the computer hardware.

4.1 Effectiveness evaluation

In order to evaluate the effectiveness of the proposed EFHT method, measurements were conducted on a color flat plate as shown in Fig. 8, which had a nonuniform-reflectivity surface with six different colors.

 figure: Fig. 8.

Fig. 8. Image of the color flat plate.

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One of the captured phase-shifting fringe patterns is shown in Fig. 9(a), and one cross section of the intensity on the dashed line in Fig. 9(a) is plotted in Fig. 9(b). The saturation level is approximate to 24.65%. The amplitude varies from color to color, and the intensity at the boundary between the two colors is not smooth. The background normalization was applied to the captured fringe patterns, and the corresponding BN fringe patterns were obtained as shown in Fig. 9(c). The intensity of the BN fringe pattern on the dashed line in Fig. 9(b) is shown in Fig. 9(d), which shows that the background is removed and the intensity is normalized to be smooth. The spectrum of the original fringe pattern and that of the BN fringe pattern are shown in Figs. 9(e) and (f), respectively, which further verified the effectiveness of the background normalization.

 figure: Fig. 9.

Fig. 9. Background normalization: (a) one captured phase-shifting fringe pattern, (b) intensity cross section of dashed line in (a), (c) BN fringe pattern corresponding to (a), (d) intensity cross section of dashed line in (c), (e) spectrum of (a), and (f) spectrum of (c).

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Then FT and HT operations were sequentially applied to the BN fringe patterns to obtain the FT-filtered phase map (Fig. 10(a)) and HT-based phase map (Fig. 10(b)), respectively. The FHT-compensated phase map was obtained by averaging the two phase maps, as shown in Fig. 10(c). The self-correction method was applied to obtain the self-corrected FHT-compensated phase as shown in Fig. 10(d). To be more intuitive, the cross sections of the FT-filtered phase and HT-based phase on the line are plotted in Fig. 10(e) and the cross sections of the phase before and after self-correction on the line are plotted in Fig. 10(f). The phase in the blue dashed box and the green dashed box are zoomed in as shown in Figs. 10(g) and (h). There was phase shift between the FT-filtered phase and the HT-based phase, leading to false phase in the FHT-compensated phase at the wrapped phase edges as shown in the circles. The results demonstrated that the phase-shift errors were removed effectively by the self-correction method.

 figure: Fig. 10.

Fig. 10. Self-correction for phase-shift error: (a) FT-filtered phase map, (b) HT-based phase map, (c) FHT-compensated phase before self-correction, (d) FHT-compensated phase after self-correction, (e) cross sections of FT-filtered phase and HT-based phase on the line in (a) and (b), (f) cross sections of FHT-compensated phase before and after self-correction on the line in (c) and (d), (g) phase in blue dashed box, and (h) phase in green dashed box.

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After self-correction for phase-shift error, the self-corrected phase error was detected as shown in Fig. 11(a), and the large fringe-edge errors distributed at the both ends of the fringes. The histogram of the detected self-corrected phase error is shown in Fig. 11(b), and the variance of the phase error was calculated as 0.05 ($\sigma$), thus the value of 0.15 ($3\sigma$) was used as the criterion for detecting the fringe-edge area, as shown in Fig. 11(c). One cross section of the wrapped phase at the fringe edge before and after the fringe-edge error correction are plotted in Fig. 11(d).

 figure: Fig. 11.

Fig. 11. True phase extraction: (a) self-corrected phase error, (b) histogram of phase error, (c) detected fringe-edge area, and (d) wrapped phase before and after fringe-edge error correction.

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The phase error distributions based on the ground truth (measured by twelve-step phase-shifting algorithm using unsaturated images) are shown in Fig. 12. Figure 12(a) shows the error distribution using the original captured fringe patterns without compensation, and the ripple-like saturation-induced errors appeared in the saturated regions. Figure 12(b) is the result using the conventional FHT compensation method, which shows errors due to the nonuniform reflectivity, large phase-shift errors at some wrapped phase edges, and fringe-edge errors at both ends of the fringe edges. The phase error after employing the proposed EFHT method is shown in Fig. 12(c), which had been reduced mostly under 0.1 rad. To give a better illustration, the cross sections of the phase errors on the dashed line in Figs. 12(a)–(c) are plotted in Fig. 12(d), and the results in the left edge region (in purple box) and in the severe saturated region (in blue box) are zoomed in as shown in Figs. 12(e) and (f). It demonstrated that the conventional FHT method reduced the saturation-induced phase error but with some residual errors. In contrast, the saturation-induced error and three types of residual errors were significantly reduced by the proposed EFHT method.

 figure: Fig. 12.

Fig. 12. Phase error distribution: (a) without compensation, (b) using conventional FHT method, (c) using EFHT method, (d) phase error cross sections on the line in (a)–(c), (e) phase errors in purple box, and (f) phase errors in blue box.

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It should be noted that the normalized intensity in the saturated region is not equal around 1 (Fig. 9(d)), which is because the background intensity calculated by Eq. (10) is not ideal when the fringe patterns are saturated. For comparison, the BN of the saturated region was implemented using the background intensity obtained by the maximum intensity 255 and the minimum intensity ${I_{\min }}$: ${I_{bg}} = (255 - {I_{\min }})/2$ (BN method 2). The intensity cross section corresponding to Fig. 9(d) is presented in Fig. 13(a), where the intensity of the saturated fringe was equal around 1. The corresponding phase errors without compensation, with the BN method 2 and the proposed BN method are shown in Fig. 13(b). The phase errors of the two methods in the saturated regions are enlarged as shown in Figs. 13(c) and (d), respectively, which were very close. The results demonstrated that the proposed BN operation on the saturated region would not influence the final reconstruction.

 figure: Fig. 13.

Fig. 13. Comparison measurement: (a) intensity cross section, (b) phase errors, (c) enlarged phase errors in blue box, and (d) enlarged phase errors in orange box.

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4.2 Comparison measurement

To further evaluate the performance of the proposed method, experiments were carried out using the conventional FHT method [32], complementary phase method [28] and the proposed EFHT method. Measurements were performed on a geometry model as shown in Fig. 14(a), and one of the captured phase-shifting patterns is shown in Fig. 14(b), where the saturation level is approximate to 26.76%. The phase reconstructions without compensation, with conventional FHT method, with complementary phase method, and with EFHT method are shown in Figs. 14(c)–(f), respectively. The corresponding phase error distributions were extracted as shown in Figs. 14(g)–(j), respectively. The results showed that the conventional FHT method leaded to some residual phase errors, whereas the complementary phase method and the proposed EFHT method reduced the phase errors more effectively. To quantitatively evaluate the methods, the maximum absolute error (MAX), mean absolute error (MAE) and the RMS error are compared in Table 1. The “Number” in the table means the number of the input images. The performance of the complementary phase method and the proposed EFHT method was better than that of the conventional FHT method. In addition, the proposed method has the similar high measurement accuracy as the complementary phase method, but with the advantage of reducing the number of the required images by half.

 figure: Fig. 14.

Fig. 14. Results of comparison measurement: (a) image of the geometry model, (b) one captured phase-shifting fringe pattern, phase reconstruction (c) without compensation, (d) with conventional FHT method, (e) with complimentary phase method, (f) with proposed EFHT method, (g)–(j) corresponding phase error of phase in (c)–(f).

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Table 1. Comparison of measurement results with different methods.

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Since the saturated fringe signal is considered as the aliased signal with high-order harmonics, which is similar to the nonlinear fringe signal, experiments were conducted using a HT-based nonlinear correction method [33] for comparison. The spectrum of the background normalized fringe pattern using the HT-based nonlinear correction method and the proposed method are shown in Figs. 15(a) and (b), which verified the proposed method had a better performance on suppressing the background intensity. The phase reconstruction using the HT-based nonlinear correction method is shown in Fig. 15(c), and there were significant residual errors on the object surface. One cross section of the phase error on the 700th row using the HT-based nonlinear correction method and the proposed method are plotted in Fig. 15(d). The results demonstrated that the proposed method performed better than the HT-based nonlinear correction method, since the saturated fringe signal is destroyed more severely than the nonlinear one.

 figure: Fig. 15.

Fig. 15. Results of comparison measurement: spectrum of background normalized fringe pattern (a) using HT-based nonlinear correction method, (b) using proposed method, (c) phase reconstruction using HT-based nonlinear correction method, (d) compared cross sections of phase error.

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4.3 Measurement of HDR scenario

To verify the proposed EFHT method to measure surfaces with different reflection properties in HDR scenario, measurement was performed simultaneously on multiple isolated objects, a plastic letter block, a metal workpiece, and a plaster dog as shown in Fig. 16(a). One of the captured phase-shifting fringe patterns is shown in Fig. 16(b), where the saturation level is approximate to 42.85%. The intensity cross sections of the lines on the three objects are shown in Fig. 16(c), which indicated the different reflections. The results without compensation, and with the conventional FHT compensation are shown in Figs. 16(d) and (e), respectively. The white points on the reconstructed surface of the metal workpiece in Fig. 16(d) were invalid points due to the intensity saturation of the high reflective surface, which were suppressed using conventional FHT method, but with large phase-shift errors. To suppress the phase-shift errors, comparison measurements were conducted by the method of averaging the unwrapped phase maps and the proposed method of averaging the wrapped phase maps of FT-filtered phase and HT-based phase. The results of the two methods are shown in Figs. 16(f) and (g), respectively. Some unwrapping errors occurred on the metal surface, whereas the phase was reconstructed successfully using the proposed method. The details of the metal workpiece in blue dashed box and the plaster dog in orange dashed box in Figs. 16(d)–(g) are zoomed in for better illustration, as shown in Figs. 17(a)–(h). The results also demonstrated that the proposed EFHT method performed well on the measurement of surfaces with different reflection properties in HDR scenario.

 figure: Fig. 16.

Fig. 16. Measurement results of an HDR scenario: (a) image of three isolated objects, (b) one captured phase-shifting fringe pattern, (c) intensity cross sections of the lines in (b); phase reconstruction (d) without compensation, (e) with conventional FHT method, (f) with method of averaging unwrapped phase maps, (g) with proposed EFHT method.

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 figure: Fig. 17.

Fig. 17. Details of the measurement results of an HDR scenario: zoomed-in metal workpiece in blue dashed box (a)–(d) without compensation, with conventional FHT method, with method of averaging unwrapped phase maps, and with proposed EFHT method; zoomed-in plaster dog in orange dashed box (e)–(h) without compensation, with conventional FHT method, with method of averaging unwrapped phase maps, and with proposed EFHT method.

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5. Conclusion

This paper presents an enhanced Fourier-Hilbert-transform method (EFHT) to suppress the saturation-induced phase error, and three types of residual errors: nonuniform-reflectivity error, phase-shift error, and fringe-edge error. The captured fringe patterns were first background normalized to suppress the effect of the nonuniform reflectivity. Next, the phase-shift error in the FHT-compensated phase was corrected by the proposed self-correction method. Then, the FEA was located by the statistical analysis of the self-corrected phase error, and the true phase within the FEA was computed using the iteration algorithm based on the sub-period phase error model. Experimental results demonstrated the effectiveness of the proposed EFHT method in reducing the saturation-induced phase error. The proposed method could significantly suppress the three residual errors compared to the conventional FHT method, and it has similar high measurement accuracy compared to the complementary phase method but reducing required images by half. Moreover, this method has the ability to measure the surfaces with different reflection properties in HDR scenarios.

Appendix: Fourier transform and Hilbert transform of fringe patterns

A. Fourier transform of fringe patterns

The saturated fringe can be considered as the aliased signal with fundamental harmonic and high-order harmonics, as described by Eq. (3). When Fourier transform is applied to the saturated fringe with respect to x, the spectrum of the fringe signal can be obtained as [36]:

$${G^s}({f_x},y) = {Q_0}({f_x},y) + \sum\limits_{j = 1}^p {[{{Q_j}({f_x} - {f_0},y) + {Q_j}^\ast ({f_x} + f,y)} ]} ,$$
where ${f_0}$ is the spatial-carrier frequency, ${f_x}$ is the spatial frequency in the x direction, ${Q_0}({f_x},y)$ is the DC component, ${Q_j}({f_x},y)$ is the jth order spectrum component, and * denotes a complex conjugate. Due to the intensity saturation, the spectrum of the fringe signal consists of many high-order harmonics. Low-pass filter with proper window size can be employed to eliminate the high-order harmonics, and N-order harmonics are considered after filtering [32], which can be written as:
$${G^s}({f_x},y) = {Q_0}({f_x},y) + \sum\limits_{j = 1}^N {[{{Q_j}({f_x} - {f_0},y) + {Q_j}^\ast ({f_x} + f,y)} ]} .$$
Then an inverse Fourier transform is employed to ${G^s}({f_x},y)$, and the FT-filtered fringe can be obtained as:
$$I_n^F = A + \sum\limits_{j = 1}^N {{B_j}} \cos [j(\varphi + 2\pi n/N)].$$

B. Hilbert transform of fringe patterns

Hilbert transform operation transforms a function $\mu (t)$ into another function by convolution as [37]:

$$H(\mu )(t) = \frac{1}{\pi }\int_{ - \infty }^{ + \infty } {\frac{{\mu (t)}}{{t - \tau }}} \textrm{d}\tau .$$
It can be considered as a multiplier operator in frequency domain:
$$FT[{H(\mu )(\omega )} ]= {\delta _H}(\omega ) \times FT(\mu )(\omega ),$$
where FT represents the Fourier transform operator, and ${\delta _H}(\omega )$ can be expressed as:
$${\delta _H}(\omega ) = .\left\{ \begin{array}{ll} {e^{i\pi /2}},&\textrm{ if }\omega < 0\\ 0,&\textrm{ if }\omega = 0\\ {e^{ - i\pi /2}},&\textrm{ if }\omega > 0 \end{array} \right..$$
Thus, Hilbert transform can shift the phase of negative and positive frequency components by ${\pi / 2}$ and ${{ - \pi } / 2}$, respectively. When Hilbert transform is applied to the FT-filtered fringe in Eq. (23), the cosine signal will be transformed into sine signal, which is called HT-based fringe in this paper and can be expressed as:
$$I_n^H = A^{\prime} + \sum\limits_{j = 1}^N {{B_j}} \sin [j(\varphi + 2\pi n/N)].$$

Funding

National Natural Science Foundation of China (61960206010); Fundamental Research Funds for the Central Universities (A0920502052201).

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (17)
Fig. 1.
Fig. 1. Effect of the nonuniform reflectivity: (a) intensity on the surface with two different reflectivity, and (b) phase error by employing FHT method before and after background normalization.

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Fig. 2.
Fig. 2. Self-correction for phase-shift error: (a) FT-filtered phase and HT-based phase, (b) FHT-compensated phase, (c) phase error, and (d) FHT-compensated phase before and after self-correction.

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Fig. 3.
Fig. 3. phase error before and after fringe-edge error correction.

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Fig. 4.
Fig. 4. Workflow of proposed EFHT method.

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Fig. 5.
Fig. 5. Simulation results: (a) one phase-shifting fringe pattern, (b) intensity cross section, (c) upper to lower: wrapped phase without compensation, with conventional FHT method, and with proposed method, and (d) upper to lower: phase error without compensation, with conventional FHT method, and with proposed method.

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Fig. 6.
Fig. 6. Phase errors: (a) with different noise, and (b) with different cutoff frequencies of the low-pass filter.

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Fig. 7.
Fig. 7. Phase errors comparison under different saturation levels.

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Fig. 8.
Fig. 8. Image of the color flat plate.

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Fig. 9.
Fig. 9. Background normalization: (a) one captured phase-shifting fringe pattern, (b) intensity cross section of dashed line in (a), (c) BN fringe pattern corresponding to (a), (d) intensity cross section of dashed line in (c), (e) spectrum of (a), and (f) spectrum of (c).

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Fig. 10.
Fig. 10. Self-correction for phase-shift error: (a) FT-filtered phase map, (b) HT-based phase map, (c) FHT-compensated phase before self-correction, (d) FHT-compensated phase after self-correction, (e) cross sections of FT-filtered phase and HT-based phase on the line in (a) and (b), (f) cross sections of FHT-compensated phase before and after self-correction on the line in (c) and (d), (g) phase in blue dashed box, and (h) phase in green dashed box.

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Fig. 11.
Fig. 11. True phase extraction: (a) self-corrected phase error, (b) histogram of phase error, (c) detected fringe-edge area, and (d) wrapped phase before and after fringe-edge error correction.

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Fig. 12.
Fig. 12. Phase error distribution: (a) without compensation, (b) using conventional FHT method, (c) using EFHT method, (d) phase error cross sections on the line in (a)–(c), (e) phase errors in purple box, and (f) phase errors in blue box.

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Fig. 13.
Fig. 13. Comparison measurement: (a) intensity cross section, (b) phase errors, (c) enlarged phase errors in blue box, and (d) enlarged phase errors in orange box.

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Fig. 14.
Fig. 14. Results of comparison measurement: (a) image of the geometry model, (b) one captured phase-shifting fringe pattern, phase reconstruction (c) without compensation, (d) with conventional FHT method, (e) with complimentary phase method, (f) with proposed EFHT method, (g)–(j) corresponding phase error of phase in (c)–(f).

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Fig. 15.
Fig. 15. Results of comparison measurement: spectrum of background normalized fringe pattern (a) using HT-based nonlinear correction method, (b) using proposed method, (c) phase reconstruction using HT-based nonlinear correction method, (d) compared cross sections of phase error.

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Fig. 16.
Fig. 16. Measurement results of an HDR scenario: (a) image of three isolated objects, (b) one captured phase-shifting fringe pattern, (c) intensity cross sections of the lines in (b); phase reconstruction (d) without compensation, (e) with conventional FHT method, (f) with method of averaging unwrapped phase maps, (g) with proposed EFHT method.

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Fig. 17.
Fig. 17. Details of the measurement results of an HDR scenario: zoomed-in metal workpiece in blue dashed box (a)–(d) without compensation, with conventional FHT method, with method of averaging unwrapped phase maps, and with proposed EFHT method; zoomed-in plaster dog in orange dashed box (e)–(h) without compensation, with conventional FHT method, with method of averaging unwrapped phase maps, and with proposed EFHT method.

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Tables (1)

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Table 1. Comparison of measurement results with different methods.

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Equations (27)

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I n ( x , y ) = A ( x , y ) + B ( x , y ) cos ⁡ [ φ ( x , y ) + 2 π n / N ] ,   n = 1 , 2 , … , N ,
φ ( x , y ) = arctan ⁡ ∑ n = 1 N I n ( x , y ) sin ⁡ ( 2 π n / N ) ∑ n = 1 N I n ( x , y ) cos ⁡ ( 2 π n / N ) .
I n s = A + ∑ j = 1 p B j cos ⁡ [ j ( φ + 2 π n / N ) ] ,
I n F = real{ IFT{ filter ⋅ [FT( I n s )]} } ,
Δ φ F  =  φ F − φ   ≈ arctan ⁡ − C N − 1 sin ⁡ ( N φ ) 1 + C N − 1 cos ⁡ ( N φ ) ,   ≈ − C N − 1 sin ⁡ ( N φ )
I n H = HT( I n F ) =  A ′ + ∑ j = 1 p B j sin ⁡ [ j ( φ + 2 π n / N ) ] ,
Δ φ H  =  φ H − φ   ≈ arctan ⁡ C N − 1 sin ⁡ ( N φ ) 1 + C N − 1 cos ⁡ ( N φ ) ,   ≈ C N − 1 sin ⁡ ( N φ )
φ F H T = ( φ F + φ H ) / 2.
I n B N = I n s − I b g I b g ,
I b g = ∑ n = 1 N I n s / N .
I b g = A + ∑ i = 1 q B i N cos ⁡ ( i N φ ) ,
I n B N = ∑ j = 1 p B j cos ⁡ [ j ( φ + 2 π n / N ) ] − ∑ i = 1 q B i N cos ⁡ [ i N ( φ + 2 π n / N ) ] A + ∑ i = 1 q B i N cos ⁡ ( i N φ ) .
I n B N = ∑ j = 1 p D j cos ⁡ [ j ( φ + 2 π n / N ) ] − ∑ i = 1 q D i N cos ⁡ [ i N ( φ + 2 π n / N ) ] ,
Δ φ = ( φ H − φ F ) / 2.
φ F H T C = { φ F H T − π ,  if  Δ φ > T h 1 φ F H T + π ,  if  Δ φ < − T h 1 φ F H T ,  others ,
Δ φ C = { Δ φ − π ,  if  Δ φ > T h 1 Δ φ + π ,  if  Δ φ < − T h 1 Δ φ ,  others .
φ k + 1  =  φ F + K sin ⁡ ( N φ k ) ,
I n = 127.5 + 127.5 cos ⁡ [ 2 π x / T + 2 π n / N ] ,
I n r = R { 127.5 + 127.5 cos ⁡ [ 2 π x / T + 2 π n / N ] } ,
I n s = { I n r ,  I n r ≤ 255 255,  I n r > 255 ,
G s ( f x , y ) = Q 0 ( f x , y ) + ∑ j = 1 p [ Q j ( f x − f 0 , y ) + Q j ∗ ( f x + f , y ) ] ,
G s ( f x , y ) = Q 0 ( f x , y ) + ∑ j = 1 N [ Q j ( f x − f 0 , y ) + Q j ∗ ( f x + f , y ) ] .
I n F = A + ∑ j = 1 N B j cos ⁡ [ j ( φ + 2 π n / N ) ] .
H ( μ ) ( t ) = 1 π ∫ − ∞ + ∞ μ ( t ) t − τ d τ .
F T [ H ( μ ) ( ω ) ] = δ H ( ω ) × F T ( μ ) ( ω ) ,
δ H ( ω ) = . { e i π / 2 ,  if  ω < 0 0 ,  if  ω = 0 e − i π / 2 ,  if  ω > 0 .
I n H = A ′ + ∑ j = 1 N B j sin ⁡ [ j ( φ + 2 π n / N ) ] .
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