Self-calibration phase-shifting algorithm with interferograms containing very few fringes based on Fourier domain estimation

Yi Wang; Xiaoxu Lu; Yicheng Liu; Jindong Tian; Liyun Zhong

doi:10.1364/OE.25.029971

1. Introduction

Optical interferometry is a powerful non-contact tool to quantitatively and accurately exhibit the property of micro or macro object [1]. Generally speaking, for different measuring requirements, there are two most important solutions: the off-axis configuration [2–4] for dynamic analysis and vibrational robustness, the on-axis configuration [5, 6] for full spatial bandwidth utilization and high accuracy measurement. And they choose different phase encoding methods: the temporal coding is utilized in on-axis configuration while the spatial coding is employed in off-axis configuration. The accuracy advantage of temporal coding method compared with spatial coding method comes from the fact that the temporal phase stability of interferograms is far better than the spatial phase stability of interferograms, because the former can be guaranteed by the experimental equipment but the latter is usually broken due to the large phase gradient of tested object. From another perspective, the advantages of temporal coding method also can be explained as the sufficient utilization of CCD bandwidth. Among on-axis phase retrieval methods [5, 7, 8], although the phase-shifting interferometry (PSI) has significant advantage in convenience due to its linear mathematical property, an obvious error named as the miscalibration error usually appears.

The miscalibration error, which is induced by the incorrect estimation of phase shifts [9], usually leads to a serious rippling oscillation distortion into the retrieved phase. As we know, the initial PSI algorithms are based on a simple model, in which the phase shifts are known [10] or linearly distributed in the integral period of $2 π$ [11]. However, in the practical application, the above assumption cannot be easily satisfied, especially in the case that only a small amount of phase-shifting interferograms are captured. To address this, the self-calibration algorithm, which can help us accurately estimate the unknown phase shifts, is proposed. Unfortunately, in the current self-calibration algorithms, though the phase retrieval can be achieved without the prior knowledge of phase shifts, their accuracy is closely related with the fringe number in interferograms [12–22], and this will greatly limit their applications. Actually, the ultra-sparse fringe patterns (USFP), in which the fringe number in interferograms is less than one, usually appear in some modern interference techniques. For example, in white light phase-shifting interferometry [23, 24], the fringe number in interference patterns is limited by the coherence length of white light; and in the quadratic phase eliminated phase-shifting microscopic system [25], the fringe number is reduced by the objective matching to achieve sufficient utilization of spatial bandwidth of CCD, thus the more detail of tested object can be reserved. In our previous work [26], though the solution for the fringe pattern with non-uniform distribution is proposed, but it still cannot work well for USFPs.

In this study, we propose a novel solution to perform retrieval phase of USFPs through using mid-band spatial spectrum matching (MSSM) algorithm, in which only three-frame phase-shifting interferograms with unknown phase shifts are enough. Specially, although the proposed MSSM algorithm is conceived as a solution for USFP, it also reveals high accuracy for the fringe patterns with many fringes. What’s more, the proposed algorithm has rapid speed because the iterative procedure is not needed. Following, we will introduce the proposed MSSM algorithm in detail.

2. Principle

In PSI, the intensity of the nth-frame phase-shifting interferograms can be described as

I_{n} (x, y) = A (x, y) + B (x, y) \cos [ϕ (x, y) + δ_{n}] + o_{n} (x, y) n = 1...... N .

where x and y denote the transverse and longitude coordinates, respectively; A(x, y) and B(x, y) represent the background intensity and modulation amplitude, respectively;

ϕ (x, y)

is the encoded phase distribution ;

δ_{n}

represents the phase shifts, in which

δ_{1} = 0

, and n = 1,2,3……N are the index number of a sequence of phase-shifting interferograms;

o_{n} (x, y)

represents the noise.

Actually, the phase demodulation procedure from N-frame phase-shifting interferograms is to solve Eqs. (1). However, when $N \leq 3$ , the total variable number in Eqs. (1) will exceed the equations number, then the self-calibration algorithm is introduced to solve Eqs. (1). Apparently, a reasonable approximation is needed to increase the number of known parameters in Eqs. (1) . But in the USFP, the widely adopted assumption, corresponding to the statistical average property of phase distribution, will become invalid [13, 16, 18], and another important approximation, the spatial constant of A(x, y) and B(x, y) [12, 17], are not easy to be satisfied in practical application. Specially, in current self-calibration algorithms, though the advanced iterative algorithm (AIA) [17] reveals relative robust in adapting the nonconstant of A(x, y) and B(x, y), but its accuracy stability against the spatial variation of A(x, y) and B(x, y) still requires many fringes in interferograms. Besides that, the recently developed ellipse fitting algorithm is also treated as a promising solution [15, 19, 21], but it is in fact a non-iterative compromised AIA solution for saving time because it’s fitting strategy is similar with the second step of AIA, which will make the global fitting of constant-in-assumption. What’s more, the number increasing of parameters in these fitting algorithms, i.e., 6-parameter in ellipse fitting algorithm but only 2-parameter in AIA, will lead to the accuracy decreasing [27]. Further, for the USFP with nonconstant A(x, y) and B(x, y), the above algorithms cannot work well.

Clearly, to achieve the accurate phase in USFP, it is needed to search for a more reasonable approximation. As we know, before Zernike invented phase contrast microscope in 1935 [28], the microscopic phase information cannot be directly observed. Actually, the phase contrast is a phenomenon induced by the self-interference, reflecting that A(x, y) and B(x, y) in interference pattern might be non-constant, but the spatial intensity fluctuation in interference pattern is mainly induced by the interference term. Fourier spectrum analysis is an effective tool to study the degree of intensity fluctuation. When the interference signal is generated, more high frequency components will appear in interference pattern. By performing Fourier transform (FT) operation $ζ$ to both sides of Eqs. (1), and if N = 3, we have that

\begin{array}{l} I_{1} (f_{x}) = A (f_{x}) + ζ [B (x) \cos ϕ (x)] + o_{1} (f_{x}) \\ = A (f_{x}) + T_{1} (f_{x}) + o_{1} (f_{x}) . \end{array}

\begin{array}{l} I_{2} (f_{x}) = A (f_{x}) + ζ {B (x) \cos [ϕ (x) + δ_{1}]} + o_{2} (f_{x}) \\ = A (f_{x}) + \cos δ_{1} T_{1} (f_{x}) - \sin δ_{1} T_{2} (f_{x}) + o_{2} (f_{x}) . \end{array}

\begin{array}{l} I_{3} (f_{x}) = A (f_{x}) + ζ {B (x) \cos [ϕ (x) + δ_{2}]} + o_{3} (f_{x}) \\ = A (f_{x}) + \cos δ_{2} T_{1} (f_{x}) - \sin δ_{2} T_{2} (f_{x}) + o_{3} (f_{x}) . \end{array}

Here, for simplicity, we have neglected the y dependence, in which

A (f_{x}), T_{1} (f_{x}), T_{2} (f_{x})

and

o_{i} (f_{x})

denote the spatial frequency spectra of

A (x), B (x) \cos ϕ (x), B (x) \sin ϕ (x)

and

o_{i} (x)

, respectively. As shown in Fig. 1, the whole spatial frequency spectra can be divided into three components: low-frequency, mid-frequency and high-frequency bands, which are mainly occupied by

A (f_{x})

,

T_{1} (f_{x})

and

T_{2} (f_{x})

, respectively. If we intercept only the mid-frequency component, the information of

A (f_{x})

and

o (f_{x})

can be suppressed, the variable number in Eqs. (1) will be reduced. And then we have that

{\tilde{I}}_{1} (f_{x}) \approx {\tilde{T}}_{1} (f_{x}) .

{\tilde{I}}_{2} (f_{x}) \approx {\tilde{T}}_{1} (f_{x}) ​ c o s δ_{1} - {\tilde{T}}_{2} (f_{x}) \sin δ_{1} .

{\tilde{I}}_{3} (f_{x}) \approx {\tilde{T}}_{1} (f_{x}) ​ \cos δ_{2} - {\tilde{T}}_{2} (f_{x}) \sin δ_{2} .

Here,

f_{x} \in (f_{2}^{-}, f_{1}^{-}), (f_{1}^{+}, f_{2}^{+})

, where

f_{2}^{-}, f_{1}^{-}

and

f_{1}^{+}, f_{2}^{+}

denote the intercepted mid-band spatial spectral range in the positive and negative frequency domains, respectively. And

{\tilde{T}}_{1} (f_{x}), {\tilde{T}}_{2} (f_{x})

represents the intercepted component of

T_{1} (f_{x})

and

T_{2} (f_{x})

, respectively.

Fig. 1 Schematic of the spectral division, in which the background spectrum $A (f_{x})$ , noise spectrum $o (f_{x})$ and interference term $T_{1} (f_{x}), T_{2} (f_{x})$ are located in the low-frequency band, high-frequency band and mid-frequency band, respectively.

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Note that Eqs. (3) represent three different combination modes of $T_{1} (f_{x})$ and $T_{2} (f_{x})$ . By using the least-square fitting algorithm, we can achieve C₁ and C₂ by

C_{1} {\tilde{I}}_{2} (f_{x}) + C_{2} {\tilde{I}}_{3} (f_{x}) = {\tilde{I}}_{1} (f_{x}) .

After

C_{1}, C_{2}

are determined, from Eqs. (3), we have that

C_{1} \cos δ_{1} + C_{2} \cos δ_{2} = 1.

C_{1} \sin δ_{1} = - C_{2} \sin δ_{2} .

Thus,

δ_{1}

and

δ_{2}

can be achieved from Eqs. (5). Subsequently,

ϕ (x)

also can be determined by the temporal least square fitting algorithm as

(\begin{matrix} A (x, y) \\ B (x, y) \cos ϕ (x, y) \\ B (x, y) \sin ϕ (x, y) \end{matrix}) = {(\begin{matrix} 1 & \cos δ_{1} & - \sin δ_{1} \\ 1 & \cos δ_{2} & - \sin δ_{2} \\ 1 & \cos δ_{3} & - \sin δ_{3} \end{matrix})}^{- 1} (\begin{matrix} I_{1} \\ I_{2} \\ I_{3} \end{matrix}) .

ϕ (x, y) = arc \tan [\frac{B (x, y) \sin ϕ (x, y)}{B (x, y) \cos ϕ (x, y)}] .

Note that the high-pass component of the filtering function plays an important role in the proposed MSSM algorithm. If the background

A (f_{x}, f_{y})

cannot be eliminated, then a residual term

\tilde{A} (f_{x}, f_{y})

will appear and the accuracy is decreased. For this reason, we define two parameters to describe the error level as

l_{1} = \frac{\sum_{f_{x}, f_{y}} a b s [\tilde{A} (f_{x}, f_{y})]}{\sum_{f_{x}, f_{y}} a b s [{\tilde{T}}_{1} (f_{x}, f_{y})]} .

l_{2} = \frac{\sum_{f_{x}, f_{y}} a b s [\tilde{A} (f_{x}, f_{y})]}{\sum_{f_{x}, f_{y}} a b s [{\tilde{T}}_{2} (f_{x}, f_{y})]} .

Here, l₁ and l₂ represent the spectrum energy ratio between the background residual term

\tilde{A} (f_{x}, f_{y})

and the filtering signal term

{\tilde{T}}_{1} (f_{x}, f_{y}), {\tilde{T}}_{2} (f_{x}, f_{y})

, respectively. If

l_{1}

and

l_{2}

is small enough, an accurate result can be achieved.

It is also benifitial to notice that the empirical mode decomposition method [29–31] has been recently applied to interferograms analysis to achieve accurate background removal, in which the fringe-pattern is adaptively decomposed into several components, i.e., IMFs (instrinsic mode functions), corresponding to the background and noise terms, respectively. And it is found that this empirical mode decomposition reveals high accuracy relative to Gaussian filtering based background removal method in many practical situations. As consequence, the proposed MSSM algorithm will present further accuracy improvement through combining with the empirical mode decomposition method.

3. Simulation

Following, we perform the quantitative calculation to verify the effectiveness of the proposed MSSM algorithm. Three-frame simulated fringe-patterns with the phase shifts of 0rad, 1.2rad, and 2.6rad are generated, and the corresponding size are 500 × 500 with pixel interval of 0.01mm. The background and modulation amplitude are set as A(x, y) = 80exp[-0.05(x² + y²)] + 80 and B(x, y) = 80exp[-0.05(x² + y²)], respectively; $ϕ_{1} = 0.03 π (x^{2} + y^{2})$ and $ϕ_{2} = π (x^{2} + y^{2})$ , corresponding to the encoded phase in the USFP and fringe-pattern with many fringes. The fitting band is intercepted by a band-pass spatial filter of $h (f_{x}, f_{y}) = h_{1} (f_{x}, f_{y}) h_{2} (f_{x}, f_{y})$ , in which $h_{1} (f_{x}, f_{y}) = \exp (- \frac{f_{x}^{2} + f_{y}^{2}}{2 σ_{1}^{2}}), h_{2} (f_{x}, f_{y}) = 1 - \exp (- \frac{f_{x}^{2} + f_{y}^{2}}{2 σ_{2}^{2}})$ ( $σ_{1} = 30, σ_{2} = 3$ ). Random noise with signal noise ratio (SNR) of 50db is added into the fringe patterns. For comparison, the result achieved with PCA and AIA algorithms are also presented. As shown in Fig. 2, from 3-frame fringe-patterns, it is found that the AIA or PCA algorithms can work only when A(x, y) and B(x, y) are quasi-constant and many fringes exist in fringe-patterns. In contrast, using the proposed MSSM algorithm, whether for the USFP or fringe-patterns with many fringes, the accurate phase can be achieved.

Fig. 2 (a1)(a2)Two-frame simulated interference patterns encoded with phase $ϕ_{1}$ and $ϕ_{2}$ , respectively; (b1)(b2) the theoretical phase distributions of $ϕ_{1}$ and $ϕ_{2}$ , respectively; (c1) (e1) (g1) the retrieved phases $ϕ_{1}$ with MSSM, PCA and AIA algorithms, respectively; (d1) (f1) (h1) the differences between (b1) and (c1), (e1), (g1), respectively; (c2) (e2) (g2) the retrieved phase $ϕ_{2}$ with MSSM, PCA and AIA algorithms, respectively; (d2) (f2) (h2) the differences between (b2) and (c2), (e2), (g2), respectively.

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To further present the advantage of the proposed MSSM algorithm, Fig. 3(a)-3(c) show the variation curves of root mean square errors (RMSEs) of phase retrieval with the parameters of $t, k$ and SNR, in which Matlab function $k \cdot p e a k s (500)$ is utilized as the phase model, A(x, y) = B(x, y) = 80exp[-0.01t(x² + y²)], $δ_{1} = l_{1}, δ_{2} = l_{2}$ and $S N R = p$ . Here, t is utilized to describe the uniformity of A(x, y) and B(x, y), k represents the fringe density, respectively. In the first curve, t is changed from 1 to 40 and k = 0.2, l₁ = 1.2 rad, l₂ = 2.6rad, p = 50db ; in the second curve, k is changed from 0.2 to 2.2 and t = 10, l₁ = 1.2 rad, l₂ = 2.6rad, p = 50db ; in the third curve, p is changed from 25db to 64db, and k = 0.2, t = 10, l₁ = 1.2 rad, l₂ = 2.6rad . For clarity, we mark a individual point of each curve and present the corresponding fringe pattern, as shown in Figs. 3(a)-3(c); and Figs. 3(d)-3(f) also give the corresponding 2-dimentional distribution variation of RMSEs with $δ_{1}$ and $δ_{2}$ achieved with different algorithms, in which k = 0.6, t = 10 and p = 40db. Here, the phase shifts are chosen as $1.3 r a d > δ_{1} > 0.3 r a d, 2.8 r a d > δ_{2} > 1.8 r a d$ to make sure that the error induced by the incorrect phase shifts estimation is not overwhelmed by the error induced by random noise. Obviously, we can find that the proposed MSSM algorithm exhibits several promising properties: First, the non-constant of A(x, y) and B(x, y) are allowed, as shown in Fig. 3(a), and the variation of RMSE of phase retrieval with MSSM algorithm is very small when A(x, y) and B(x, y) are greatly changed from 80exp[-0.01(x² + y²)] to 80exp[-0.4(x² + y²)]; second, the result is non-sensitive to the variation of fringe number in interference pattern, as shown in Fig. 3(b). When the phase model is changed from $0.2 \cdot p e a k s (500)$ to $2.2 \cdot p e a k s (500)$ , respectively corresponding to the USFP and dense fringe pattern, it is observed that the using the AIA and PCA algorithms, the RMSEs reveal large fluctuation while almost unchanged with MSSM algorithm. These results demonstrate that by using the proposed MSSM algorithm, the error induced by the unknown phase shifts, acting as a ripple phase component, has been effectively eliminated for both the USFPs and dense fringe pattern.

Fig. 3 Variation curves of RMSE with different parameters (a) t ; (b) k; (c) SNR; the 2-dimentional distribution variation of RMSEs with $δ_{1}, δ_{2}$ achieved by different algorithms (d) MSSM; (e) PCA; (f) AIA.

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Of course, in the proposed MSSM algorithm, the filtering window parameter of high-pass filter is related with the accuracy, but the key issue is that the accuracy cannot be too sensitive to this blind parameter. To prove the accuracy of phase retrieval with the proposed MSSM algorithm is not sensitive to the filtering window parameter (especially for USFP), a group of simulation result is shown, in which A(x, y) = 80exp[-0.05(x² + y²)] + 80 and B(x, y) = 80exp[-0.05(x² + y²)], $ϕ = 0.2 \cdot p e a k s (500)$ and the filtering function $h (f_{x}, f_{y}) = h_{1} (f_{x}, f_{y}) h_{2} (f_{x}, f_{y})$ , $h_{1} (f_{x}, f_{y}) = \exp (- \frac{f_{x}^{2} + f_{y}^{2}}{2 \cdot 30^{2}})$ , $h_{2} (f_{x}, f_{y}) = 1 - \exp (- \frac{f_{x}^{2} + f_{y}^{2}}{2 \cdot σ^{2}}) .$ The SNR of fringe-patterns is 50db and the phase shifts are 0rad, 1.2rad and 2.6rad, respectively. Figure 4 shows the variation curves of RMSEs and l₁, l ₂ with the filtering window $σ$ .

Fig. 4 Variation curves of (a) RMSE and (b) the error level (l₁ and l₂) with the filtering window $σ$ .

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We can see though the fringe-patterns have very few fringes and nonuniform background, by using a wide-range of filtering window, the residual background occupies only 10%-20% of total energy, and the RMSE of phase retrieval is very stable and small(0.01 to 0.015rad). These results demonstrate that the proposed MSSM method can work well even when the fringe-patterns have very few fringes and nonuniform background. In addition, Figs. 4(a) and 4(b) are strong correlative, further indicating that l₁ and l ₂ are suitable for describing the accuracy of proposed MSSM algorithm.

According to the spatial spectral analysis, a more obvious aspect of the advantage of the proposed method compared with AIA is presented. For the second step of AIA, when the $ϕ (x, y)$ is known, it is assumed that A(x, y) and B(x, y) are constants, and the phase shifts $δ_{i}$ , A₀, B₀, can be achieved by the spatial fitting, which is essentially equivalent to the spectral fitting, and the proposed MSSM algorithm also can perform the inverse transform and achieve parameters fitting in spatial domain. Actually, the background removal procedure is also existed in AIA during phase shifts determination, which is equivalent to a high-pass filter of $h (f_{x}, f_{y}) = 1 - \exp (- \frac{f_{x}^{2} + f_{y}^{2}}{2 \cdot σ^{2}})$ with $σ \to 0$ , suppressing only zero-frequency component, so the more residual values of A(x, y) will remain and l₁ and l₂ becomes larger than those of the proposed MSSM algorithm. In the case that A(x, y) is nonuniform, the accuracy of MSSM algorithm will sharply increase when the $σ$ becomes larger from $σ = 0$ . This result presents that the traditional constant assumption of A(x, y), which is adopted by many algorithms, including the AIA or Lissajous figure algorithm, should be amended. That is to say, to achieve the smaller l₁ and l₂, it should be reasonable to allow a slight extension of the above assumption in spatial spectrum. As a result, AIA or Lissajous figure algorithms might achieve an essential improvement in accuracy by this consideration.

4. Experiment

Experiments are also carried to exhibit the expansibility of the proposed MSSM algorithm in both USFPs and fringe-patterns with many fringes. First, 300-frame phase-shifting interferograms with circular phase distribution are captured and the reference phase is achieved by using temporal Fourier transform method, in which a large amount of interferograms can guarantee the accuracy of phase retrieval. A He-Ne stabilized frequency laser is utilized as light source, a CCD with size of 768 × 576 pixels and pixel size of 0.01mm is employed to capture interferograms. Second, a local area of interferogram with size of 350 × 350 pixels is intercepted to achieve phase retrieval, in which several fringes exist in interferogram while A(x, y) and B(x, y) are not constant. As shown in Fig. 5, we can see that using the MSSM algorithm, the rippling oscillation error can be more effectively eliminated relative to the PCA and AIA algorithms, no obvious miscalibration error appears in Fig. 5(d). And Table 1 gives the quantitative calculation result of phase retrieval.

Fig. 5 Experimental result of fringe-pattern with many fringes (a) one-frame fringe-pattern; (b) the intercepted calculation area from (a); (c) the reference phase distribution of (b); the retrieved phases and the corresponding differences between the reference phase and the achieved phases with different algorithms (d) (g) MSSM; (e) (h) PCA; (f) (i)AIA, respectively.

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Table 1. RMSE, PVE and calculation time of phase retrieval with different algorithms in Fig. 5

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Subsequently, we decrease the size of interferogram to achieve the USFP (80 × 80 pixels), as shown in Fig. 6 and Table 2. For the USFP, it is found that the proposed MSSM algorithm can achieve accurate circular phase distribution while the PCA and AIA algorithms cannot work well in this case. Note that the average B(x, y) in Fig. 6(b) is larger than that in Fig. 5(b), which will make the SNR of the former is better than the latter due to the dark noise of CCD. Correspondingly, using the MSSM algorithm, the RMSE of phase retrieval from Fig. 6 (0.024rad) is lower than that of Fig. 5 (0.037rad). Unfortunately, both PCA and AIA algorithms cannot benefit from this advantage due to the serious miscalibration error.

Fig. 6 Experimental results of USFP (a) One-frame interferogram; (b) the intercepted USFP from (a); (c) the reference phase map of (b); the retrieved phases and the corresponding differences between the reference phase and the achieved phases with different algorithms (d) (g) MSSM; (e) (h) PCA; (f) (i)AIA, respectively.

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Table 2. RMSE, PVE and calculation time of phase retrieval with different algorithms in Fig. 6

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The above results present that AIA and PCA algorithms can work well only for the interferograms containing many fringes while the MSSM algorithm is more robust. Further, the proposed MSSM algorithm is utilized to on-axis phase-shifting digital holographic microscopy system, as shown in Fig. 7, in which the phase aberration is eliminated by the objective matching technique, and two same objectives MO1, MO2 with magnification of 40 × are placed in the reference arm and object arm, respectively. A T-lymphocyte cell is chosen as the tested sample. Figure 8 and Table 3 give the corresponding results of phase retrieval with MSSM, AIA and PCA algorithms, respectively.

Fig. 7 Experimental setup of on-axis phase-shifting digital holographic microscopy system. MO1,MO2: microscope objective; PZT: piezoelectric ceramic transducer; M1, M2: mirror; BS1, BS2: beam splitter.

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Fig. 8 Experimental results of a T-lymphocyte cell (a) three-frame interferograms; (b) the reference phase of (a) achieved by temporal Fourier transform algorithm; the unwrapped phase distributions and the corresponding differences between the reference phase and the wrapped phases achieved with different algorithms (c)(f) PCA; (d)(g) MSSM; (e)(h) AIA; (i) the cross-section curves of the 170th row in (f), (g) and (h), respectively.

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Table 3. RMSE, PVE and calculation time of phase retrieval with different algorithms in Fig. 8

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In Fig. 8, due to the fringes number in interferogram is too little to satisfy the condition requirement, we can see that the PCA algorithm cannot work well; moreover, the absolute constant assumption of A(x, y) and B(x, y) is also not satisfied, so the accuracy of AIA is very low. In contrast, using the proposed MSSM algorithm, the phase retrieval reveals high accuracy. From these results, we can conclude that the proposed MSSM algorithm is robust in both the fringe number and the non-constant of A(x, y) and B(x, y).

5. Conclusion

In conclusion, based on Fourier domain estimation technique and a small amount phase-shifting interferograms with USFPs, a novel self-calibration MSSM algorithm is proposed, in which the phase shifts is estimated through performing the spectral division of interferograms. The achieved results demonstrate that although the proposed MSSM algorithm is conceived as a solution for USFP, it also reveals high accuracy for fringe-patterns with many fringes. What’s more, the proposed method has rapid calculation speed because the iterative procedure is not needed. Specially, due to no requirement for the fringe number, the proposed MSSM algorithm should be a good candidate for the recently developed interferometry methods, including white light phase-shifting interferometry, the more sufficient utilizing of spatial-bandwidth of CCD through the fringe number reducing. And this will greatly extend the applications of proposed MSSM algorithm.

Funding

National Natural Science Foundation of China (61475048, 61727814 and 61575069).

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Algorithms	AIA	PCA	MSSM
PVE(rad)	0.785	0.678	0.661
RMSE(rad)	0.097	0.047	0.037
Time(s)	1.980	0.039	0.052

Algorithms	AIA	PCA	MSSM
PVE(rad)	0.560	0.663	0.129
RMSE(rad)	0.135	0.341	0.024
Time(s)	1.342	0.014	0.024

Algorithms	AIA	PCA	MSSM
PVE(rad)	0.886	1.051	0.795
RMSE(rad)	0.160	0.350	0.095
Time(s)	1.042	0.021	0.041

Algorithms	AIA	PCA	MSSM
PVE(rad)	0.785	0.678	0.661
RMSE(rad)	0.097	0.047	0.037
Time(s)	1.980	0.039	0.052

Algorithms	AIA	PCA	MSSM
PVE(rad)	0.560	0.663	0.129
RMSE(rad)	0.135	0.341	0.024
Time(s)	1.342	0.014	0.024

Self-calibration phase-shifting algorithm with interferograms containing very few fringes based on Fourier domain estimation

Abstract

1. Introduction

2. Principle

3. Simulation

4. Experiment

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Tables (3)

Equations (14)

Optics Express