Abstract

Hybrid single shot algorithm for accurate phase demodulation of complex fringe patterns is proposed. It employs empirical mode decomposition based adaptive fringe pattern enhancement (i.e., denoising, background removal and amplitude normalization) and subsequent boosted phase demodulation using 2D Hilbert spiral transform aided by the Principal Component Analysis method for novel, correct and accurate local fringe direction map calculation. Robustness to fringe pattern significant noise, uneven background and amplitude modulation as well as local fringe period and shape variations is corroborated by numerical simulations and experiments. Proposed automatic, adaptive, fast and comprehensive fringe analysis solution compares favorably with other previously reported techniques.

© 2016 Optical Society of America

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2016 (1)

B. Deepan, C. Quan, and C. J. Tay, “Determination of phase derivatives from a single fringe pattern using Teager-Hilbert-Huang transform,” Opt. Commun. 359, 162–170 (2016).
[Crossref]

2015 (4)

B. Deepan, C. Quan, and C. J. Tay, “Phase retrieval from a single fringe pattern using Teager-Hilbert- Huang transform,” Proc. SPIE 9302, 93020Q (2015).
[Crossref]

A. Kus, W. Krauze, and M. Kujawinska, “Active limited-angle tomographic phase microscope,” J. Biomed. Opt. 20(11), 111216 (2015).
[Crossref] [PubMed]

M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
[Crossref] [PubMed]

S. Dehaeck, Y. Tsoumpas, and P. Colinet, “Analyzing closed-fringe images using two-dimensional Fan wavelets,” Appl. Opt. 54(10), 2939–2952 (2015).
[Crossref] [PubMed]

2014 (6)

Z. Zhang and H. Guo, “Principal-vector-directed fringe-tracking technique,” Appl. Opt. 53(31), 7381–7393 (2014).
[Crossref] [PubMed]

H. Wang, Q. Kemao, R. Liang, H. Wang, M. Zhao, and X. He, “Oriented boundary padding for iterative and oriented fringe pattern denoising techniques,” Sig. Proc. 102(9), 112–121 (2014).
[Crossref]

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
[Crossref]

M. Trusiak, M. Wielgus, and 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. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

2013 (5)

2012 (4)

2011 (3)

2010 (3)

2009 (5)

H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. 34(8), 1141–1143 (2009).
[Crossref] [PubMed]

J. Villa, J. A. Quiroga, and I. De la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34(11), 1741–1743 (2009).
[Crossref] [PubMed]

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009).
[Crossref] [PubMed]

X. Yang, Q. Yu, and S. Fu, “Determination of skeleton and sign map for phase obtaining from a single ESPI image,” Opt. Commun. 282(12), 2301–2306 (2009).
[Crossref]

F. Zhang, W. Liu, J. Wang, Y. Zhu, and L. Xia, “Anisotropic partial differential equation noise-reduction algorithm based on fringe feature for ESPI,” Opt. Commun. 282(12), 2318–2326 (2009).
[Crossref]

2008 (4)

2007 (5)

Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007).
[Crossref] [PubMed]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46(21), 4613–4624 (2007).

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274(2), 286–292 (2007).
[Crossref]

X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273(1), 60–66 (2007).
[Crossref]

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

2006 (1)

2005 (7)

2004 (5)

2003 (1)

2002 (5)

2001 (4)

1999 (2)

1998 (3)

Q. Yu, X. Liu, and X. Sun, “Generalized spin filtering and an improved derivative-sign binary image method for the extraction of fringe skeletons,” Appl. Opt. 37(20), 4504–4509 (1998).
[Crossref] [PubMed]

J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15(6), 1536–1544 (1998).
[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. A 454(1971), 903–995 (1998).
[Crossref]

1997 (1)

1996 (2)

Q. Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 35(20), 3783–3790 (1996).
[Crossref] [PubMed]

K. Creath and J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24(5–6), 365–379 (1996).
[Crossref]

1995 (1)

M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34(8), 2459–2466 (1995).
[Crossref]

1994 (2)

1991 (1)

M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[Crossref]

1988 (1)

1986 (1)

1982 (1)

Andresen, K.

Arnold, J. F.

Arola, D. D.

D. Zhang, M. Ma, and D. D. Arola, “Fringe skeletonizing using an improved derivative sign binary method,” Opt. Lasers Eng. 37(1), 51–62 (2002).
[Crossref]

Baird, J. P.

Barnes, T. H.

Bone, D. J.

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Brémand, F.

Burton, D. R.

Cai, Y.

Carazo, J. M.

Chang, Y.

Chen, Z.

Colinet, P.

Creath, K.

K. Creath and J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24(5–6), 365–379 (1996).
[Crossref]

Crespo, D.

Cuevas, F.

Cuevas, F. J.

Cui, X.

D’Acquisto, L.

A. M. Siddiolo and L. D’Acquisto, “A direction/orientation- based method for shape measurement by shadow Moire,” IEEE Trans. Instrum. Meas. 57(4), 843–849 (2008).
[Crossref]

Dalmau-Cedeño, O. S.

de la Rosa, I.

Debnath, S. K.

Deepan, B.

B. Deepan, C. Quan, and C. J. Tay, “Determination of phase derivatives from a single fringe pattern using Teager-Hilbert-Huang transform,” Opt. Commun. 359, 162–170 (2016).
[Crossref]

B. Deepan, C. Quan, and C. J. Tay, “Phase retrieval from a single fringe pattern using Teager-Hilbert- Huang transform,” Proc. SPIE 9302, 93020Q (2015).
[Crossref]

Dehaeck, S.

Du, Y.

Estrada, J. C.

Feng, G.

Fu, S.

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Q. Yu, X. Yang, S. Fu, and X. Sun, “Two improved algorithms with which to obtain contoured windows for fringe patterns generated by electronic speckle-pattern interferometry,” Appl. Opt. 44(33), 7050–7054 (2005).
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H. Wang, Q. Kemao, R. Liang, H. Wang, M. Zhao, and X. He, “Oriented boundary padding for iterative and oriented fringe pattern denoising techniques,” Sig. Proc. 102(9), 112–121 (2014).
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H. Wang, Q. Kemao, R. Liang, H. Wang, M. Zhao, and X. He, “Oriented boundary padding for iterative and oriented fringe pattern denoising techniques,” Sig. Proc. 102(9), 112–121 (2014).
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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).
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F. Zhang, W. Liu, J. Wang, Y. Zhu, and L. Xia, “Anisotropic partial differential equation noise-reduction algorithm based on fringe feature for ESPI,” Opt. Commun. 282(12), 2318–2326 (2009).
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M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
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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).
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M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
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K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
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M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
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M. Trusiak, M. Wielgus, and 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).
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K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

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M. Trusiak, K. Patorski, and 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).
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F. Zhang, W. Liu, J. Wang, Y. Zhu, and L. Xia, “Anisotropic partial differential equation noise-reduction algorithm based on fringe feature for ESPI,” Opt. Commun. 282(12), 2318–2326 (2009).
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Yang, X.

X. Yang, Q. Yu, and S. Fu, “Determination of skeleton and sign map for phase obtaining from a single ESPI image,” Opt. Commun. 282(12), 2301–2306 (2009).
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X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273(1), 60–66 (2007).
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Q. Yu, S. Fu, X. Liu, X. Yang, and X. Sun, “Single-phase-step method with contoured correlation fringe patterns for ESPI,” Opt. Express 12(20), 4980–4985 (2004).
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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).
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X. Yang, Q. Yu, and S. Fu, “Determination of skeleton and sign map for phase obtaining from a single ESPI image,” Opt. Commun. 282(12), 2301–2306 (2009).
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X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273(1), 60–66 (2007).
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X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274(2), 286–292 (2007).
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Q. Yu, X. Yang, S. Fu, and X. Sun, “Two improved algorithms with which to obtain contoured windows for fringe patterns generated by electronic speckle-pattern interferometry,” Appl. Opt. 44(33), 7050–7054 (2005).
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Q. Yu, S. Fu, X. Yang, X. Sun, and X. Liu, “Extraction of phase field from a single contoured correlation fringe pattern of ESPI,” Opt. Express 12(1), 75–83 (2004).
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Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41(14), 2650–2654 (2002).
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Q. Yu, X. Liu, and X. Sun, “Generalized spin filtering and an improved derivative-sign binary image method for the extraction of fringe skeletons,” Appl. Opt. 37(20), 4504–4509 (1998).
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Q. Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 35(20), 3783–3790 (1996).
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Q. Yu, X. Liu, and K. Andresen, “New spin filters for interferometric fringe patterns and grating patterns,” Appl. Opt. 33(17), 3705–3711 (1994).
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Q. Yu and K. Andresen, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33(29), 6873–6878 (1994).
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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).
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Zhang, D.

D. Zhang, M. Ma, and D. D. Arola, “Fringe skeletonizing using an improved derivative sign binary method,” Opt. Lasers Eng. 37(1), 51–62 (2002).
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Zhang, F.

F. Zhang, W. Liu, J. Wang, Y. Zhu, and L. Xia, “Anisotropic partial differential equation noise-reduction algorithm based on fringe feature for ESPI,” Opt. Commun. 282(12), 2318–2326 (2009).
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Zhang, Z.

Zhao, M.

H. Wang, Q. Kemao, R. Liang, H. Wang, M. Zhao, and X. He, “Oriented boundary padding for iterative and oriented fringe pattern denoising techniques,” Sig. Proc. 102(9), 112–121 (2014).
[Crossref]

Zhou, D.

Zhou, S.

Zhou, X.

Zhu, X.

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
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X. Zhu, Z. Chen, and C. Tang, “Variational image decomposition for automatic background and noise removal of fringe patterns,” Opt. Lett. 38(3), 275–277 (2013).
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Zhu, Y.

F. Zhang, W. Liu, J. Wang, Y. Zhu, and L. Xia, “Anisotropic partial differential equation noise-reduction algorithm based on fringe feature for ESPI,” Opt. Commun. 282(12), 2318–2326 (2009).
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Appl. Opt. (21)

J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests,” Appl. Opt. 40(13), 2081–2088 (2001).
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Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
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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. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997).
[Crossref] [PubMed]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46(21), 4613–4624 (2007).

Q. Yu, “Spin filtering processes and automatic extraction of fringe centerlines in digitial interferometric patterns,” Appl. Opt. 27(18), 3782–3784 (1988).
[Crossref] [PubMed]

Q. Yu, X. Liu, and K. Andresen, “New spin filters for interferometric fringe patterns and grating patterns,” Appl. Opt. 33(17), 3705–3711 (1994).
[Crossref] [PubMed]

Q. Yu and K. Andresen, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33(29), 6873–6878 (1994).
[Crossref] [PubMed]

Q. Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 35(20), 3783–3790 (1996).
[Crossref] [PubMed]

Q. Yu, X. Liu, and X. Sun, “Generalized spin filtering and an improved derivative-sign binary image method for the extraction of fringe skeletons,” Appl. Opt. 37(20), 4504–4509 (1998).
[Crossref] [PubMed]

Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41(14), 2650–2654 (2002).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Synthetic fringe pattern (a), fringe orientation map derived directly from the PCA algorithm (b), continuous regions separation result obtained from Fig. 1(b) by segmentation process (c), distribution of relative difference of two eigenvalues of covariance matrix (d).

Fig. 2
Fig. 2

Horizontal (a) and vertical (b) fringe orientation angles determined basing on the preliminary orientation map Fig. 1(b); regions indicating potential zero-phase-gradient spots (“orientation vortices”) (c).

Fig. 3
Fig. 3

Region (white pixels) indicating both horizontal and vertical fringe orientations (a), local distribution of relative eigenvalues difference (b) and preliminary orientation map (c); five starting pixels are marked in blue whereas final outcome of iterative algorithm in terms of local minimum of relative eigenvalue difference is marked in green. In this case each starting pixel produced the same outcome (minimum).

Fig. 4
Fig. 4

The flow-chart of minima detection algorithm.

Fig. 5
Fig. 5

Fringe orientation angle variability analysis along the sides of the rectangle (pixels marked in red) encapsulating the minima of relative eigenvalues difference (in green).

Fig. 6
Fig. 6

Continuous (in black and white) orientation angle sign regions separated by masks (in magenta) obtained from preliminary orientation map (Fig. 1(b)) (a), fringe direction map calculated using proposed PCA based conditioned region matching algorithm (b), simulated ground truth fringe direction map (c).

Fig. 7
Fig. 7

Erroneous fringe direction map obtained for fringes in Fig. 1(a) using the skeletoning algorithm [58] with masked pixels marked in gray - mask calculated for threshold value 0.75 (a), mask (black areas) obtained for threshold value 0.97 enabling the algorithm to avoid passing through regions marked by blue ellipses (b), the biggest continuous unmasked region of Fig. 7(b) selected for orientation unwrapping by the algorithm presented in [58] (c), orientation (d) and direction (e) map obtained by combined gradient and plane-fit method [47].

Fig. 8
Fig. 8

Fringe pattern enhancement results obtained by the Gaussian (a), ASR (b) and the proposed ASRm method (c); fringe local direction map obtained for the enhanced patterns by combined method (d), skeletoning (e) and the proposed PCA method (f). In other words results of A1 algorithm are depicted in (a) and (d), A2 in (b) and (e), HHT-PCA in (c) and (f). It is worth to note the direction map estimation improvement made after fringe enhancement in terms of QI: (a, d) 0.4047 (0.3772), (b, e) 0.4790 (0.4567), (c, f) 0.6461 (0.5220).

Fig. 9
Fig. 9

Erroneous phase map obtained using direction-less Hilbert transform (a), ideal simulated wrapped phase map (b), wrapped phase maps obtained using the proposed HHT-PCA method (c), A1 (d) and A2 (e); unwrapped phase distribution with linear carrier term removed calculated by A1 (f) and the proposed HHT-PCA scheme (g); ideal unwrapped phase distribution (h) and the phase error of the HHT-PCA (i).

Fig. 10
Fig. 10

Second simulated fringe pattern (a), enhancement results obtained using the Gaussian filtering (b), ASR (c) and ASRm (d) methods; phase error maps for unwrapped phase distributions obtained using the full A1 (e), A2 (f), and HHT-PCA (g) processing.

Fig. 11
Fig. 11

Experimental fringe pattern with closed fringes (a), reference fringes (b), enhancement results of A1 (c), A2 (d) and the proposes HHT-PCA (e) methods; unwrapped phase error maps of A1 (f), A2 (g) and the proposed HHT-PCA (h) algorithms calculated using reference unwrapped phase distribution (i). Reference phase values were obtained using 9-frame AIA algorithm with adaptive low-pass filtering for noise reduction. Note that reference fringes (b) are simply generated as cosine of unwrapped reference phase (i).

Fig. 12
Fig. 12

Experimental fringe pattern with closed fringes (a), reference fringes (b), enhancement results of A1 (c), A2 (d) and the proposed HHT-PCA (e) methods; unwrapped phase error maps of A1 (f), A2 (g) and the proposed HHT-PCA (h) algorithms calculated using reference unwrapped phase distribution (i). Reference phase values were obtained using 5-frame AIA reference algorithm with adaptive band-pass filtering for noise and background reduction. Note that reference fringes (b) are simply generated as cosine of unwrapped reference phase (i).

Tables (2)

Tables Icon

Table 1 Summary of the quantitative evaluation using synthetic fringe patterns.

Tables Icon

Table 2 Summary of the quantitative evaluation using experimental fringe patterns.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

C = [ σ x 2 σ x y σ x y σ y 2 ] ,
Δ = λ 1 λ 2 λ 1 + λ 2 .
θ d i f f = | θ 1 θ 2 |
β = R M ( θ ) ,
β = W R [ U W R ( 2 θ ) / 2 ] ,
I = a + b cos ( φ ) + n ,
I F = b cos ( φ ) ,
S P F ( ζ 1 , ζ 2 ) = ζ 1 + i ζ 2 ζ 1 2 + ζ 2 2 ,
Q I F = i exp ( i β ) F 1 { S P F ( ζ 1 , ζ 2 ) F [ I F ( x , y ) ] } = b sin ( φ ) ,
A F P = I F + i Q I F ,
b = a b s ( A F P ) ,
φ = a n g l e ( A F P ) .
I = i = 1 m B I M F i + r e s .
n = i = 1 k B I M F i , a = i = l + 1 m B I M F i + r e s , b cos ( φ ) = i = k + 1 l B I M F i .
I F ( x , y ) = B I M F H M ( x , y ) .
I F = i = 2 m k b B I M F i B I M F i ,

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