Abstract

To overcome the phase shift error in phase shifting interferometry, a three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization (DN&DDVN) is proposed. It does not need pre-filtering for the interferograms and can obtain relatively accurate phase distribution with a simple process and less computational time. This simulation and experiment verify the correctness and feasibility of DN&DDVN.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (2)

2016 (3)

C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

2015 (1)

2014 (2)

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (4)

2010 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

2009 (2)

2008 (2)

2007 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

2006 (2)

2004 (2)

2003 (2)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

1997 (1)

1995 (1)

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

1988 (1)

1974 (1)

Belenguer, T.

Brangaccio, D. J.

Bruning, J. H.

Cai, L. Z.

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[Crossref] [PubMed]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Carazo, J. M.

Chai, L.

Chen, Q.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Chen, W.

Chen, Y. C.

Cheng, X. C.

de Groot, P. J.

Deck, L. L.

Deng, J.

Dong, G. Y.

Estrada, J. C.

Farrant, D. I.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Gallagher, J. E.

Han, B.

Hao, J.

Herriott, D. R.

Hibino, K.

Hou, X.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Ishikawa, K.

Jin, W.

Kinnstaetter, K.

Larkin, K. G.

Lee, C. M.

Li, C.

Liang, C. W.

Liang, R.

Lin, P. C.

Liu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Liu, Q.

Liu, S.

Lohmann, A. W.

Lu, X.

Luo, C.

Lv, X.

Ma, S.

Meng, X. F.

Oikawa, Y.

Oreb, B. F.

Patorski, K.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Qin, J.

Quiroga, J. A.

Rosenfeld, D. P.

Schwider, J.

Servin, M.

Shen, X. X.

Sorzano, C. O.

Streibl, N.

Sun, W. J.

Tian, C.

Tian, X.

Trusiak, M.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Vargas, J.

Wan, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wang, H.

Wang, J.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wang, K.

Wang, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Wang, Z.

White, A. D.

Wu, D.

Wu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wu, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Xu, J.

Xu, Q.

Xu, X. F.

Yan, R. S.

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

Yang, X. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Yatabe, K.

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

Zhang, Y.

Zhong, L.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

J. Opt. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Optics-UK (1)

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Meas. Sci. Technol. (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Opt. Commun. (1)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Opt. Express (8)

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[Crossref] [PubMed]

C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Hyper ellipse fitting in subspace method for phase-shifting interferometry: Practical implementation with automatic pixel selection,” Opt. Express 25(23), 29401–29416 (2017).
[Crossref]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

Y. Zhang, X. Tian, and R. Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15071 (2018).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[Crossref] [PubMed]

J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

Opt. Lett. (10)

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Optik (Stuttg.) (1)

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

Other (2)

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007), Chap. 1–7.

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007), Chap. 14.

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Figures (9)

Fig. 1
Fig. 1 Geometrical relationship between two normalized vectors D 1k , D 2k and the corresponding diagonal vectors.
Fig. 2
Fig. 2 Simulated interferograms, background intensity, modulation amplitude and phase distribution. (a), (b) and (c) the three phase shifted interferograms, (d), (e) and (f) the simulated background intensity, modulation amplitude, and theoretical phase distribution.
Fig. 3
Fig. 3 Simulated results. (a), (b), (c), and (d) the extracted phase distributions using DN&DDVN (RMS = 6.658 rad), GS3 (RMS = 6.667 rad), PCA (RMS = 6.657 rad), and AIA (RMS = 6.658 rad), (e), (f), (g), and (h) the phase error distributions after using DN&DDVN, GS3, PCA, and AIA.
Fig. 4
Fig. 4 RMS phase errors of different methods with different levels of noises.
Fig. 5
Fig. 5 Simulated results of complex fringes using DN&DDVN. (a) One of the simulated complex interferograms, (b), (c) and (d) the reference phase distribution, extracted phase distribution, and phase error distribution.
Fig. 6
Fig. 6 Experimental results of the circular fringes. (a) One of the phase shifted interferograms, (b), (c), and (d) the extracted phase distributions obtained by DN&DDVN (RMS = 5.0283 rad), GS3 (RMS = 5.0525 rad) and PCA (RMS = 5.0284 rad), (e) the reference phase distribution obtained by AIA (RMS = 5.0283 rad).
Fig. 7
Fig. 7 The differences between the reference and extracted phase distributions for circular fringes: (a)DN&DDVN, (b) GS3, and (c) PCA.
Fig. 8
Fig. 8 Experimental results of the complex fringes. (a) One of the phase shifted interferograms, (b), (c), and (d) the extracted phase distributions obtained by DN&DDVN (RMS = 8.8850 rad), GS3 (RMS = 8.8925 rad) and PCA (RMS = 8.8850 rad), (e) the reference phase distribution obtained by AIA (RMS = 8.8851 rad).
Fig. 9
Fig. 9 The differences between the reference and extracted phase distributions for complex fringes: (a)DN&DDVN, (b) GS3, and (c) PCA.

Tables (7)

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Table 1 RMS phase errors and computational time of the different methods

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Table 2 RMS phase and RMS phase errors with different numbers of the phase shifted fringe patterns using DN&DDVN

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Table 3 RMS phase errors with different phase shifts using DN&DDVN method

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Table 4 RMS phase errors and processing time with different levels of noises using different methods

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Table 5 RMS phase errors and computational time of the different methods for complex fringes

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Table 6 Computational time of the different methods for circular fringes

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Table 7 Computational time of the different methods for complex fringes

Equations (20)

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I 1k = a k + b k cos( φ k + θ 1 ) I 2k = a k + b k cos( φ k + θ 2 ) I 3k = a k + b k cos( φ k + θ 3 )
D 1k = I 1k I 2k =2 b k sin( θ 2 2 )sin( φ k + θ 2 2 )=2 b k sin( θ 2 2 )cos( Φ k π 2 )=2 b k sin( θ 2 2 )cos( Φ k )
D 2k = I 1k I 3k =2 b k sin( θ 3 2 )sin( φ k + θ 3 2 )=2 b k sin( θ 3 2 )cos( Φ k +Δ π 2 )=2 b k sin( θ 3 2 )cos( Φ k +Δ )
u =u/ u,u =u/ u
D 1k = D 1k D 1k = b k cos( Φ k ) k=1 K b 2 k cos 2 ( Φ k )
D 2k = D 2k D 2k = b k cos( Φ k +Δ ) k=1 K b 2 k cos 2 ( Φ k +Δ )
k=1 K b 2 k cos 2 ( Φ k ) k=1 K b 2 k cos 2 ( Φ k +Δ )
D 1k = c k cos( Φ k )
D 2k = c k cos( Φ k +Δ )
D ^ dif = D 1k D 2k =2 c k sin( Φ k + Δ 2 )sin Δ 2
D ^ sum = D 1k + D 2k =2 c k cos( Φ k + Δ 2 )cos Δ 2
D ˜ dif = c k sin( Φ k + Δ 2 ) k=1 K c 2 k sin 2 ( Φ k + Δ 2 )
D ˜ sum = c k cos( Φ k + Δ 2 ) k=1 K c 2 k cos 2 ( Φ k + Δ 2 )
k=1 K c 2 k sin 2 ( Φ k + Δ 2 ) k=1 K c 2 k cos 2 ( Φ k + Δ 2 )
D ˜ dif = d k sin( Φ k + Δ 2 )
D ˜ sum = d k cos( Φ k + Δ 2 )
Φ k =arctan( D ˜ dif D ˜ sum ) Δ 2
Φ k = Φ k + Δ 2 =arctan( D ˜ dif D ˜ sum )
a=1.3exp[ 0.02( x 2 + y 2 ) ],b=1.2exp[ 0.02( x 2 + y 2 ) ] φ=5π( x 2 + y 2 ), θ 1 =0rad, θ 2 =1rad, θ 3 =3rad
φ=5π( x 2 + y 2 + x 3 + y 3 )+10peaks( 401 ) 5π( x 2 + y 2 + x 3 + y 3 )+10peaks( 401 )

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