Abstract

We introduce the algorithm for the direct phase estimation from the single noisy interferometric pattern. The method, named implicit smoothing spline (ISS), can be regarded as a formal generalization of the smoothing spline interpolation for the case when the interpolated data is given implicitly. We derive the necessary equations, discuss the properties of the method and address its application for the direct estimation of the continuous phase in both classical interferometry and digital speckle pattern interferometry (DSPI). The numerical illustrations of the algorithm performance are provided to corroborate the high quality of the results.

© 2014 Optical Society of America

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  17. M. A. Gdeisat, D. R. Burton, D. R. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45(34), 8722–8732 (2006).
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  22. M. Zhong, W. Chen, T. Wang, X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. 51(10), 1138–1142 (2013).
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  23. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
    [CrossRef]
  24. N. E. Huang, Z. H. Wu, “A review on Hilbert-Huang transform method and its applications to geophysical studies,” Rev. Geophys. 46(2), 1–23 (2008).
    [CrossRef]
  25. Y. Lei, J. Lin, Z. He, M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. 35(1–2), 108–126 (2013).
    [CrossRef]
  26. M. Trusiak, K. Patorski, M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
    [CrossRef] [PubMed]
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  33. P. Craven, G. Wahba, “Smoothing noisy data with spline functions estimating the correct degree of smoothing by the method of generalized cross validation,” Numer. Math. 31, 377–403 (1979).
    [CrossRef]
  34. M. J. D. Powell, “A hybrid method for nonlinear equations,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon and Breach, 1970), pp. 87–114.
  35. M. J. D. Powell, “A Fortran subroutine for solving systems of nonlinear algebraic equations,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon and Breach, 1970), pp. 115–161.
  36. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 2 (Cambridge University, 1992).
  37. J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50(16), 2425–2430 (2011).
    [CrossRef] [PubMed]
  38. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2007).
  39. G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. 42(7), 2010–2014 (2003).
    [CrossRef]
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    [CrossRef]
  41. C. de Boor, “Calculation of the smoothing spline with weighted roughness measure,” this paper can be downloaded at http://www.cs.wisc.edu .

2014 (1)

M. Trusiak, M. Wielgus, K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52(1), 230–240 (2014).
[CrossRef]

2013 (4)

Y. Lei, J. Lin, Z. He, M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. 35(1–2), 108–126 (2013).
[CrossRef]

K. Pokorski, K. Patorski, “Processing and phase analysis of fringe patterns with contrast reversals,” Opt. Express 21(19), 22596–22614 (2013).
[CrossRef] [PubMed]

M. Zhong, W. Chen, T. Wang, X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. 51(10), 1138–1142 (2013).
[CrossRef]

L. Kai, Q. Kemao, “Improved generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express 21(20), 24385–24397 (2013).
[CrossRef] [PubMed]

2012 (4)

2011 (3)

2010 (1)

2009 (1)

2008 (2)

N. E. Huang, Z. H. Wu, “A review on Hilbert-Huang transform method and its applications to geophysical studies,” Rev. Geophys. 46(2), 1–23 (2008).
[CrossRef]

A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[CrossRef]

2007 (1)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principle, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[CrossRef]

2006 (2)

Z. Wang, H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

M. A. Gdeisat, D. R. Burton, D. R. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45(34), 8722–8732 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

K. Patorski, L. Salbut, “Simple polarization phase-stepping scatterplate interferometry,” Opt. Eng. 43(2), 393–397 (2004).
[CrossRef]

2003 (1)

G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. 42(7), 2010–2014 (2003).
[CrossRef]

1999 (1)

1998 (1)

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

1997 (1)

1996 (1)

K. Creath, J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[CrossRef]

1987 (1)

1985 (1)

O. Y. Kwon, D. M. Sough, “Multichannel grating phase-shift interferometers,” Proc. SPIE 599, 273–279 (1985).
[CrossRef]

1982 (1)

1979 (1)

P. Craven, G. Wahba, “Smoothing noisy data with spline functions estimating the correct degree of smoothing by the method of generalized cross validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

1977 (1)

1972 (1)

1967 (1)

C. Reinsch, “Smoothing by spline functions,” Numer. Math. 10, 177–183 (1967).
[CrossRef]

Barnes, T.

Burton, D.

S. Fernandez, M. A. Gdeisat, J. Salvi, D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011).
[CrossRef]

Burton, D. R.

Chen, W.

M. Zhong, W. Chen, T. Wang, X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. 51(10), 1138–1142 (2013).
[CrossRef]

Craven, P.

P. Craven, G. Wahba, “Smoothing noisy data with spline functions estimating the correct degree of smoothing by the method of generalized cross validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

Creath, K.

K. Creath, J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[CrossRef]

Cuevas, F. J.

de Boor, C.

C. de Boor, A Practical Guide to Splines (Springer, 1994).

C. de Boor, “Calculation of the smoothing spline with weighted roughness measure,” this paper can be downloaded at http://www.cs.wisc.edu .

Dursun, A.

A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[CrossRef]

Ecevit, F. N.

A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[CrossRef]

Etchepareborda, P.

Federico, A.

Fernandez, S.

S. Fernandez, M. A. Gdeisat, J. Salvi, D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 2 (Cambridge University, 1992).

Gdeisat, M. A.

S. Fernandez, M. A. Gdeisat, J. Salvi, D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011).
[CrossRef]

M. A. Gdeisat, D. R. Burton, D. R. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45(34), 8722–8732 (2006).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2007).

He, Z.

Y. Lei, J. Lin, Z. He, M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. 35(1–2), 108–126 (2013).
[CrossRef]

Hoang, T.

Huang, N. E.

N. E. Huang, Z. H. Wu, “A review on Hilbert-Huang transform method and its applications to geophysical studies,” Rev. Geophys. 46(2), 1–23 (2008).
[CrossRef]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Ichioka, Y.

Ina, H.

Inuiya, N.

Kai, L.

Kaufmann, G. H.

Kemao, Q.

Kobayashi, S.

Kwon, O. Y.

O. Y. Kwon, D. M. Sough, “Multichannel grating phase-shift interferometers,” Proc. SPIE 599, 273–279 (1985).
[CrossRef]

Lalor, D. R.

Lei, Y.

Y. Lei, J. Lin, Z. He, M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. 35(1–2), 108–126 (2013).
[CrossRef]

Li, Y. J.

Lin, J.

Y. Lei, J. Lin, Z. He, M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. 35(1–2), 108–126 (2013).
[CrossRef]

Liu, D.

Liu, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Long, S. R.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Luo, Y. J.

Luu, L.

Ma, H.

Z. Wang, H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

Ma, J.

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

Marroquin, J. L.

Ozder, S.

A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[CrossRef]

Pan, B.

Patorski, K.

M. Trusiak, M. Wielgus, K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52(1), 230–240 (2014).
[CrossRef]

K. Pokorski, K. Patorski, “Processing and phase analysis of fringe patterns with contrast reversals,” Opt. Express 21(19), 22596–22614 (2013).
[CrossRef] [PubMed]

M. Trusiak, K. Patorski, M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
[CrossRef] [PubMed]

K. Patorski, L. Salbut, “Simple polarization phase-stepping scatterplate interferometry,” Opt. Eng. 43(2), 393–397 (2004).
[CrossRef]

Peyrovian, M. J.

Pokorski, K.

Powell, M. J. D.

M. J. D. Powell, “A hybrid method for nonlinear equations,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon and Breach, 1970), pp. 87–114.

M. J. D. Powell, “A Fortran subroutine for solving systems of nonlinear algebraic equations,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon and Breach, 1970), pp. 115–161.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 2 (Cambridge University, 1992).

Reinsch, C.

C. Reinsch, “Smoothing by spline functions,” Numer. Math. 10, 177–183 (1967).
[CrossRef]

Roddier, F.

Rodier, C.

Salbut, L.

K. Patorski, L. Salbut, “Simple polarization phase-stepping scatterplate interferometry,” Opt. Eng. 43(2), 393–397 (2004).
[CrossRef]

Salvi, J.

S. Fernandez, M. A. Gdeisat, J. Salvi, D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011).
[CrossRef]

Sarac, Z.

A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[CrossRef]

Sawchuk, A. A.

Schmit, J.

K. Creath, J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[CrossRef]

Servin, M.

Shen, Z.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Shih, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Sough, D. M.

O. Y. Kwon, D. M. Sough, “Multichannel grating phase-shift interferometers,” Proc. SPIE 599, 273–279 (1985).
[CrossRef]

Su, X.

M. Zhong, W. Chen, T. Wang, X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. 51(10), 1138–1142 (2013).
[CrossRef]

Takeda, M.

Tan, S.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 2 (Cambridge University, 1992).

Tian, C.

Topkara, H. S.

A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[CrossRef]

Trusiak, M.

M. Trusiak, M. Wielgus, K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52(1), 230–240 (2014).
[CrossRef]

M. Trusiak, K. Patorski, M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
[CrossRef] [PubMed]

Tung, C. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Vadnjal, A. L.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 2 (Cambridge University, 1992).

Vo, M.

Wahba, G.

P. Craven, G. Wahba, “Smoothing noisy data with spline functions estimating the correct degree of smoothing by the method of generalized cross validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

Wang, G.

Wang, H.

Wang, T.

M. Zhong, W. Chen, T. Wang, X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. 51(10), 1138–1142 (2013).
[CrossRef]

Wang, Z.

J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50(16), 2425–2430 (2011).
[CrossRef] [PubMed]

Z. Wang, H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

Watkins, L.

Watkins, L. R.

L. R. Watkins, “Review of fringe pattern phase recovery using 1-D and 2-D continuous wavelet transforms,” Opt. Lasers Eng. 50(8), 1015–1022 (2012).
[CrossRef]

Wielgus, M.

M. Trusiak, M. Wielgus, K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52(1), 230–240 (2014).
[CrossRef]

M. Trusiak, K. Patorski, M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
[CrossRef] [PubMed]

Wu, M. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Wu, Z. H.

N. E. Huang, Z. H. Wu, “A review on Hilbert-Huang transform method and its applications to geophysical studies,” Rev. Geophys. 46(2), 1–23 (2008).
[CrossRef]

Yang, Y. Y.

Yen, N.-C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Zheng, Q.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A 454, 903–995 (1998).
[CrossRef]

Zhong, M.

M. Zhong, W. Chen, T. Wang, X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. 51(10), 1138–1142 (2013).
[CrossRef]

Zhou, H. Ch.

Zhuo, Y. M.

Zuo, M.

Y. Lei, J. Lin, Z. He, M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. 35(1–2), 108–126 (2013).
[CrossRef]

Appl. Opt. (9)

C. Rodier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987).
[CrossRef]

Y. Ichioka, N. Inuiya, “Direct phase detecting system,” Appl. Opt. 11(7), 1507–1514 (1972).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Smoothing spline intensity denoising (a) and phase calculation (b) compared with the ISS-estimated phase (d) and the corresponding intensity (c).

Fig. 2
Fig. 2

Flowchart of the proposed phase demodulation algorithm based on the ISS.

Fig. 3
Fig. 3

Synthetic pattern spoiled with noise (a); underlying phase distribution (e); result and residual error of Gaussian filtration and cosine inversion result (b),(f); ISS averaged 1D rasters result and error (c),(g); 2D ISS result and error (d),(h). Errors in radians.

Fig. 4
Fig. 4

Synthetic pattern spoiled with noise (a); pattern skeleton used for the interpolation (e); result and error of the TPS method (b),(f); result and residual error of the CWT method (c),(g); ISS result and error (d),(h). Errors in radians.

Fig. 5
Fig. 5

Synthetic pattern spoiled with noise (a); ideal noiseless intensity distribution (f); result and error of the FFT method (b),(g); result and residual error of the TPS method (c),(h); result and residual error of the CWT method (d),(i); ISS result and error (e),(j). Errors in radians.

Fig. 6
Fig. 6

Synthetic DSPI pattern (a); ideal noiseless intensity distribution (e); result and error of the TPS method (b),(f); result and residual error of the CWT method (c),(g); ISS result and error (d),(h). Errors in radians.

Fig. 7
Fig. 7

Synthetic DSPI pattern (a); ideal noiseless intensity distribution (e); result and error of the TPS method (b),(f); result and residual error of the CWT method (c),(g); ISS result and error (d),(h). Errors in radians.

Fig. 8
Fig. 8

Experimental DSPI pattern [39] as used by the ISS algorithm (a); enhanced contrast DSPI pattern, shown solely for demonstratory purposes (b); rewrapped CWT-processed pattern (c); rewrapped ISS-processed pattern (d).

Fig. 9
Fig. 9

Experimental DSPI pattern as used by the ISS algorithm (a); rewrapped CWT result (b); rewrapped ISS-processed pattern (c).

Fig. 10
Fig. 10

Experimental interferometic pattern [40], as used by the ISS algorithm (a); rewrapped ISS-processed pattern (b); partially found pattern skeleton used to produce ISS initial phase guess (c).

Tables (3)

Tables Icon

Table 1 Errors comparison for the numerical tests 1 and 2

Tables Icon

Table 2 Errors comparison for the tests 3–5

Tables Icon

Table 3 Three popular PSS parametrizations

Equations (17)

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P ( S ) = i = 0 N [ Y i S i ] 2 + p Ω [ S ( x ) ] 2 d x ,
Ω [ S ( x ) ] 2 d x = S T K S .
P ( S ) S i = 2 ( Y S ) + 2 p K S .
( I + p K ) S ^ = Y ,
P f ( S ) = i = 0 N [ Y i f ( S i ) ] 2 + p Ω [ S ( x ) ] 2 d x .
f ( S ^ ) [ Y f ( S ^ ) ] = p K S ^ ,
sin ( 2 S ^ ) 2 Y sin ( S ^ ) = 2 p K S ^ ,
I ( x ) = b ( x ) + m ( x ) cos [ ϕ ( x ) ] ,
I ^ ( x ) cos [ ϕ ( x ) ] .
[ Δ I ( x ) ] 2 = [ I 2 ( x ) I 1 ( x ) ] 2 [ sin ( θ ) cos ( ϕ / 2 ) ] 2 = 0.5 sin 2 ( θ ) [ 1 + cos ( ϕ ) ] ,
sin ( 2 S ^ ) 2 ( Y 1 ) sin ( S ^ ) = 2 p K S ^ ,
K = Q T G 1 Q
G i i = h i + h i + 1 ; G i 1 , i = G i , i 1 = h i 2
Q i i = 1 h i ; Q i , i + 1 = h i + h i + 1 h i h i + 1 ; Q i , i + 2 = 1 h i + 1
PSS 1 ( S ) = i = 1 N [ Y i S i ] 2 + p Ω [ S ( x ) ] 2 d x
PSS 2 ( S ) = ρ i = 1 N [ Y i S i ] 2 + Ω [ S ( x ) ] 2 d x
PSS 3 ( S ) = λ i = 1 N [ Y i S i ] 2 + ( 1 λ ) Ω [ S ( x ) ] 2 d x

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