Abstract

Unified interpretation for the real and pseudo moiré phenomena using the concept of biased and unbiased frequency pairs in the Fourier spectrum is given. Intensity modulations are responsible for pseudo moiré appearance in the image plane rather than average intensity variations dominating real moiré. Detection of pseudo moiré necessitates resolving superimposed structures in the image plane. In the case of the product type superimposition generating both real and pseudo moiré, our interpretation utilizes the Fourier domain information only. The moiré pattern characteristics such as an effective carrier, modulation and bias intensity distributions can be readily predicted. We corroborate them using two-dimensional continuous wavelet transform and fast adaptive bidimensional empirical mode decomposition methods as complementary image processing tools.

© 2011 OSA

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References

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  1. S. Kobayashi, Handbook on Experimental Mechanics, 2nd ed., (SEM, Bethel, 1993).
  2. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
  3. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66(2), 87–94 (1976).
    [CrossRef]
  4. I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
    [CrossRef]
  5. I. Amidror, The Theory of the Moiré Phenomenon (Springer-Verlag, London, 2009).
  6. O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. 64(10), 1287–1294 (1974).
    [CrossRef]
  7. O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. 65(6), 685–694 (1975).
    [CrossRef]
  8. K. Patorski, S. Yokozeki, and T. Suzuki, “„Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
    [CrossRef]
  9. R. Eschbach, “Generation of moiré by nonlinear transfer characteristics,” J. Opt. Soc. Am. A 5(11), 1828–1835 (1988).
    [CrossRef]
  10. Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
    [CrossRef]
  11. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45(34), 8722–8732 (2006).
    [CrossRef] [PubMed]
  12. K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49(19), 3640–3651 (2010).
    [CrossRef] [PubMed]
  13. K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt. 50(5), 773–781 (2011).
    [CrossRef] [PubMed]
  14. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt. 47(14), 2592–2598 (2008).
    [CrossRef] [PubMed]
  15. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
    [CrossRef] [PubMed]
  16. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry signals presenting low-modulated regions by means of the bidimensional empirical mode decomposition,” Appl. Opt. 50(5), 641–647 (2011).
    [CrossRef] [PubMed]
  17. M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50(28), 5513–5523 (2011).
    [CrossRef] [PubMed]
  18. S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 1313–1316 (2008).
  19. S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process. ID728356, 1–18 (2008).
    [CrossRef]
  20. N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
    [CrossRef]
  21. S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12(10), 701–704 (2005).
    [CrossRef]
  22. D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22(4), 469–483 (1996).
    [CrossRef]
  23. J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
    [CrossRef]
  24. X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2008).
    [CrossRef]

2011 (3)

2010 (1)

2009 (2)

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
[CrossRef] [PubMed]

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

2008 (3)

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process. ID728356, 1–18 (2008).
[CrossRef]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt. 47(14), 2592–2598 (2008).
[CrossRef] [PubMed]

X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2008).
[CrossRef]

2006 (2)

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

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

2005 (1)

S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12(10), 701–704 (2005).
[CrossRef]

2003 (1)

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

1998 (1)

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

1996 (1)

D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22(4), 469–483 (1996).
[CrossRef]

1988 (1)

1976 (2)

O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66(2), 87–94 (1976).
[CrossRef]

K. Patorski, S. Yokozeki, and T. Suzuki, “„Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

1975 (1)

1974 (1)

Adhami, R. R.

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process. ID728356, 1–18 (2008).
[CrossRef]

Amidror, I.

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

Barber, D.

D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22(4), 469–483 (1996).
[CrossRef]

Bernini, M. B.

Bhuiyan, S. M. A.

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process. ID728356, 1–18 (2008).
[CrossRef]

Bouaoune, Y.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

Bryngdahl, O.

Bunel, Ph.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

Burton, D. R.

Damerval, S.

S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12(10), 701–704 (2005).
[CrossRef]

Delechelle, E.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

Dobkin, D. P.

D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22(4), 469–483 (1996).
[CrossRef]

Eschbach, R.

Federico, A.

Gdeisat, M. A.

Guanlei, X.

X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2008).
[CrossRef]

Hersch, R. D.

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

Huang, N. E.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Huhdanpaa, H.

D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22(4), 469–483 (1996).
[CrossRef]

Kaufmann, G. H.

Khan, J. F.

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process. ID728356, 1–18 (2008).
[CrossRef]

Lalor, M. J.

Liu, H. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Long, S. R.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Ma, H.

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

Meignen, S.

S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12(10), 701–704 (2005).
[CrossRef]

Niang, O.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

Nunes, J. C.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

Patorski, K.

Perrier, V.

S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12(10), 701–704 (2005).
[CrossRef]

Pokorski, K.

Sheng, Z.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Shih, W. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Suzuki, T.

K. Patorski, S. Yokozeki, and T. Suzuki, “„Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Tung, C. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Wang, Z.

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

Wielgus, M.

Wu, M. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Xiaogang, X.

X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2008).
[CrossRef]

Xiatong, W.

X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2008).
[CrossRef]

Yen, N. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Yokozeki, S.

K. Patorski, S. Yokozeki, and T. Suzuki, “„Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Zeng, Q.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

ACM Trans. Math. Softw. (1)

D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22(4), 469–483 (1996).
[CrossRef]

Appl. Opt. (7)

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

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt. 47(14), 2592–2598 (2008).
[CrossRef] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
[CrossRef] [PubMed]

K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49(19), 3640–3651 (2010).
[CrossRef] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry signals presenting low-modulated regions by means of the bidimensional empirical mode decomposition,” Appl. Opt. 50(5), 641–647 (2011).
[CrossRef] [PubMed]

K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt. 50(5), 773–781 (2011).
[CrossRef] [PubMed]

M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50(28), 5513–5523 (2011).
[CrossRef] [PubMed]

EURASIP J. Adv. Signal Process. (1)

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process. ID728356, 1–18 (2008).
[CrossRef]

IEEE Signal Process. Lett. (1)

S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12(10), 701–704 (2005).
[CrossRef]

Image Vis. Comput. (1)

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[CrossRef]

J. Mod. Opt. (1)

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “„Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Opt. Eng. (1)

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

Pattern Recognit. (1)

X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2008).
[CrossRef]

Proc. R. Soc. Lond. A (1)

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998).
[CrossRef]

Other (4)

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 1313–1316 (2008).

I. Amidror, The Theory of the Moiré Phenomenon (Springer-Verlag, London, 2009).

S. Kobayashi, Handbook on Experimental Mechanics, 2nd ed., (SEM, Bethel, 1993).

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

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Figures (10)

Fig. 1
Fig. 1

Computer generated cosinusoidal gratings to generate moiré patterns; (a) reference grating common to all studied moiré images; (b) second grating used for multiplicative and additive superimposition to form real and pseudo moiré patterns, respectively; and c) half frequency second grating used for generating the product superimposition pseudo moiré.

Fig. 2
Fig. 2

Left column: (a) product type superimposition of cosinusoidal amplitude gratings of equal periods (real moiré); (b) product type superimposition of gratings of period ratio 2:1 (pseudo-moiré); (c) conventional additive superimposition of gratings of equal periods (pseudo moiré). Corresponding intensity distribution diagonal cross-sections are presented in the right column.

Fig. 3
Fig. 3

Fourier spectrum of the cases shown in Fig. 2. The dots represent the locations of fundamental harmonics of superimposed structures and the bias term (f1 and f2 denote spatial frequencies of the first and second grating, respectively; in (a) and (c) f1 = f2, in (b) f2 = f1/2). Small circles represent generated beat frequencies (in parentheses). Notations in parentheses indicate harmonics forming particular beat frequencies, not their coordinates in the frequency plane. For example, pseudo-moiré modulation bands in case (b) are generated by unbiased frequency pairs [ +f2; ( +f1, –f2)] and [-f2; (-f1, +f2)] contributing the lowest value beat frequencies, see text for details.

Fig. 4
Fig. 4

(a) CWT filtered real moiré image, (b) CWT calculated modulation map, (c) bias distribution (Gaussian filter convolution processing), and (d) diagonal cross-section of part (c).

Fig. 6
Fig. 6

(a) CWT filtered conventional additive moiré, (b) CWT calculated modulation map, (c) diagonal cross-section of part (b), and (d) bias distribution (Gaussian filter convolution processing).

Fig. 9
Fig. 9

Pseudo moiré carrier fringes obtained by decomposing BIMF3, Fig. 8(c), four times and taking the resulting BIMF2 as the starting image for next decomposition. The FABEMD OSFWtype2 algorithm was used. Compare the results with Fig. 5(a).

Fig. 5
Fig. 5

(a) CWT filtered pseudo moiré image, (b) CWT determined modulation map, (c) bias distribution (Gaussian filter convolution processing), and (d) diagonal cross-section of part (c).

Fig. 7
Fig. 7

First five BIMFs and residual part (first five BIMFs subtracted from real moiré) obtained by decomposing the real moiré pattern using the modified FABEMD (OSFWtype1) method.

Fig. 8
Fig. 8

First five BIMFs and residual part [understood as in Fig. 7(f)] obtained by decomposing the pseudo moiré pattern using the modified FABEMD (OSFWtype2) method.

Fig. 10
Fig. 10

First four BIMFs and the residual part [understood as in Fig. 7(f)] obtained by decomposing the conventional additive moiré pattern using the modified FABEMD (OSFWtype1) method.

Tables (3)

Tables Icon

Table 1 Decomposition parameter values used for processing the product type real moiré using the modified FABEMD OSFW type 1 method.

Tables Icon

Table 2 Decomposition parameter values used for processing the product type pseudo moiré using the modified FABEMD OSFW type 2 method.

Tables Icon

Table 3 Decomposition parameter values used for processing the additive type pseudo moiré using the modified FABEMD OSFW type 1 method.

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