Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression

ARTICLE IN PRESS Optics and Lasers in Engineering 48 (2010) 684–689

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Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression Qian Kemao , Haixia Wang, Wenjing Gao, Lin Feng, Seah Hock Soon School of Computer Engineering, Nanyang Technological University, Singapore 639798, Singapore

a r t i c l e in f o

a b s t r a c t

Article history: Received 26 November 2009 Received in revised form 12 January 2010 Accepted 18 January 2010 Available online 6 February 2010

Extracting phase distribution from arbitrary phase-shifted fringe patterns, if possible, is very useful in phase-shifting interferometry. The advanced iterative algorithm (AIA) is introduced and the windowed Fourier ridges and least squares fitting (WFRLSF) is proposed. Both algorithms are sensitive to noise, which limits their applications to almost perfect fringe patterns. The windowed Fourier filtering (WFF) algorithm is proposed for both pre-filtering and post-filtering to suppress the noise. Simulation results show that with the effective noise suppression, the phase error is reduced to less than 0.1 rad. Experimental examples are also given for verification. The almost identical results produced by the AIA and the WFRLSF suggest that both algorithms can be used for phase extraction with cross-validation. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Phase-shifting interferometry Advanced iterative algorithm Windowed Fourier transform Least squares fitting Noise suppression

1. Introduction Phase-shifting interferometry is a powerful technique for fullfield, accurate and non-contact measurements [1–6]. To extract phase distribution from fringe patterns with known phase-shifts, the accuracy of the phase-shifts is essential but often difficult to be guaranteed [1–4]. Consequently error-compensating phaseshifting algorithms were proposed, assuming that the phase-shifts are around their nominal values [5,6]. This solution is straightforward and has been well accepted. An alternative solution is to extract the phase distribution without knowing the phase-shift values. One example is the advanced iterative algorithm (AIA) proposed by Wang and Han [7]. The AIA evolved from Okada et al.’s work [8] which was extended from Greivenkamp’s work [9]. The AIA has attracted some research interests recently due to its high effectiveness [10–12]. Another method called WFRLSF is proposed in this paper. It first estimates phase-shifts by a windowed Fourier ridges (WFR) algorithm [13] and then estimates phase distribution by a least squares fitting (LSF) [7–9]. For the AIA, noise affects the convergence of iterations and consequently the accuracy of the extracted phase. For the WFRLSF, though the WFR algorithm is able to accurately estimate the phase-shifts from noisy fringe patterns, the subsequently extracted phase is still noisy. It will be seen that in both algorithms, the phase error increases with noise level. To improve

 Corresponding author.

E-mail address: [email protected] (Q. Kemao). 0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.01.009

immunity to noise, a windowed Fourier filtering algorithm (WFF) [14–16] is proposed for both pre-filtering before phase extraction and post-filtering after phase extraction. Simulation shows that with the noise suppression, both the AIA and the WFRLSF produce phase errors less than 0.1 rad. It will be observed, interestingly and surprisingly, that the AIA and the WFRLSF give almost identical results, though their principles are quite different. Thus the WFRLSF, having no convergence problem, can serve as an empirical way to show the convergence of the AIA which has not been proven yet. The contributions of this paper include (i) The WFRLSF for phase extraction from arbitrary phaseshifted fringe patterns is proposed; (ii) The WFF which will be shown to be very effective as prefiltering and post-filtering for both the AIA and the WFRLSF is proposed; (iii) The WFRLSF can be used to show the convergence of the AIA due to their very similar performances.

The rest of the paper is organized as follows. The AIA is introduced in Section. 2. The WFF and the WFR are introduced in Section 3. The WFRLSF is proposed in Section 4. Using the WFF to improve the performances of the AIA and the WFRLSF is proposed in Section 5. Simulated and experimental examples are given for verification. Spatially non-uniform phase-shifts and high order harmonics as error sources in phase extraction are discussed. The paper is concluded in Section 6.

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2. Advanced iterative algorithm (AIA) [7] The AIA introduced in this section is directly adapted from Ref. [7] for completeness of this paper. The notations are similar to those in [7] for consistency. A part of the AIA, Eqs. (4–5), will also be used for the WFRLSF proposed in Section 4. Phase-shifted fringe patterns can be written as follows: fij ¼ Aij þ Bij cosðjj þ di Þ þ nij ;

ð1Þ

where the subscript i=1, 2, y, M is used as a frame index and M is the total frame number; the subscript j= 1, 2, y, N is used as a pixel index and N is the total pixel number; fij, Aij, Bij and nij are fringe intensity, background intensity, fringe amplitude, and noise of frame i and pixel j, respectively; jj is the phase value of pixel j; and di is the phase-shift value of frame i. Eq. (1) can be rewritten as fij ¼ aj þbj cosdi þ cj sindi þnij ;

ð2Þ

where aj = Aij, bj = Bij cos jj and cj = Bij sin jj are assumed to be constant in different frames. If the phase-shifts di are known, for each pixel j, aj, bj and cj can be solved from Eq. (2) by the LSF, i.e., by minimizing the residual error Ej ¼

algorithm. Its convergence is not theoretically proven. However, the method works well especially when the noise level is very low, which is evident from Ref. [7] and subsequent publications [10–12].

3. Windowed Fourier transform [14–16] In this section, the windowed Fourier ridges (WFR) and the windowed Fourier filtering (WFF), two algorithms based on windowed Fourier transform, are briefly introduced. They are directly adapted from Refs. [14–16] for the completeness of this paper. The notations are similar to those in [14–16] for consistency. The WFR will be used in developing the WFRLSF, while the WFF will be adopted to suppress noise in both the AIA and the WFRLSF. A windowed Fourier transform pair can be expressed as a forward and an inverse transforms as follows: Z 1Z 1  f ðx; yÞgu;v; ð10Þ Sf ðu; v; x; ZÞ ¼ x;Z ðx; yÞdxdy; 1 1

f ðx; yÞ ¼

M X

ðaj þ bj cos di þ cj sin di fij Þ2 ;

685

1 4p2

Z

1Z 1Z 1Z 1

Sf ðu; v; x; ZÞgu;v;x;Z ðx; yÞdxdZdudv;

1 1 1 1

ð3Þ

ð11Þ

i¼1

which leads to the following simplified calculation: 2

M X

M 6 6 2 3 6 6 M aj 6X 6b 7 6 cos di 4 j5¼6 6i¼1 6 cj 6X 6 M 4 sin di i¼1

cos di

i¼1 M X

cos2 di

i¼1 M X

cos di sin di

i¼1

M X

31 2 sin di

7 7 i¼1 7 7 M 7 X cos di sin di 7 7 7 i¼1 7 7 M X 7 2 sin di 5

3 M X fij 7 6 7 6 i¼1 7 6 7 6 M 7 6X 6 fij cos di 7 7; 6 7 6i¼1 7 6 7 6X 7 6 M 4 fij sin di 5

i¼1

i¼1

ð4Þ where the superscript 1 denotes the matrix inverse. The phase can then be calculated as 1

jj ¼ tan ðcj =bj Þ:

ð5Þ

Similarly Eq. (1) can be rewritten as fij ¼ a0i þb0i cos jj þ ci0 sin jj þ nij ;

ð6Þ

where a0i ¼ Aij ; b0i ¼ Bij cos di and ci0 ¼ Bij sin di are assumed to be constant for all the pixels. If the phase values of jj are known, for each frame i, a0i , b0i and ci0 can be solved from Eq. (6) by the LSF, i.e., by minimizing the residual error E0i ¼

N X

ða0i þ b0i cos jj þci0 sin jj fij Þ2 ;

ð7Þ

where f(x, y) is an input fringe pattern; Sf(u, v;x, Z) is the windowed Fourier spectrum of f(x, y); (x, y) and (u, v) are spatial coordinates; (x, Z) is a frequency coordinate; the symbol * denotes a complex conjugate operator; the windowed Fourier element gu,v;x,Z(x, y) is a windowed harmonic that is spatially centered at (u, v) and tuned to a frequency of (x, Z). The WFR algorithm searches for a windowed Fourier element which is the most similar to each portion of a fringe pattern covered by the window. Consequently, frequencies of the fringe pattern at (u, v) along x and y directions, ox(u, v) and oy(u, v), can be determined as jSf ðu; v; x; ZÞj ; ½ox ðu; vÞ; oy ðu; vÞ ¼ arg max x;Z

ð12Þ

arg max

where means that the arguments x and Z which x;Z maximize jSf ðu; v; x; ZÞj are taken as ox(u, v) and oy(u, v), respectively. The local frequency [ox(u, v), oy(u, v)] is also called the ridge location. Subsequently phase distribution can be extracted from the ridge information:

jðu; vÞ ¼ anglefSf ½u; v; ox ðu; vÞ; oy ðu; vÞgþ ox ðu; vÞu þ oy ðu; vÞv: ð13Þ When the WFR is used to process a single fringe pattern, it should be highlighted that

j¼1

which leads to the following simplified calculation: 2 6 N 6 2 03 6 6 ai 6X N 6 b0 7 6 cos jj 4 i5¼6 6 6j¼1 ci0 6 6X N 6 4 sin jj j¼1

N X

cos jj

j¼1 N X

cos2 jj

j¼1 N X

cos jj sin jj

j¼1

N X

31 2 sin jj

7 7 j¼1 7 7 7 N X 7 cos jj sin jj 7 7 j¼1 7 7 7 N X 2 7 sin jj 5 j¼1

3

N X 7 6 fij 7 6 j¼1 7 6 7 6 7 6X N 7 6 6 fij cos jj 7; 7 6 7 6j¼1 7 6 7 6X N 7 6 4 fij sin jj 5 j¼1

ð8Þ The phase-shifts can then be calculated as

di ¼ tan1 ðci0 =b0i Þ:

ð9Þ

Both phase values and phase-shifts are estimated by alternately using Eqs. (4 and 5) and Eqs. (8 and 9), which is iterated until a certain condition is satisfied [7]. This forms the AIA

(i) The WFR gives results with sign ambiguity, i.e., if ox(u, v), oy(u, v) and j(u, v) is a solution, so is  ox(u, v),  oy(u, v) and  j(u, v); (ii) The WFR does not give accurate estimation when both |ox(u, v)| and |oy(u, v)| are low. The WFF algorithm is another algorithm based on the windowed Fourier transform, which assumes that the windowed Fourier spectrum of noise in a fringe pattern permeates the entire windowed Fourier domain with small spectrum coefficients. The WFF algorithm hence filters a fringe pattern by thresholding its windowed Fourier spectrum, which can be written as Z 1 Z 1 Z Zh Z xh 1 f ðx; yÞ ¼ Sf ðu; v; x; ZÞgu;v;x;Z ðx; yÞdx dZ du dv; 4p2 1 1 Zl xl ð14Þ

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with ( Sf ðu; v; x; ZÞ ¼

Sf ðu; v; x; ZÞ

if

jSf ðu; v; x; ZÞj Z thr

0

if

jSf ðu; v; x; ZÞj o thr

ð15Þ

where Sf ðu; v; x; ZÞ denotes the thresholded spectrum; f ðx; yÞ denotes the filtered fringe pattern; thr denotes a threshold; only the spectrum of (x,Z)A[xl,xh]  [Zl,Zh] is used for reconstruction to pre-exclude unnecessary frequency components to save computing time.

4. Windowed Fourier ridges and least squares fitting (WFRLSF) In this section, the WFRLSF is proposed for phase extraction from fringe patterns with arbitrary phase-shifts. It consists of two steps, phase-shift estimation using the WFR, and phase estimation using the LSF. The first step has been separately proposed in an earlier publication as a technique for phase-shifter calibration [13]. Thus, it is quite straightforward to propose the WFRLSF with the preparations in Sections 2 and 3 and in Ref. [13], but it has not been seen in the literature, to the best of our knowledge. As both the LSF in Section 2 and WFR in Section 3 will be used in the WFRLSF, the notations in Section 3 are adapted to be consistent with those in Section 2 and the adaptation should be apparent. For example, ox(u, v)extracted from ith fringe pattern is written as (ox)ij. Details of the WFRLSF are as follows. The first step of the WFRLSF is to estimate the phase-shifts by the WFR. As explained in Section 3, the WFR is able to extract local frequency and phase information from the (i 1)th single fringe pattern, which could be [(ox)(i  1)j,(oy)(i  1)j,jj + di  1] or [ (ox)(i  1)j,  (oy)(i  1)j,  jj  di  1] due to the sign ambiguity problem. Similarly from the ith single fringe pattern, the WFR gives [(ox)ij,(oy)ij,jj + di] or [  (ox)ij,  (oy)ij,  jj  di]. There are four possibilities of the phase difference between two consecutive fringe patterns: (i) (jj + di) (jj + di  1) = di  di  1, (ii) (jj + di) (  jj  di  1)=2jj + di + di  1, (iii) (  jj  di) (jj + di  1)= 2jj  di  di  1, and (iv) (  jj  di)  (jj  di  1) =  dI + di  1. Since phase-shifted fringe patterns share same local frequencies at each pixel, by forcing the signs of (ox)ij and (ox)(i  1)j to be the same, possibilities of (ii) and (iii) are eliminated. It should be noted that the extracted phase at pixels with ox around zero are not reliable because oy may get its sign randomly due to noise. In this case, oy from two consecutive frames is further forced to be the same. There are still two possible values of the phase difference left in (i) and (iv), but their absolute values are the same. If the phase-shift is assumed to be di  di  1A[0, p), it can be uniquely determined as

dij dði1Þj ¼ jdij dði1Þj j ¼ jðjj þ di Þðjj þ di1 Þj:

ð16Þ

In the same manner, all the phase-shifts between any two consecutive frames can be estimated. The phase-shift between mth frame and the first frame can consequently be calculated as 



dmj ¼ dmj d1j ¼ dmj dðm1Þj þ    þ ðd2j d1j Þ ¼

m  X



dij dði1Þj ;

i¼2

ð17Þ where d1j is set to zero. The WFR can thus give phase-shifts at all the pixels. If the phaseshifts are assumed to be spatially constant, averaging the phase-shift values from all the pixels yields a more reliable estimation. However, the following pixels are excluded from the averaging: (i) Pixels in low fringe density areas that can be characterized as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð18Þ ðox Þ2ij þ ðoy Þ2ij o othr ;

where othr =0.2 rad/pixel is selected for all the examples in this paper. The reason is that the WFR is not reliable when both |ox(u, v)| and |oy(u, v)| are low, as mentioned in Section 3. (ii) Pixels near image borders due to edge effects. The second step of the WFRLSF is to estimate phase distribution by the LSF. After all the phase-shifts have been determined in the first step, the LSF expressed in Eqs. (4 and 5) can be directly used to estimate the phase. Though the WFR already provides the phase distribution, it is not reliable when both |ox(u, v)| and |oy(u, v)| are low. Thus, the LSF is adopted for the phase estimation. Unlike the AIA, the WFRLSF only needs the LSF once because the phase-shifts have been determined by the WFR algorithm accurately [13]. It is interesting to note that, in Ref. [16], a sampling interval not larger than 0.025 rad/pixel is suggested for ox(u, v) and oy(u, v) in the WFR algorithm, in order to reduce the errors of the local frequency and phase extraction. These errors will be canceled when the phase difference is calculated in Eq. (16). This is because the same pixel in different phase-shifted fringe patterns has the same local frequencies. Thus, the sampling interval can be larger so that the computation can be faster. The sampling interval of 0.1 rad/pixel is used in this paper. Similarly, since a pixel has the same curvature of the phase, the correction of the phase error due to the curvature of the phase [16] is unnecessary as they will also be automatically canceled in Eq. (16).

5. Noise suppression in the AIA and the WFRLSF 5.1. Noise suppression using the WFF Noise is unavoidable in fringe patterns. It can be naturally expected that when the fringe patterns are noisy, the phase extracted by the AIA is also noisy. Thus post-filtering is necessary. However, when the noise is severe, post-filtering alone is not sufficient because the AIA may fail to estimate the phase properly and it is too late for the post-filtering to rescue. To prevent such failure, pre-filtering is also needed. We propose to use the WFF for pre-filtering and/or postfiltering. The WFF is chosen for filtering due to (i) Its high effectiveness. It has been theoretically proven that the standard deviation of noise can be reduced typically by about 30 times [16]. In filtering a fringe pattern [17], the WFF result is often similar to or better than the coherence enhancing diffusion [18] which is similar to the spin filter [19]. In filtering a wrapped phase map [20], its equivalence to the regularized phase tracking [21] and superiority to the sine/ cosine average filter [22] have also been shown. (ii) Its high flexibility. The same WFF algorithm can be used to filter both intensity fringe patterns (open or closed) [17] and wrapped phase maps [20] without changing programming codes. The codes have been provided in [15] and [23]. For the AIA, the pre-filtering processes intensity fringe patterns while the post-filtering processes the wrapped phase map, which can be easily realized by a single WFF algorithm. The continuous wavelet transform which usually applies to open intensity fringe patterns is not suitable [24]. Once the WFF is chosen for filtering, we have the following three choices for the AIA: (i) Only post-filtering is applied, i.e., AIA+ WFF; (ii) Only pre-filtering is applied, i.e., WFF+ AIA;

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The following remarks can be made based on Figs. 1 and 2:

(iii) Both pre-filtering and post-filtering are applied, i.e., WFF+AIA +WFF. As mentioned, the AIA+WFF sometimes gives large error as the post-filtering is not always able to prevent the errors caused by the AIA. Meanwhile, the WFF+AIA and the WFF+AIA+WFF give similar results. However, the post-filtering using the WFF does help to further reduce the noise that survived from the previous steps. Our simulation shows that the WFF+AIA+WFF generally gives the most satisfactory results. The simulation results for the above conclusion are not illustrated in order to keep this paper concise. Analogically, the WFF+ WFRLSF+ WFF is proposed to suppress the noise in the WFRLSF. Due to the inherent noise immunity of the WFR algorithm, the pre-filtering using WFF is not very necessary for the WFRLSF. Nevertheless, it is generally not harmful to include it, and it sometimes helps to improve the WFRLSF performance [16].

5.2. Verifications by simulation and experimental examples We use simulation to see the influence of noise in the AIA and the WFRLSF and noise suppression by the WFF+ AIA+ WFF and the WFF+ WFRLSF+WFF. A fringe pattern is generated as follows: f ðx; yÞ ¼ 1þ cosf0:002½ðx128Þ2 þ ðy128Þ2 g þL n0 ðx; yÞ; 1 rx; yr 256

ð19Þ

where n0(x, y) is Gaussian noise with mean of zero and standard deviation of one; L=0, 0.1, 0.2, y, 2 are various noise levels for testing. Phase-shifts of 711, 2091 and 3111 are introduced into Eq. (19) to generate another three phase-shifted fringe patterns. For quantitative evaluation, the following measures are defined: (i) Phase difference 1X jðjj ÞS ðjj ÞT j; Dp ðS; TÞ ¼ N j

687

(i) Both the AIA and the WFRLSF give noisy phase distributions. Thus, the AIA or the WFRLSF alone can only be used in limited cases with very low noise levels; (ii) Both the WFF+AIA+WFF and the WFF+WFRLSF+WFF give accurate phase extraction and dramatically improve the performance of the AIA and the WFRLSF against noise. With the WFF for noise suppression, the AIA and the WFRLS are less sensitive to noise and their applications will be much wider. (iii) It is interesting and surprising that the AIA and the WFRLSF give almost identical results, though their principles are quite different. The WFF+ AIA+ WFF and the WFF+ WFRLSF+ WFF also give almost identical results. As the principle of the WFRLSF can be proven, it can serve as an empirical way to show the convergence and effectiveness of the AIA. Two experimental fringe pattern sets are also tested. The first set consists of projected fringe patterns with good quality. One of four fringe patterns is shown in Fig. 3(a). Figs. 3(b–e) are the results of the AIA, the WFRLSF, the WFF +AIA +WFF, and the WFF+WFRLSF+ WFF, respectively. It shows that when the noise is not severe, all the methods can extract the phase distribution effectively. Quantitatively, Dp(‘WFF+ AIA +WFF’,‘WFF +WFRLSF + WFF’) and Dps(‘WFF+AIA + WFF’,‘WFF+ WFRLSF +WFF’) are 0.0035 and 0.026 rad, respectively. The second set consists of speckle correlation fringe patterns. One of four fringe patterns is shown in Fig. 4(a). The noise is much more severe. Figs. 4(b–e) are the results of the AIA, the WFRLSF, the WFF+AIA +WFF, and the WFF+ WFRLSF+ WFF, respectively. The results by the AIA and the WFRLSF are hardly useful, while the WFF+ AIA+ WFF and the WFF+ WFRLSF+WFF significantly improve their performances against noise. Quantitatively, Dp(‘WFF+ AIA+ WFF’,‘WFF+ WFRLSF +WFF’) and Dps(‘WFF+ AIA+ WFF’,‘WFF+WFRLSF + WFF’) are 0.21 and 0.29 rad, respectively. This discrepancy is largely due to the very low quality of the original fringe patterns.

ð20Þ

where S and T represent any two algorithms selected from the AIA, the WFRLSF, the WFF+AIA+WFF, the WFF+WFRLSF+WFF, or ‘‘ground truth’’; correspondingly (jj)s and (jj)T are the phases extracted by these algorithms. (ii) Phase-shift difference 1 X jðdi ÞS ðdi ÞT j ð21Þ Dps ðS; TÞ ¼ M1 i where (di)S and (di)T are the phase-shifts extracted by these algorithms. The AIA, the WFRLSF, the WFF+ AIA+ WFF, and the WFF+ WFRLSF+WFF are applied to the simulated fringe patterns. The phase errors with respect to different noise levels by the four algorithms are shown in Fig. 1. Fringe patterns with simulated speckle noise are also tested. The speckle radius is 2 pixels. The details of speckle noise simulation can be found in [25]. Fig. 2(a) shows one of four fringe patterns with the speckle noise. Figs. 2(b–e) are the results of the AIA, the WFRLSF, the WFF+ AIA+ WFF, and the WFF+ WFRLSF+ WFF, respectively. Among all the simulated cases, the speckle noise gives the largest errors where both Dp(‘WFF+ AIA+ WFF’,‘true’) and Dp(‘WFF+ WFRLSF+ WFF’,‘true’) are slightly less than 0.1 rad; they are almost identical because Dp(‘WFF+AIA +WFF’,‘WFF + WFRLSF+ WFF’) is less than 0.01 rad; Dps(‘WFF+ AIA+ WFF’,‘true’) and Dps(‘WFF+WFRLSF + WFF’,‘true’) are 0.06 and 0.01 rad, respectively.

5.3. Discussions In both the AIA and the WFRLSF, the phase-shifts are assumed to be spatially constant. One error source is that the phase-shifts are spatially non-uniform. However, the fringe patterns can be divided into several blocks so that the AIA can still be used for each block [10]. For the WFRLSF, in the WFR step, after the phase-shifts at all pixels have been estimated, instead of averaging the phase shifts in the valid area, they can be fitted as a plane or quadric surface to respond to the non-uniformity of the phase-shifts. In the LSF step, the phase estimation can be done block by block, similar to Ref. [10]. The pre-filtering and post-filtering using the WFF are still applicable to the ‘‘modified’’ AIA and WFRLSF as the WFF is applied to the fringes and is not related to phase-shifts. Another error source for phase extraction is high order harmonics in fringe patterns. For the AIA, it can be reformulated by taking these high order harmonics into optimization [11]. For the WFRLSF, in the WFR step, the high order harmonics can be avoided automatically and the phase-shift estimation is not affected [26]. The LSF step can be reformulated similar to Ref. [11]. The pre-filtering and post-filtering using the WFF are again applicable to the ‘‘modified’’ AIA and WFRLSF. The computation of the AIA is simple but needs several iterations. For the WFRLSF, the computation cost of the WFR is high, but it can be reduced towards real-time if a Graphics Processing Unit (GPU) is used [27]. The GPU is getting popular recently. The GPU acceleration is also applicable to the WFF [27].

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1.4 1.2

AIA WFRLS WFF+AIA+WFF WFF+WFRLS+WFF

Phase error (rad)

1 0.8 0.6 0.4 0.2 0

0

0.5

1

1.5

2

Fig. 1. Error of extracted phase with respect to additive noise level.

Fig. 2. A simulated example with speckle noise: (a) one of four fringe patterns; (b) the AIA result; (c) the WFRLSF result; (d) the WFF+ AIA+ WFF result; (e) the WFF+ WFRLSF+ WFF result.

Fig. 3. An experimental example with low noise: (a) one of four fringe patterns; (b) the AIA result; (c) the WFRLSF result; (d) the WFF+ AIA+ WFF result; (e) the WFF+ WFRLSF+ WFF result.

The AIA can be applied to any fringe pattern sets. However, the WFRLSF needs at least a part of the fringe pattern to be dense enough to give accurate phase-shift estimation. Further, the phase-shifts between two consecutive fringe

patterns need to be within [0, p). Thus the AIA is more flexible than the WFRLSF. Since the WFRLSF has no convergence problem, it can still be used along with the AIA for crossvalidation.

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689

Fig. 4. An experimental example with severe noise: (a) one of four fringe patterns; (b) the AIA result; (c) the WFRLSF result; (d) the WFF+ AIA+ WFF result; (e) the WFF+ WFRLSF+ WFF result.

6. Conclusion In this paper, we discuss the influence of noise in the advanced iterative algorithm (AIA) and windowed Fourier ridges and least squares fitting (WFRLSF) algorithm for phase extraction from arbitrary phase-shifted fringe patterns. Windowed Fourier filtering (WFF) is proposed as both pre-filtering and post-filtering to suppress the noise in both the AIA and the WFRLSF. Simulation results show that the phase errors by the WFF+ AIA+ WFF and the WFF+ WFRLSF+WFF are almost identical and the phase errors are less than 0.1 rad, which dramatically improve the performances of the AIA and the WFRLS. Experimental results are also given for verification.

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