Phase-shifting algorithms with known and unknown phase shifts: comparison and hybrid

Yuchi Chen; Qian Kemao

doi:10.1364/OE.452583

1. Introduction

Optical metrology plays an important role in precision engineering [1,2]. Among various techniques of optical metrology, phase-shifting interferometry [3,4] is known for its capability of full-field, non-contact, and high accuracy measurement, and has been intensively studied and widely used in both industry and academia for more than half a century [5–9]. In phase-shifting interferometry, the phase is directly related to the measured quantities such as shape [10], deformation [11], and refractive index [12]. A proper phase-shifting algorithm (PSA) that accurately extracts the phase from phase-shifted fringe patterns is essential, especially when one or multiple of the following common error sources present [6]: (i) the phase-shift errors caused by mis-calibration or imperfection of the phase-shifter [13], (ii) the intensity harmonics caused by nonlinearity of cameras and multi-beam reflections [14,15], (iii) the non-uniform phase-shift distributions caused by vibrations or air turbulence [16,17], and (iv) the random additive intensity noise caused by components of the optical system especially the detectors [18]. Although such error sources can be directly suppressed by high-quality components and/or optimized systems [8], choosing a PSA that is insensitive to these error sources is often a preferred solution. In fact, a big pool of PSAs with a wide variety has been developed [6–9].

The majority of PSAs adopt known phase shifts, extract the phase using trigonometric identities [5–9], and are abbreviated as kPSAs for convenience. They are simple and widely used. The digital Fourier transform algorithms (DFT) [4], the least squares algorithms (LSA) [19,20], the averaging algorithms (AVE) [21–23], the (N+1)-bucket algorithms (N+1) [24,25], the windowed digital Fourier transform algorithms (WDFT) [26] and the generic error-compensating algorithms (GEC) [27] are such examples.

The other PSAs, which are abbreviated as uPSAs, adopt unknown phase shifts and thus are naturally immune to phase-shift errors. They can be further divided into three groups:

(i) The first group uses trigonometric identities for derivation, with the help of a regular phase shifts, e.g., the Carré’s algorithm (Carré) [3] and the Stoilov’s algorithm [28], or the statistical properties of fringe patterns, e.g., the Lissajous figure method [29,30], the phase statistics method [31] and the Euclidean matrix norm method [32]. Note that the forms of Carré and the Stoilov’s algorithm are very similar to the kPSAs, and thus often evaluated together with kPSAs.
(ii) The second group adopts an optimization approach. The Okada’s algorithm [33] uses an alternative linear least squares fitting framework to iteratively estimate the phase and the phase shifts. The advanced iterative algorithm (AIA) [34] improves the Okada’s algorithm to make the phase extraction simple, robust and accurate. The Xu’s algorithm [35], the Hoang’s algorithms [36], the phase-tilt iteration algorithm [37] and the general iterative algorithm (GIA) [38] have been developed to further improve the AIA, by considering various error sources in the framework. With a similar approach, the model-based phase-shifting interferometry has been implemented in industrial applications [39].
(iii) The third group decomposes the fringe patterns into different components through linear algebra manipulations, such as the principal component analysis (PCA) [40], the Gram–Schmidt orthonormalization [41], the hyper-accurate ellipse fitting in subspace [42] and the VU factorization (VU) [43].

With a large number of PSAs available, examining them becomes critical for proper algorithm selection and performance boosting, and has been an important research topic. There are three main approaches on the evaluation of kPSAs. In the first approach, the performances of some selected PSAs with respect to various error sources are directly compared through simulation, including Creath’s early study [5] and Buytaert el al.’s comprehensive comparison of 84 algorithms [7], among others [5–9]. In the second approach, Freischlad and Koliopoulos proposed the Fourier description method to evaluate the immunities of kPSAs to the phase-shift errors, the intensity harmonics and the random noise [44]. The Fourier description method was further generalized by Frequency Transfer Function method [45–47] which can be used to evaluate the immunity to the phase-shift error, the intensity harmonics and the random additive intensity noise. In the third approach, the kPSA’s analysis and design are seamlessly combined through Larkin et al.’s [25] and de Groot’s [48] frequency domain evaluation, Surrel’s characteristic polynomials [26,49] and Hibino et al.’s equation solving [27,50–52], making the systematic algorithm design of error-compensating algorithms possible. Meanwhile, Servin et al. proposed a systemic approach to design kPSAs with irregular phase shifts by using the Frequency Transfer Function [53]. As for the uPSA, the simulation examination on the AIA’s performance [54], the theoretical analysis on the PCA [55] and the modeling and optimization framework of the GIA [38] can be seen as examples of the above three evaluation approaches, respectively. However, such works are much fewer than those for the kPSA.

Therefore, while there is sufficient evaluation within kPSAs, there is an obvious lack of comprehensive evaluations and comparisons within uPSAs and between uPSAs and kPSAs. More specifically, the following aspects are not clear and need to be answered: (i) Are there any restrictions on the fringe patterns when we use the uPSAs? (ii) Which PSA has the best performance when the fringe patterns are with errors? and (iii) Is it possible to further improve the PSAs by combining the merits of the kPSAs and the uPSAs?

To answer these questions, we conduct a detailed examination of PSAs including the widely used kPSAs and the more recent uPSAs. For kPSAs, we select twelve algorithms with 3 to 15 steps [4,21,22,24,26,27,50,56,57] by considering their performances and representativeness, with their details given in Sec. 2.1. For the uPSAs, we select Carré [3] due to its simplicity, the AIA [34] and the PCA [40] due to their fundamental ideas, the GIA [38] and the VU [43] due to their robustness and outstanding accuracy. The following contributions are made:

(i) A detailed simulation evaluation to clarify the pre-requisites of the uPSAs on fringe pattern parameters is carried out;
(ii) Comprehensive comparisons within uPSAs and also between the uPSAs and the kPSAs are given, both for the first time, to provide a clear and big picture of their advantages and disadvantages;
(iii) A hybrid kPSA-GIA which use the kPSAs to initialize the GIA, and use the GIA to extract the phase is proposed.

The rest of the paper is arranged as follows. The selected PSAs in different categories are introduced in Sec. 2. The pre-requisites of the uPSAs are studied in Sec. 3, based on which, comprehensive comparisons within the uPSAs and between the uPSAs and the kPSAs are given in Sec. 4 and Sec. 5, respectively. The hybrid kPSA-GIA is proposed and validated in Sec. 6. An experiment is carried out in Sec. 7. The conclusions are drawn in Sec. 8. As many acronyms are used in this paper, they are listed in Table 1 for easy reference.

Table 1. List of acronyms.

View Table | View all tables in this article

2. Selected PSAs

In this section, the PSAs selected for analysis and comparison are briefly introduced.

2.1 kPSAs

The fringe patterns used in PSAs are generally modeled as [6]

(1)$$I({x,y;i} )= A({x,y} )+ B({x,y} )\cos [{\varphi ({x,y} )+ \delta ({x,y;i} )} ],$$

where (x,y;i) represents the spatial-temporal coordinates with x = 0,1,…,N_x−1 and y = 0,1,…,N_y−1 corresponding to pixel location index and i = 0,1,…,F−1 corresponding to frame index; N_x is the total number of rows; N_y is the total number of columns; F is the total number of frames; I is the fringe intensity; A and B are the background intensity and the fringe amplitude, respectively; φ is the phase information; and δ is a phase shift which can be introduced actively or passively.

In the kPSAs, the pixel-independent phase shifts are actively introduced as

(2)$$\delta ({x,y;i} )= \delta (i )= ({i - \varGamma } )\omega ,$$

where ω is the phase-shift increment, i.e. the phase shift difference between any two consecutive frames, and Γ is the parameter which controls the phase origin, i.e. there is no shift of phase at the Γ-th frame. We call this special frame as the frame of the phase origin [9]. Most kPSAs use the first frame, i.e., Γ =0, or the middle frame, i.e., Γ=(F-1)/2 as the frame of phase origin. Based on trigonometric identities, the phase is extracted by a weighted combination of the phase-shifted fringe patterns as [9]

(3)$$\varphi ({x,y} )= {\tan ^{ - 1}}\left[ {\frac{{S({x,y} )}}{{C({x,y} )}}} \right] = {\tan ^{ - 1}}\left[ {\frac{{\sum\nolimits_{i = 0}^{F - 1} {s(i )I({x,y;i} )} }}{{\sum\nolimits_{i = 0}^{F - 1} {c(i )I({x,y;i} )} }}} \right],$$

where S and C are the numerator and denominator of the arctangent function, respectively; and s(i) and c(i) are the weights for the i-th frame. As can be seen, the kPSAs are pixel-wise and has the flexibility to work with different fringe densities, different background intensities and different fringe amplitudes. Thus, its only pre-requisite on fringe pattern parameter is that the phase shifts should be known. Since the kPSAs have been well developed [26,27,44–46,49–52] and intensively evaluated [5–9], only the following typical kPSAs are selected as representatives for a benchmarking purpose:

(i) The classic kPSAs with small step numbers (F = 3∼5) which are convenient to use. We select the 3-step and the 4-step algorithms by Bruning et al. [4] and the 5-step algorithm by Schwider et al. [21]. These kPSAs are listed as the first three algorithms in Table 2.
(ii) The popular families of the kPSAs specified with a median step number of F = 7. The noticeable families included the DFT [4], the AVE [22], the N+1 [24], the WDFT [26] and the GEC [27]. Since they are specified at the same frame number, they can be fairly compared to showcase the interesting variety and to highlight the importance of the algorithm selection in phase-shifting interferometry. These kPSAs are listed as the next five algorithms in Table 2.
(iii) The kPSAs with good performances and large step numbers (F = 9∼15). We select the 9-step algorithm by Hibino et al. [27] which has a good immunity to phase-shift non-uniformities, the 10-step algorithm by Surrel [26] and the 11-step algorithm by Hibino et al. [50,56] (with a minus sign as indicated by [26]) which has a good immunity to both up to the forth order intensity harmonics and the linear phase-shift errors, and the 15-step algorithm by De Groot et al. [57] which has a good immunity to even order intensity harmonics and random noise. These kPSAs are listed as the last four algorithms in Table 2.

Table 2. Selected kPSAs for evaluation.

View Table | View all tables in this article

The Surrel’s characteristic polynomial plots of these kPSAs which are computed based on [26] are also shown in the last column of Table 2. These plots show the roots of the characteristic polynomials as illustrated by circles and can be used to quickly understand the kPSAs’ error immunity. The two most useful conclusions of the characteristic polynomial plots are: (i) when there is no phase-shift error, a kPSA is insensitive to an intensity harmonic of k-th order (k > 1) if there are roots at both k and 2π/ω-k at the unit circle; and (ii) when there is a linear phase-shift error, a kPSA is insensitive to an intensity harmonic of k-th order (k > 1) if there are multiple roots at both k and 2π/ω-k, and more roots mean better immunity. In both cases, when k = 1, only 2π/ω-1 needs to be considered. For example, for the 5-step algorithm in Table 2, ω=π/2 and 2π/ω=4. This algorithm is insensitive to the second order harmonic (k = 2) when there is no phase-shift error, because there is a root at k = 2 and 2π/ω-k = 4-2 = 2. It is also insensitive to a linear phase-shift error since there are two roots at 2π/ω-1 = 4-1 = 3. Besides, seven kPSAs in Table 2 are analyzed in [49], while the other five are either from other famous families (the DFT, the AVE and the GEC) or newly proposed (9-step and 15-step).

A special PSA worth mentioning is Carré [3]. The fringe model for this algorithm follows Eqs. (1) and (2) with F = 4, Γ=1.5 and an unknown ω. By using triangular identities and assuming ω∈[0, π], it has been derived that [58]

(4)$$\varphi \left( {x,y} \right) = {\tan ^{ - 1}}\left\{{\begin{matrix} {{\begin{matrix} \frac{\sqrt{{\begin{array}{l} \left\{ {3\left[ {I\left( {x,y;2} \right) - I\left( {x,y;3} \right)} \right] - \left[ {I\left( {x,y;1} \right) - I\left( {x,y;4} \right)} \right]} \right\}\\ \times \left[ {I\left( {x,y;2} \right) - I\left( {x,y;3} \right)} \right] + \left[ {I\left( {x,y;1} \right) - I\left( {x,y;4} \right)} \right] \end{array}} }}{{\left[ {I\left( {x,y;2} \right) + I\left( {x,y;3} \right)} \right] - \left[ {I\left( {x,y;1} \right) + I\left( {x,y;4} \right)} \right]}}\end{matrix}}} \end{matrix}} \right\}.$$

In this algorithm, the best phase increment is ω=1.92 rads (110°) [59]. Nevertheless, we set ω=π/2 rad, which is most often used. Noting that, since the phase shifts are unknown, Carré is a uPSA. However, in the rest of this paper, Carré is treated as one of the kPSAs for convenience due to the following reasons: (i) Comparing Eq. (4) and Eq. (3) shows that the form of Carré is similar to the other kPSAs; (ii) Furthermore, although the phase-shift increment ω is unknown, it remains constant; and (iii) Finally, Carré is a 4-step algorithm, unlike other uPSAs which can be applied to any sufficient numbers of frames.

2.2 uPSAs

Although less used, the uPSAs emerge as good competitors of the kPSAs. In this work, we select the AIA, the GIA, the PCA and the VU as representatives for evaluation.

2.2.1 AIA [34]

The AIA uses the fringe pattern model shown in Eq. (1) with unknown pixel-independent phase shifts, i.e. δ(x,y;i)=δ(i). In the AIA, we set the frame of the phase origin at the first frame, i.e. δ(0) = 0. The AIA iteratively estimates the phase φ and the phase shifts δ, until a convergence condition is satisfied.

The first step of the AIA estimates the phase in a pixel-wise manner. By treating the phase shifts as known and assuming that both the background intensity and the fringe amplitude are constant across frames, the fringe pattern model can be rewritten as

(5)$$I({x,y;i} )= a({x,y} )+ b({x,y} )\cos [{\delta (i )} ]+ c({x,y} )\sin [{\delta (i )} ],$$

where a(x,y)=A(x,y); b(x,y)=B(x,y)cos[φ(x,y)] and c(x,y) = −B(x,y)sin[φ(x,y)]. For each pixel, the unknowns a(x,y), b(x,y) and c(x,y) could be obtained from multiple linear equations with different index i and can be easily solved by linear least squares fitting with an inversion of a 3×3 matrix. Afterwards, the phase φ(x,y) can be calculated from b(x,y) and c(x,y).

The second step estimates the phase shifts in a frame-wise manner. By treating the phase as known and assuming that both the background intensity and the fringe amplitude are constant across pixels in each frame, the fringe pattern model can be rewritten as

(6)$$I({x,y;i} )= a^{\prime}(i )+ b^{\prime}(i )\cos [{\varphi ({x,y} )} ]+ c^{\prime}(i )\sin [{\varphi ({x,y} )} ],$$

where a′(i)=A(i); b′(i)=B(i)cos[δ(i)] and c′(i) = −B(i)sin[δ(i)]. Again, the unknowns a′(i), b′(i) and c′(i) can be estimated by linear least squares fitting. Then, the phase shift of the i-th frame δ(i) can be calculated from the b′(i) and c′(i).

The AIA initializes the phase shifts based on prior knowledge or as random values casted within [0, 2π]. The stopping criteria is either the absolute phase shift changes of all the frames are sufficiently small, i.e.,

(7)$$|{[{{\delta^m}(i )- {\delta^m}(0 )} ]- [{{\delta^{m - 1}}(i )- {\delta^{m - 1}}(0 )} ]} |< \varepsilon \textrm{,}$$

or the maximum iteration number is reached

(8)$$m > M\textrm{,}$$

where |·| takes the absolute value, ε is a pre-set error tolerance, the superscript m indicates the number of iterations, and M is the pre-set maximum iteration number.

The AIA requires at least half a fringe within a frame for convergence [54]. As the AIA assumes the uniformity of the background intensity and the fringe amplitude, an error will occur when this assumption is violated, but it is not seen to be significant [60].

2.2.2 GIA [38]

It is interesting to see that the AIA estimates the unknown phase shifts together with the phase, making the algorithm insensitive to the phase-shift error. However, the AIA is still sensitive to the other types of error sources. Hence, following a similar framework, the recently proposed GIA more ambitiously incorporates all the concerned error sources into the fringe model and estimates them together with the phase.

Based on their different natures, the unknowns in the GIA are classified into three groups: (i) the background intensity and the fringe amplitude, (ii) the phase and (iii) the phase-shifts related parameters. Subsequently, these three groups of unknowns are optimized alternatively by the Levenberg-Marquart method, which has demonstrated excellent convergence and accuracy. In addition, the background intensity and the fringe amplitude can be separately estimated within a small window with a window size of (2w+1)×(2w+1), so that they are no longer required to be constant in the entire image, yet robust to noise. A typical size is w = 5∼10 and we use w = 8 in this paper. The GIA can be conveniently initialized by the AIA or other PSAs. In later sections, we use the AIA for initialization thus need more than half a fringe within the frame. Meanwhile, we also notice that the GIA has a phase error when the background intensity and the fringe amplitude are discontinues which will be shown in Sec. 3.

2.2.3 PCA [40]

The PCA follows the AIA’s fringe model, except that there is no specification on the frame of the phase origin. The PCA first removes the background intensity by subtracting the temporal average intensity, i.e., $A({x,y} )\approx \mathop \sum \limits_{i = 0}^{F - 1} I({x,y;i} )/F$. Furthermore, if the following approximation is well satisfied,

(9)$$\sum\nolimits_{y = 0}^{{N_y} - 1} {\sum\nolimits_{x = 0}^{{N_x} - 1} {\{{B({x,y} )\cos [{\varphi ({x,y} )} ]} \}\{{B({x,y} )\sin [{\varphi ({x,y} )} ]} \}} } \approx 0,$$

then, viewing from the entire image, Bsin(φ) and Bcos(φ) are orthogonal to each other and can be considered as two most significant principal components of the background-intensity-removed fringe patterns [40]. Thus, the task of the phase extraction becomes the estimation of these two principal components, which can be easily achieved by PCA.

To do so, the background-intensity-removed fringe pattern intensities of the i-th frame are taken column-wisely to form a row vector with a length of N_x×N_y. Next, the F row vectors from the F fringe patterns are stacked to form an F×(N_x×N_y) matrix D whose entries are D_ij= I(x,y;i)-A(x,y) with j is correlated to the pixel location (x,y) through j = N_yx + y. Subsequently, the following covariance matrix C = D×D^T with a size of F×F can be diagonalized through singular value decomposition where the superscript T represents matrix transpose. Next, based on the Hotelling transform [61], the principal components of D can be extracted. The two principal components with the largest eigenvalues are corresponding to the sine and cosine terms of phase and can be used to extract the phase.

The PCA has two restrictions as indicated in its evaluation [55]: (i) There should be more than one fringe in the image so that the orthogonality assumption in Eq. (9) is valid; and (ii) the phase shifts must be well distributed in the [0 2π] interval range so that the temporal averaging well estimates the background intensity and the principle components can be well decomposed [55]. These two restrictions will be evaluated through simulations in Sec. 3. Besides, the PCA has a mean error problem since the frame of the phase origin cannot be specified.

2.2.4 VU [43]

The VU views fringe patterns in a similar way as the PCA, but the background intensity is not pre-removed. As such, the VU rewrites the fringe model with reshaped matrices as follows,

(10)$${\mathbf I} = {\mathbf V}{{\mathbf U}^T} = \left( {\begin{array}{ccc} {\mathbf a}&{\mathbf c}&{ - {\mathbf s}} \end{array}} \right)\left( {\begin{array}{c} {{{\mathbf 1}^T}}\\ {{{\mathbf u}^T}}\\ {{{\mathbf v}^T}} \end{array}} \right),$$

where the matrix V has a dimension of (N_x×N_y)×3; the three vectors a, c and s have respective entries a_j= A(j), c_j= B(j)cos[φ(j)] and s_j= B(j)sin[φ(j)]; j has the same definition as the one in the PCA; the matrix U has a dimension of 3×F; the vector 1 is a vector filled with ones; the vectors u and v have respective entries of u_i= cos[δ(i)] and v_i= sin[δ(i)].

However, the VU does not follow the PCA’s concept of principle components. Instead, it iteratively estimates two matrices U and V through pseudoinverse with the assumption that the other matrix is known, reminding us the framework of the AIA. The VU also needs the restrictions on fringe density which will be confirmed in Sec. 3.

3. Pre-requisites of the uPSAs

A significate merit of the uPSAs is that they can extract the phase from fringe patterns with random unknown phase shifts. However, this advantage is achieved at a cost. In fact, the uPSAs may result in significant phase errors due to specific fringe density, phase-shift distribution, background intensity or fringe amplitude. In contrast, the kPSAs’ performances are not affected by these factors. Thus, it is necessary to understand these pre-requisites of the uPSAs on fringe pattern parameters (Sec. 3.2) before we carry out detailed comparisons. The main approach for our PSA evaluation is through simulation, which has been widely used and well-recognized. Details are given (Sec. 3.1) as the preparation for the rest of the paper.

3.1. Codes, fringe patterns and error measures

The implementation of the kPSAs is according to Eq. (3) and the weight parameters in Table 2, which is straightforward. Carré is implemented following Eq. (4) and Refs. [3,58]. The AIA and the GIA are coded following Refs. [34] and [38], respectively. The codes of the PCA and the VU have been made available by the respective original authors [40,43], which are directly adopted in this paper. The settings of the parameters in the AIA, the GIA and the VU follow Refs. [34,38,43].

Due to the complexity of our evaluation, we start with the following default specifications on the fringe patterns as a common ground: (i) The frame number is set as eight; (ii) The image size is set as N_x = N_y = 1024; (iii) The background intensity and the fringe amplitude are set as A = 0.5 and B = 0.4; (iv) The phase is simulated as

(11)$$\varphi ({x,y} )= {\omega _p} \times \textrm{peaks}_\textrm{s}({{N_x},{N_y}} )+ {\omega _x}x + {\omega _y}y,$$

where ω_p= 10, ω_x= 0 and ω_y= 0.25, peaks_s uses the MATLAB function “peaks_s” and linearly scales the value into the range of [0, 2π]; (v) The phase shifts are randomly set as [38]

(12)$${\delta _0} = \left[ {\begin{array}{cccccccc} {0}&{0.440}&{0.795}&{1.246}&{1.601}&{2.665}&{2.695}&{3.449} \end{array}} \right].$$

The phase map and the first fringe pattern of the common ground are shown in Fig. 1.

Fig. 1. The simulation parameters of the common ground. (a) The phase; (b) the first fringe pattern.

Aberrations	Explanation	Aberrations	Explanation
AIA	Advanced iterative algorithm	N+1	(N+1)-bucket algorithm
AVE	Averaging algorithm	PCA	Phase-shifting algorithm based on principal component analysis
Carré	Carré’s algorithm	PSA	Phase-shifting algorithm
DFT	Digital Fourier transform algorithm	PVE	Peak and valley of a phase error
GEC	Generic error-compensating algorithm	STDE	Standard deviation value of a phase error
GIA	General iterative algorithm	STD	Standard deviation
kPSA	Phase-shifting algorithm with known phase shifts	uPSA	Phase-shifting algorithm with unknown phase shifts
LSA	Least squares algorithm	VU	Phase-shifting algorithm based on VU factorization
ME	Mean value of a phase error	WDFT	Windowed digital Fourier transform algorithm

Name	F	ω (rad)	Γ	Parameter s and c
3-step [4]	3	2π/3	1	$\frac{\sqrt{3} \times [1, 0, - 1]}{1, 2, 1}$
4-step [4]	4	π/2	0	$\frac{0, 1, 0, 1}{1, 0, 1, 0}$
5-step [21]	5	π/2	0	$\frac{0, - 2, 0, 2, 0}{1, 0, 2, 0, 1}$
DFT [4]	7	2π/7	0	$\frac{sin (i \times \frac{2 π}{7}) (i = 0 - 6)}{cos (i \times \frac{2 π}{7}) (i = 0 - 6)}$
AVE [22]	7	π/2	0	$\frac{1, 6, 5, 20, 5, 6, 1}{1, 4, 15, 0, 15, 4, 1}$
N+1 [24]	7	π/3	0	$\frac{1, 3, 3, 0, 3, 3, 1}{\sqrt{3} \times [1, 1, - 1, - 2, - 1, 1, 1]}$
WDFT [26]	7	π/2	3	$\frac{1, 0, 3, 0, 3, 0, 1}{0, 2, 0, 4, 0, 2, 0}$
GEC [27]	7	2π/5	3	$\frac{0.263, 0.325, 0.951, 0, 0.951, 0.325, 0.263}{0.0854, 0.724, 0.309, 1, 0.309, 0.724, 0.0854}$
9-step [27]	9	π/2	4	$\frac{- 1, 2, 14, 18, 0, - 18, - 14, - 2, 1}{- 2, - 8, - 8, 8, 20, 8, - 8, - 8, - 2}$
10-step [26]	10	π/3	0	$\frac{\sqrt{3} \times [- 1, - 3, - 3, 1, 6, 6, 1, - 3, - 3, - 1]}{1, 1, 7, 11, 6, 6, 11, 7, 1, 1}$
11-step [50,56]	11	π/3	5	$\frac{- \sqrt{3} \times [0, - 1, - 4, - 7, - 6, 0, 6, 7, 4, 1, 0]}{- 2, - 5, - 6, - 1, 8, 12, 8, - 1, - 6, - 5, - 2}$
15-step [57]	15	π/2	7	$\frac{1, 0, 9, 0, 21, 0, 29, 0, 29, 0, 21, 0, 9, 0, 1}{0, 4, 0, 15, 0, 26, 0, 30, 0, 26, 0, 15, 0, 4, 0}$

Fringe parameters		AIA	GIA	PCA	VU
Phase	Fringe density	≥0.5	≥0.5 (initialized by AIA);	≥2	≥0.5
Phase	Fringe density	≥0.5	Free (initialized by the kPSAs)	≥2	≥0.5
Background intensity and fringe amplitudes	Non-uniformities			Slightly sensitive	Slightly sensitive
	Discontinuities		Sensitive (in general); Insensitive (if a 1×1 window is used or fringe patterns are segmented)

	AVE	WDFT	AIA	GIA	PCA	VU	AVE-GIA	WDFT-GIA
STD	0.098	0.087	0.060	0.054	0.060	0.065	0.054	0.054
Time	0.083	0.079	14.84	713.12	0.52	1.13	1293.91	1392.74

Abstract

1. Introduction

2. Selected PSAs

2.1 kPSAs

2.2 uPSAs

2.2.1 AIA [34]

2.2.2 GIA [38]

2.2.3 PCA [40]

2.2.4 VU [43]

3. Pre-requisites of the uPSAs

3.1. Codes, fringe patterns and error measures

3.2 Pre-requisites of the uPSAs on fringe pattern parameters

3.2.1 On the phase

3.2.2 On the phase-shift distribution

3.2.3 On the background intensity and the fringe amplitude

3.2.4 Summary

4. Within the uPSAs: the performances with unknown phase shifts

4.1 Phase-shift errors

4.2 Intensity harmonics

4.3 Phase-shift non-uniformities

4.4 Random noise

4.5 More tests and the summary

5. Between the uPSAs and the kPSAs: the performance with known phase shifts

5.1 Phase-shift errors

5.2 Intensity harmonics

5.3 Phase-shift non-uniformities

5.4 Random noise

5.5 More tests and the summary

6. Hybrid kPSA-GIA

7. Experiments evaluation and the speed

8. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (30)

Tables (4)

Equations (26)

Optics Express