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Abstract
Fringe analysis in two-shot phase-shifting interferometry is important but meets challenges due to a limited number of images, corrupting noise, and background modulation. Here we propose an effective algorithm for phase retrieval from two interferograms with unknown phase shifts. The algorithm first evaluates the phase shift in a local mask through phase fitting and global optimization and then computes a full-field phase map using an arctangent function. Since the phase shift evaluation is performed within a local mask, the algorithm is fast compared with conventional optimization-based algorithms and typically needs tens of seconds to complete the processing. Computer simulation and experimental results show that the proposed algorithm has excellent performance compared with state-of-the-art algorithms. A complete software package of the algorithm in MATLAB is available at http://two-shot.sourceforge.io/.
© 2017 Optical Society of America
1. Introduction
Two-shot phase-shifting interferometry (PSI) that uses two randomly phase-shifted fringe patterns for the measurement of a range of quantities is a field being actively explored [1–4]. Two-shot PSI has many noticeable advantages. Compared with conventional multi-shot (such as three, four, and five-shot) PSI [5, 6] that requires at least three interferograms as well as known phase shifts between two consecutive interferograms, two-shot PSI needs as few as two interferograms and the phase shift can be random, unknown, and arbitrary (except singular cases 0 and π rad). These features reduce the requirements of hardware performance (e.g., repeatability and accuracy of the phase shifter) and environmental conditions (e.g., air turbulence) [7] in practical measurements. Compared with single-shot interferometry [8–12] that only uses single interferograms for measurements, two-shot PSI is capable of resolving sign ambiguity of retrieved phase and offers greater measurement precision. It is becoming popular and has found applications in many fields, such as image encryption [1], displacement measurement [2], and surface figure testing [3].
While the demand for hardware performance and environmental conditions has been much reduced compared with that in multi-shot PSI, requirements for fringe analysis algorithms in two-shot PSI have been increased. In multi-shot PSI, well-developed techniques for phase retrieval from multiple (≥ 3) fringe patterns are readily available for use [5, 6, 13–17]. However, since only two randomly phase-shifted fringe patterns are available and they might be corrupted with noise, background, and modulation [18], phase retrieval in two-shot PSI meets challenges. Fringe analysis algorithms play an essential role in two-shot PSI [19–26]. Kreis et al. pioneered the work of two-shot fringe analysis with unknown phase shifts and proposed a Fourier transform-based demodulation method [27]. The method works well when the phase shift varies from π/3 to 2π/3, but may fail for small phase shifts or noisy fringe patterns. Vargas et al. proposed a Gram-Schmidt (GS) orthonormalization method for phase retrieval by treating two fringe patterns as independent vectors [28–30]. The algorithm is simple, fast, and being able to yield very accurate results, but might be susceptible to noise and background [31, 32]. Based on the GS work, Ma et al. proposed a robust two-dimensional continuous wavelet transform algorithm for noise suppression and phase extraction from two interferograms [31]. Trusiak et al. enhanced the performance of the GS algorithm by pre-filtering noise and background of the images using Hilbert-Huang transform [32]. Niu et al. proposed a quotient of inner products algorithm based on the GS algorithm to simultaneously extract phase map and phase shift [7]. Besides the GS family algorithms, Deng et al. studied phase extraction from two fringe patterns by computing the ratio of extreme values of interference of two interferograms [33]. Liu et al. developed a Lissajous figure and ellipse fitting algorithm for simultaneous phase shift and phase map extraction [34, 35]. Wielgus et al. presented an algorithm capable of performing fringe pattern phase demodulation from two frames with unknown, linearly nonuniform phase shift [36]. All are very efficient algorithms for two-shot fringe pattern analysis.
While most state-of-the-art algorithms use mathematical transform and vector algebra to analyze fringe patterns in two-shot PSI, we recently proposed a parametric algorithm for two-shot fringe pattern analysis using orthogonal function and global optimization [37]. The optimization-based approach has advantages, such as being able to deal with singular phase shifts (0 or π rad) and small fringe numbers (e.g. less than one) and being insensitive to random noise and local defects. However, the algorithm is very slow (it generally takes several hours to finish processing) and is less likely to succeed in processing interferograms with complex fringe shapes. Based on the work, we propose a two-step method for simultaneous phase shift and phase map retrieval in two-shot PSI. In the first step, fringe patterns are segmented using a circular mask and the images in the mask are evaluated using the aforementioned parametric method for phase shift extraction. In the second step, full-field phase map is reconstructed based on the extracted phase shift using a simple arctangent function. The proposed method inherits some advantages of the parametric method, such as being robust to noise, but has much faster and more powerful processing capability.
2. Principle
2.1 Phase retrieval
Two-shot phase-shifted fringe patterns can be mathematically formulated as
where x and y are spatial coordinates; a and b are background and modulation terms, respectively; ϕ and δ are phase map and phase shift to be retrieved, and n is additive noise. By ignoring the noise and suppressing the background using a high-pass Gaussian filter G(x, y; σ) = 1 – exp[(x2 + y2)/σ2], where σ is the standard deviation (σ = 0.03 is used throughout the paper), Eqs. (1) and (2) become After applying a simple trigonometric transformation to Eq. (4), we can find the quadrature signalTaking both Eqs. (3) and (5) into account, the modulating phase ϕ can be computed from an arctangent function asTo perform the calculation, the phase shift δ needs to be recovered first, which could be realized as follows.2.2 Phase shift evaluation through optimization
In [37], we proposed a parametric approach for fringe pattern analysis in two-shot PSI. The algorithm uses a linear combination of orthogonal functions to represent full-field phase and employs an optimization algorithm to search expansion coefficients. It can simultaneously yield phase maps and phase shifts from two fringe patterns and is essentially applicable to help evaluate the phase shift here. However, since the phase fitting is performed globally, the original algorithm is computationally expensive and time-consuming. Considering that the phase shift δ is a constant across the full field, it is reasonable to hypothesize that only processing a small portion of the data using the algorithm will yield comparably accurate results of the phase shift. This strategy will greatly enhance solution convergence and computation speed and, meanwhile, preserve the evaluation accuracy and robustness.
Suppose the fringe patterns are segmented by a circular mask at the site (x0, y0) with a radius r0. The images within the mask can be normalized as [38–40]
where ϕmask is the phase within the mask. Normally, it can be decomposed into a linear combination of finite terms of orthogonal bases aswhere j is index, cj is expansion coefficient, and Zj is orthogonal basis. Here Zernike polynomials that are a complete set of orthogonal bases across a unit disk are employed. They are defined as [37, 41]where ρ ∈ [0, 1] and θ ∈ [0, 2π] are normalized radial and angular coordinates, respectively; j = (n + 1)(n + 2)/2 is the index; the radial polynomialswhere n and m are non-negative integers, representing radial degree and azimuthal frequency, respectively, and satisfying n - m = even. Based on the least-square criteria, a cost function describing the closeness between the normalized fringe patterns and their estimates is constructed asHere X = [c1, c2, …, cj, δ]T is the solution vector, at which the cost function attains its global minima,In this process, the radial degree n of the Zernike polynomials should be a number large enough to well approximate the phase in the mask. However, a large n involves more expansion coefficients (variables) to be determined, which will greatly complicate the solution space of the optimization problem and increase computation complexity. To improve computation speed and, meanwhile, ensure good fitting, a radial degree n = 3 is used through this study. In this way, eleven variables (ten expansion coefficients and one phase shift) need to be determined. By modeling the phase shift using a linear function δ = α0 + α1x + α2y, the above mathematical model could be easily adapted for processing two-shot fringe patterns with a linearly-varying phase shift [36].
The constructed cost function f is typically multimodal and has numerous local minima, which tends to trick many optimization algorithms. A differential evolution (DE) algorithm that is a powerful population-based stochastic search technique for multidimensional global optimization [37, 42] is employed to pinpoint global minima in this study. The basic idea of the DE algorithm is to evolve parameter vectors containing large population candidates towards global minima through repeated mutation, crossover, and selection. Its principle and implementation have been well described in [37, 42] and are not covered here.
Once the phase shift is extracted using the global optimizer, it is substituted back into Eq. (6) to yield the full-field phase map.
3. Simulation and experiments
3.1 Simulation
A numerical simulation was first performed to demonstrate the validity and effectiveness of the proposed algorithm. Figures 1(a) and 1(b) show two computer-generated fringe patterns with a grid of 256 × 256 pixels and a small phase shift 0.01π rad. The simulation settings are as follows: the background a(x, y) = 0, the modulation b(x, y) = 1, the theoretical phase map
where x, y ∈ [-0.9, 0.9], and the noise n(x, y) is additive Gaussian white noise with a mean 0 and a standard deviation 0.6. Figures 1(c) and 1(d) illustrate the continuous and wrapped versions of the theoretical phase map, respectively.
Fig. 1 Simulated two-shot fringe patterns [(a) and (b)] with a phase shift δ = 0.01π rad and the continuous (c) and wrapped (d) phase. The three red disks on (a) and (b) are three masks used to evaluate the phase shift. Colorbar unit: rad.
To evaluate the phase shift, three masks all with a radius of 30 pixels [red disks in Figs. 1(a) and 1(b)] were selected at different sites of the fringe pattern. Phases within the three masks were fitted with up to third-order (n = 3) Zernike polynomials and the expansion coefficients and phase shift were searched using the DE algorithm. Figure 2 shows typical evaluation results, including original fringe patterns in the masks, cost function f versus population evolution generations, computed expansion coefficients cj, reconstructed phase ϕ, and re-computed fringe intensity I using the reconstructed phase ϕ. The evaluations succeed in all three cases and the phase shifts are estimated to be (0.0095 ± 0.0002)π, (0.0103 ± 0.0003)π, and (0.01 ± 0.0003)π rad for mask 1, 2, and 3, respectively, which are close to the true value 0.01π. Note that the results are expressed as mean ± standard deviation from five independent runs since the DE algorithm is a stochastic search technique. Once the phase shift is extracted, it is substituted back into Eq. (6) to yield the full-field phase map and reconstruction error, which are illustrated in the first column of Fig. 3. The retrieved full-field phase map is in good agreement with the theoretical one in Fig. 1(d).
Fig. 2 Phase shift evaluation using fringe data in different masks. First column: fringe patterns in mask 1 (first row), 2 (second row), and 3 (third row). Second column: evaluation process showing that the value of the cost function decreases with evolution generations of the solution vector (X). Third column: optimized fit coefficients. Fourth column: computed phase using Eq. (9). Last column: computed intensity map using Eq. (7). The extracted phase shifts for mask 1, 2, and 3 are (0.0095 ± 0.0002)π, (0.0103 ± 0.0003)π, and (0.01 ± 0.0003)π rad, respectively.
Fig. 3 Comparison of retrieved phase using different algorithms. From left to right: retrieved phase maps and error maps by the proposed method, the parametric method [37], the Kreis method [27], the OF method [19], and the GS method [28]. Colorbar unit: rad.
To compare the proposed algorithm with other state-of-the-art methods, the same images were also processed by the parametric method using up to eighth-order Zernike polynomials [37], the Kreis method [27], the optical flow (OF) method [19], and the GS method [28]. Corresponding retrieved phase maps and residual errors are put side by side along with those by the proposed method in Fig. 3. Among all the algorithms, the parametric method gives the most accurate result (root mean square or RMS error 0.13 rad) since it tends to filter out high-frequency noise, but takes much longer time to finish the computation (parametric: ~7 hr, proposed ~20 sec, Kreis: ~0.1 sec, OF: ~0.2 sec, GS: ~0.01 sec). The proposed algorithm has close reconstruction accuracy to the GS method (RMS error 0.51 vs 0.49 rad) and is more accurate than the Kreis method and the OF method (RMS error 0.51 vs 0.64 rad, and 0.51 vs 0.65 rad, respectively).
Optimal selection of the mask, including the position (x0, y0) and the radius r0, is important for accurate evaluation of the phase shift. The Zernike polynomials employed to represent phase in the mask are restricted to third order in this study, which is sufficient to approximate linear (first order) and circular (second order) fringes. From the perspective, it is recommended to place the mask on linear or circular fringes in images to minimize phase fitting error (see Fig. 1). The mask size can typically be 10%-30% of the entire image, balancing local noise, phase fitting error, and computation complexity. Figure 4 illustrates a specific example of evaluation results of the phase shift when the mask is centered at mask 1 but has a varying diameter from 5% to 80% of the whole image in the simulation. The results indicate that when the mask has a moderate radius (15%-60% of the radius of the whole image), the extracted phase shift is most accurate. The estimation errors of the phase shift for radii less than 15% result from local noise and errors for radii greater than 60% are caused by insufficient fitting of the phase by the third-order Zernike polynomials.
Fig. 4 Specific example showing the relation between extracted phase shift δ and mask radius. When the mask has a moderate size of 15%-60% of the whole image, the extracted phase shift is most accurate (δ true value 0.01π). The results, shown in mean ± standard deviation, are calculated from five independent runs. The insets are snapshots of the images in the mask at different radii.
Based on the example, we also studied the influences of the background, modulation, and additive noise on the accuracy of final results. The background a(x, y) and modulation b(x, y) are modelled using the Gaussian functions a(x, y; σa) = a∙exp[(x2 + y2)/σa2] and b(x, y; σb) = b∙exp[(x2 + y2)/σb2], where a and b are amplitudes and σa and σb are standard deviations, while the additive noise is simulated using Gaussian white noise with a standard deviation σ. Figure 5 shows the relationships between the RMS values of phase retrieval errors and the amplitudes of varying background, modulation, and noise. It is seen that the reconstruction RMS errors increase almost linearly with the background amplitude a and the noise standard deviation σ, while it is hardly affected by the modulation amplitude b, which is true due to the cancellation of the modulation term during the calculation [Eqs. (3) and (5)]. Since the same high-pass Gaussian filter [Eqs. (3) and (4)] was used throughout the analysis, the reconstruction error induced by background could be greatly reduced by adjusting the kernel size of the high-pass Gaussian filter.
Fig. 5 Influences of background (a), modulation (b), and additive noise (c) on the accuracy of phase retrieval results.
3.2 Experiments
The practical performance of the proposed algorithm in two-shot PSI was first tested using a pair of experimental interferograms with closed fringes, as shown in Figs. 6(a) and 6(b). The images with dense fringes and varying spatial frequency have a size of 615 × 615 pixels. Eleven other phase-shifted images (not shown here) together with the two frames were first processed using the advanced iterative algorithm (AIA) [43] to provide the reference phase, which is shown in Fig. 6(c). A window centered at the top spanning 20% of the images was used as the mask, as shown in Fig. 6(d). The fringe patterns within the mask were evaluated using the proposed algorithm to yield the phase shift, which is 0.58π rad in this case (0.57π rad by the AIA as a comparison). Based on the extracted phase shift and high-pass Gaussian-filtered intensity maps (σ = 0.03), the phase map [Fig. 6(e)] was retrieved and the error map [Fig. 6(i)] was computed versus the reference. To compare the results with those by other algorithms, the fringe patterns were also processed by the parametric method using up to the seventh-order Zernike polynomials, the Kreis method, and the GS method. Resultant phase maps and residual error maps are illustrated in Figs. 6(f)-6(h) and Figs. 6(j)-6(l), respectively. The corresponding RMS values of the residual errors are 0.29 rad, 0.46 rad, 0.34 rad, and 0.31 rad, respectively. Although all four methods could successfully process the fringe patterns, the proposed method yields the smallest residual error in this case.
Fig. 6 Phase retrieval from two-shot experimental fringe patterns [(a) and (b)] with closed fringes. (c) Reference phase map obtained using the AIA method (13 frames) [43]. (d) Fringe pattern in the mask [red disk in (a)] for phase shift evaluation. The extracted phase shift is 0.58π rad in this case. (e) – (h) Recovered phase maps using the proposed method, the parametric method, the Kreis method, and the GS method, respectively. (i) – (l) Corresponding error maps with respect to the AIA result. Colorbar unit: rad.
The proposed algorithm was also tested using a pair of experimental interferograms with open fringes, as shown in Figs. 7(a) and 7(b). The images have a size of 365 × 365 pixels. A window centered at the top spanning 20% of the image size was used as the mask [shown in Fig. 7(d)] for phase shift evaluation. Figures 7(d) and 7(e) show retrieved phase maps by the proposed method and the AIA (four frames), respectively. Figure 7(f) illustrates the resultant error map, which has an RMS value of 0.14 rad. In this case, the extracted phase shifts by the proposed method and the AIA are 0.49π rad and 0.43π rad, respectively. The fringe patterns were high-pass Gaussian filtered (σ = 0.03) before processed by the proposed method; no filtering procedures were applied when processed by the AIA.
Fig. 7 Phase retrieval from two-shot experimental fringe patterns [(a) and (b)] with open fringes. (c) Fringe pattern in the selected mask [red disk in (a)] for phase shift evaluation. The extracted phase shift is 0.49π rad in this case. (d), (e), and (f) Retrieved phase maps using the proposed method, the AIA method (four frames), and their difference, respectively. Colorbar unit: rad.
Figure 8 demonstrates the application of the proposed algorithm to the analysis of two-shot interferograms with background modulation. The images were acquired in a non-null PSI [44, 45] and have a size of 786 × 786 pixels. The selected mask [Fig. 8(c)] is on the top and spans 20% of the image size. Figures 8(d) and 8(e) show retrieved phase maps by the proposed method and the AIA (13 frames), respectively. Figure 8(f) illustrates the resultant error map, which has an RMS value of 0.32 rad. The extracted phase shifts by the proposed method and the AIA are 0.48π rad and 0.47 rad, respectively. The fringe patterns were high-pass Gaussian filtered (σ = 0.03) before processed by the proposed method; no filtering procedures were applied when processed by the AIA. The results suggest that the proposed method could well deal with fringe patterns with background modulation.
Fig. 8 Phase retrieval from two-shot experimental fringe patterns [(a) and (b)] with background modulation. (c) Fringe pattern in the selected mask [red disk in (a)] for phase shift evaluation. The extracted phase shift is 0.48π rad in this case. (d), (e), and (f) Retrieved phase maps using the proposed method, the AIA method (13 frames), and their difference, respectively. Colorbar unit: rad.
In all the experiments presented above, the parameters used in the DE algorithm for solution search are as follows: population size NP = 15D, where D is the dimensionality and equals 11 in this study; population crossover rate CR = 1; scaling factor F = 0.3[1 + rand(0, 1)], where rand(0, 1) is a random number uniformly distributed in [0, 1].
Although all sub-aperture fringe patterns selected above for phase shift evaluation have more-than-one fringes, they principally need not be so. In other words, they can have less-than-one fringes. In this case, the phase of the sub-aperture fringe patterns could be even better approximated by low-order polynomials, and the phase shift can be robustly evaluated through global optimization. This may turn out to be a distinct advantage of the proposed algorithm.
Digital filtering plays an important role in phase retrieval from two-shot fringe patterns. In this study, although a Quiroga normalization algorithm [38] is adopted for sub-aperture fringe patterns processing, only high-pass Gaussian filtering is employed for overall background removal of two-shot fringe patterns. Using advanced filtering strategies, such as Hilbert-Huang adaptive filtering [32, 46], empirical mode decomposition based adaptive fringe pattern enhancement [8], can substantially improve the phase retrieval accuracy and yield better outcomes.
4. Conclusion
We proposed an effective algorithm for simultaneous phase map and phase shift retrieval in two-shot PSI. The implementation of the algorithm includes two steps. First mask the fringe patterns and evaluate the unknown phase shift using up to third-order Zernike polynomials and the DE optimization algorithm. Then compute full-field phase map based on the extracted phase shift. The mask is preferably placed on linear or circular fringes with a moderate size (10-30% of the whole image size) by balancing local noise, phase fitting error, and computation time. Computer simulation and experimental results suggest that the proposed algorithm is at least two-order of magnitude faster than the parametric method and is superior to the Kreis and OF methods and comparable with the GS algorithm in terms of phase retrieval accuracy. The algorithm could be extended to deal with two-frame fringe patterns with unknown, linearly-varying phase shift [36] and holds potential in a range of practical applications, such as optical shop testing and fringe projection profilometry. A complete software package in MATLAB is available at http://two-shot.sourceforge.io/.
Funding
National Natural Science Foundation of China (NSFC) under (60877043 and 61575061).
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