## Fourier domain interpretation of real and pseudo-moiré phenomena |

Optics Express, Vol. 19, Issue 27, pp. 26065-26078 (2011)

http://dx.doi.org/10.1364/OE.19.026065

Acrobat PDF (14823 KB)

### 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

## 1. Introduction

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

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

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]

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]

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]

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]

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]

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]

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

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]

## 2. Heuristic interpretation

9. R. Eschbach, “Generation of moiré by nonlinear transfer characteristics,” J. Opt. Soc. Am. A **5**(11), 1828–1835 (1988). [CrossRef]

_{2}; ( +f

_{1}, –f

_{2})] comprises plus first harmonic f

_{2}of the second (half frequency) grating and the beat frequency formed by +1 harmonic f

_{1}of the first (denser) grating and −1 harmonic -f

_{2}of the second grating. Second pair is denoted as [-f

_{2}; (-f

_{1}, +f

_{2})] and comprises relevant frequency –f

_{2}of the second grating and the beat frequency (-f

_{1}, +f

_{2}). Each frequency pair generates a difference frequency determining the period and orientation of modulation bands. Whole information, therefore, is contained in the Fourier domain plane. Note that the zero order (bias) frequency, common to both gratings, does not appear along the direction passing through just mentioned frequency pairs. Correspondingly, we can interpret pseudo-moiré fringes (modulation bands) as generated by unbiased frequency pairs forming the lowest beat frequency in the moiré configuration under consideration.

9. R. Eschbach, “Generation of moiré by nonlinear transfer characteristics,” J. Opt. Soc. Am. A **5**(11), 1828–1835 (1988). [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]

_{2}; ( +f

_{1},-f

_{2})] is perpendicular to two mutually parallel directions formed by connecting locations of the frequencies in the two following pairs: [ +f

_{2}; (-f

_{1},-f

_{2})] and [-f

_{2}; ( +f

_{1}, + f

_{2})]. Additionally, taking into consideration the above discussion on formation of modulation bands we conclude that it is necessary and enough to resolve both superimposed gratings with frequencies f

_{1}and f

_{2}.

**56**(9), 1103–1118 (2009). [CrossRef]

**56**(9), 1103–1118 (2009). [CrossRef]

_{2}; ( + f

_{1}, –f

_{2})] and [-f

_{2}; (-f

_{1}, + f

_{2})] should be smaller than frequencies of both component gratings. Last statement is universal for all superimposition type moiré phenomena.

## 3. Numerical experiments

### 3.1 Continuous wavelet transform processing

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

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]

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]

- 1. 2D CWT ridge is extracted for every point of the analyzed pattern (coefficient with highest absolute value is selected). Thus two dimensional complex ridge map is obtained.
- 2. CWT image filtration. 2D CWT of the fringe pattern is computed using a complex, directional wavelet (in our case, modified Morlet wavelet) and selected values of scale and angle parameters. Each image point real value of the ridge map is computed, giving a filtered image. Adequate selection of parameters ensures noise filtration and bias suppression since CWT acts as a bandpass filter.
- 3. CWT image modulation determination. This step processes analogically to the previous one, but, instead of the real value of the ridge map, the absolute value is taken. Computing it for every image point provides the fringe pattern modulation distribution.
- 4. Background distribution is determined by Gaussian filter convolution averaging.

*m*controlling the Morlet wavelet Gaussian window width (with no influence on the plane wave frequency) is crucial for the fringe pattern modulation determination [10

**45**(4), 045601 (2006). [CrossRef]

**50**(5), 773–781 (2011). [CrossRef] [PubMed]

*m*= 1 was used. The CWT normalization parameter was set to 1.98 to ensure uniform modulation reference for the whole image as suggested in [12

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]

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]

**49**(19), 3640–3651 (2010). [CrossRef] [PubMed]

**50**(5), 773–781 (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]

### 3.2 Processing using FABEMD approach

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]

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]

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]

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]

**66**(2), 87–94 (1976). [CrossRef]

**50**(28), 5513–5523 (2011). [CrossRef] [PubMed]

_{1},-f

_{2}); ( +f

_{1}, +f

_{2})], see Fig. 3(b). These carrier fringes are coarser than carrier fringes in the case of real moiré, Figs, 2(a) and 3(a). The modulation bands are formed by frequency beating in three groups: [(-f

_{1},-f

_{2}); -f

_{1}; (-f

_{1}, +f

_{2})]; [-f

_{2}, 0, +f

_{2}] and [( +f

_{1},−f

_{2}), + f

_{1}, ( +f

_{1}, +f

_{2})]. In all these three groups beating frequencies lie along mutually parallel directions and are mutually equidistant. Although exemplary BIMFs just discussed could be obtained by sophisticated Fourier or wavelet band pass filtering, the adaptive FABEMD approach is simpler and more straightforward. Complementary character of CWT and FABEMD methods can be readily apprehended.

## 4. Conclusions

## Acknowledgements

## References and links

1. | S. Kobayashi, |

2. | K. Patorski, |

3. | O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. |

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. |

5. | I. Amidror, |

6. | O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. |

7. | O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. |

8. | K. Patorski, S. Yokozeki, and T. Suzuki, “„Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. |

9. | R. Eschbach, “Generation of moiré by nonlinear transfer characteristics,” J. Opt. Soc. Am. A |

10. | Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. |

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. |

12. | K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. |

13. | K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt. |

14. | M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt. |

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. |

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. |

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. |

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. |

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 |

21. | S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. |

22. | D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. |

23. | J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. |

24. | X. Guanlei, W. Xiatong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. |

**OCIS Codes**

(100.2650) Image processing : Fringe analysis

(120.4120) Instrumentation, measurement, and metrology : Moire' techniques

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 30, 2011

Revised Manuscript: October 19, 2011

Manuscript Accepted: October 23, 2011

Published: December 7, 2011

**Citation**

Krzysztof Patorski, Krzysztof Pokorski, and Maciej Trusiak, "Fourier domain interpretation of real and pseudo-moiré phenomena," Opt. Express **19**, 26065-26078 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26065

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### References

- S. Kobayashi, Handbook on Experimental Mechanics, 2nd ed., (SEM, Bethel, 1993).
- K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
- O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am.66(2), 87–94 (1976). [CrossRef]
- 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]
- I. Amidror, The Theory of the Moiré Phenomenon (Springer-Verlag, London, 2009).
- O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am.64(10), 1287–1294 (1974). [CrossRef]
- O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am.65(6), 685–694 (1975). [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]
- R. Eschbach, “Generation of moiré by nonlinear transfer characteristics,” J. Opt. Soc. Am. A5(11), 1828–1835 (1988). [CrossRef]
- Z. Wang and 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, 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]
- 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]
- 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. 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]
- 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]
- 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]
- 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).
- 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]
- 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. A454(1971), 903–995 (1998). [CrossRef]
- S. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005). [CrossRef]
- D. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996). [CrossRef]
- 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]
- 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]

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