Abstract

The windowed Fourier filtering (WFF), defined as a thresholding operation in the windowed Fourier transform (WFT) domain, is a successful method for denoising a phase map and analyzing a fringe pattern. However, it has some shortcomings, such as extremely high redundancy, which results in high computational cost, and difficulty in selecting an appropriate window size. In this paper, an extension of WFF for denoising a wrapped-phase map is proposed. It is formulated as a convex optimization problem using Gabor frames instead of WFT. Two Gabor frames with differently sized windows are used simultaneously so that the above-mentioned issues are resolved. In addition, a differential operator is combined with a Gabor frame in order to preserve discontinuity of the underlying phase map better. Some numerical experiments demonstrate that the proposed method is able to reconstruct a wrapped-phase map, even for a severely contaminated situation.

© 2016 Optical Society of America

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References

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2016 (1)

D. Kitahara and I. Yamada, “Algebraic phase unwrapping based on two-dimensional spline smoothing over triangles,” IEEE Trans. Signal Process. 64, 2103–2118 (2016).
[Crossref]

2015 (3)

Q. Kemao, “Applications of windowed Fourier fringe analysis in optical measurement: a review,” Opt. Lasers Eng. 66, 67–73 (2015).
[Crossref]

Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An accelerated linearized alternating direction method of multipliers,” SIAM J. Imaging Sci. 8, 644–681 (2015).
[Crossref]

S. Ono and I. Yamada, “Signal recovery with certain involved convex data-fidelity constraints,” IEEE Trans. Signal Process. 63, 6149–6163 (2015).
[Crossref]

2014 (2)

2012 (5)

2011 (2)

S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regularizations with discrete l1-type functionals,” Commun. Math. Sci. 9, 797–827 (2011).
[Crossref]

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

2010 (2)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

C. Quan, H. Niu, and C. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
[Crossref]

2008 (4)

2007 (7)

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
[Crossref]

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

J. M. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
[Crossref]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193–1195 (2007).
[Crossref]

M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis in signal priors,” Inv. Probl. 23, 947–968 (2007).
[Crossref]

2006 (1)

H. Zou, “The adaptive lasso and its oracle properties,” J. Am. Stat. Assoc. 101, 1418–1429 (2006).
[Crossref]

2005 (1)

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[Crossref]

2004 (2)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20, 73–87 (2004).
[Crossref]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[Crossref]

2002 (1)

J. M. Bioucas-Dias and J. M. N. Leitao, “The F020F05AπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408–422 (2002).
[Crossref]

1999 (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[Crossref]

1997 (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[Crossref]

1993 (1)

E. Wesfreid and M. V. Wickerhauser, “Adapted local trigonometric transforms and speech processing,” IEEE Trans. Signal Process. 41, 3596–3600 (1993).
[Crossref]

1982 (1)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[Crossref]

Astola, J.

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (phasela) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[Crossref]

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[Crossref]

Asundi, A.

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[Crossref]

Bach, F.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Balazs, P.

P. L. Søndergaard, B. Torrésani, and P. Balazs, “The linear time frequency analysis toolbox,” Int. J. Wavelets Multiresolut. Inf. Process. 10, 1250032 (2012).
[Crossref]

Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound, Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, eds. (Springer, 2014), pp. 419–442.

Barbastathis, G.

Bauschke, H. H.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, 2011).

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[Crossref]

Bioucas-Dias, J.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

J. M. Bioucas-Dias and J. M. N. Leitao, “The F020F05AπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408–422 (2002).
[Crossref]

Boyd, S.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 127–239 (2014).
[Crossref]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[Crossref]

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[Crossref]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20, 73–87 (2004).
[Crossref]

Chartrand, R.

R. Chartrand, “Shrinkage mappings and their induced penalty functions,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2014), pp. 1026–1029.

Chen, Y.

Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An accelerated linearized alternating direction method of multipliers,” SIAM J. Imaging Sci. 8, 644–681 (2015).
[Crossref]

Chen, Z.

Christensen, O.

O. Christensen, Frames and Bases: An Introductory Course (Springer, 2008).

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Combettes, P. L.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, 2011).

P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, eds. (Springer, 2011), pp. 185–212.

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Eduardo Pasiliao, J.

Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An accelerated linearized alternating direction method of multipliers,” SIAM J. Imaging Sci. 8, 644–681 (2015).
[Crossref]

Egiazarian, K.

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (phasela) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[Crossref]

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[Crossref]

Elad, M.

M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis in signal priors,” Inv. Probl. 23, 947–968 (2007).
[Crossref]

Estrada, J. C.

Feng, L.

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[Crossref]

Gao, W.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

Holighaus, N.

Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound, Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, eds. (Springer, 2014), pp. 419–442.

Huang, H. Y. H.

Itoh, K.

Jenatton, R.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Katkovnik, V.

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (phasela) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[Crossref]

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[Crossref]

Kemao, Q.

Q. Kemao, “Applications of windowed Fourier fringe analysis in optical measurement: a review,” Opt. Lasers Eng. 66, 67–73 (2015).
[Crossref]

Q. Kemao, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420-5428 (2008).
[Crossref]

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408–5419 (2008).
[Crossref]

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

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
[Crossref]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
[Crossref]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193–1195 (2007).
[Crossref]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[Crossref]

Kitahara, D.

D. Kitahara and I. Yamada, “Algebraic phase unwrapping based on two-dimensional spline smoothing over triangles,” IEEE Trans. Signal Process. 64, 2103–2118 (2016).
[Crossref]

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing (ICASSP) (IEEE, 2015), pp. 3871–3875.

Lan, G.

Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An accelerated linearized alternating direction method of multipliers,” SIAM J. Imaging Sci. 8, 644–681 (2015).
[Crossref]

Leitao, J. M. N.

J. M. Bioucas-Dias and J. M. N. Leitao, “The F020F05AπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408–422 (2002).
[Crossref]

Li, Y.

Liu, Y.

Lo, Y.-L.

Mairal, J.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Milanfar, P.

M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis in signal priors,” Inv. Probl. 23, 947–968 (2007).
[Crossref]

Nam, L. T. H.

Navarro, M. A.

Nishiyama, S.

Niu, H.

C. Quan, H. Niu, and C. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
[Crossref]

Obozinski, G.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
[Crossref]

Ono, S.

S. Ono and I. Yamada, “Signal recovery with certain involved convex data-fidelity constraints,” IEEE Trans. Signal Process. 63, 6149–6163 (2015).
[Crossref]

Ouyang, Y.

Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An accelerated linearized alternating direction method of multipliers,” SIAM J. Imaging Sci. 8, 644–681 (2015).
[Crossref]

Parikh, N.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 127–239 (2014).
[Crossref]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Pesquet, J.-C.

P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, eds. (Springer, 2011), pp. 185–212.

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
[Crossref]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Pruša, Z.

Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound, Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, eds. (Springer, 2014), pp. 419–442.

Qian, K.

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[Crossref]

Quan, C.

C. Quan, H. Niu, and C. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
[Crossref]

Quiroga, J. A.

Rubinstein, R.

M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis in signal priors,” Inv. Probl. 23, 947–968 (2007).
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Servin, M.

Setzer, S.

S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regularizations with discrete l1-type functionals,” Commun. Math. Sci. 9, 797–827 (2011).
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Søndergaard, P. L.

P. L. Søndergaard, B. Torrésani, and P. Balazs, “The linear time frequency analysis toolbox,” Int. J. Wavelets Multiresolut. Inf. Process. 10, 1250032 (2012).
[Crossref]

Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound, Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, eds. (Springer, 2014), pp. 419–442.

Soon, S. H.

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
[Crossref]

K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[Crossref]

Steidl, G.

S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regularizations with discrete l1-type functionals,” Commun. Math. Sci. 9, 797–827 (2011).
[Crossref]

Tay, C.

C. Quan, H. Niu, and C. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
[Crossref]

Teuber, T.

S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regularizations with discrete l1-type functionals,” Commun. Math. Sci. 9, 797–827 (2011).
[Crossref]

Tian, L.

Tomioka, S.

Torrésani, B.

P. L. Søndergaard, B. Torrésani, and P. Balazs, “The linear time frequency analysis toolbox,” Int. J. Wavelets Multiresolut. Inf. Process. 10, 1250032 (2012).
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Valadao, G.

J. M. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
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Vargas, J.

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
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Wang, H.

Weng, J.-F.

Wesfreid, E.

E. Wesfreid and M. V. Wickerhauser, “Adapted local trigonometric transforms and speech processing,” IEEE Trans. Signal Process. 41, 3596–3600 (1993).
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Wickerhauser, M. V.

E. Wesfreid and M. V. Wickerhauser, “Adapted local trigonometric transforms and speech processing,” IEEE Trans. Signal Process. 41, 3596–3600 (1993).
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Wiesmeyr, C.

Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound, Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, eds. (Springer, 2014), pp. 419–442.

Xie, X.

Yamada, I.

D. Kitahara and I. Yamada, “Algebraic phase unwrapping based on two-dimensional spline smoothing over triangles,” IEEE Trans. Signal Process. 64, 2103–2118 (2016).
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S. Ono and I. Yamada, “Signal recovery with certain involved convex data-fidelity constraints,” IEEE Trans. Signal Process. 63, 6149–6163 (2015).
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D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing (ICASSP) (IEEE, 2015), pp. 3871–3875.

Yamagishi, M.

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing (ICASSP) (IEEE, 2015), pp. 3871–3875.

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R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
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H. Zou, “The adaptive lasso and its oracle properties,” J. Am. Stat. Assoc. 101, 1418–1429 (2006).
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Appl. Opt. (8)

Commun. Math. Sci. (1)

S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regularizations with discrete l1-type functionals,” Commun. Math. Sci. 9, 797–827 (2011).
[Crossref]

Found. Trends Mach. Learn. (2)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
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F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization with sparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106 (2011).
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N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 127–239 (2014).
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IEEE Trans. Geosci. Remote Sens. (1)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

IEEE Trans. Image Process. (3)

J. M. Bioucas-Dias and J. M. N. Leitao, “The F020F05AπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408–422 (2002).
[Crossref]

J. M. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (phasela) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[Crossref]

IEEE Trans. Signal Process. (3)

D. Kitahara and I. Yamada, “Algebraic phase unwrapping based on two-dimensional spline smoothing over triangles,” IEEE Trans. Signal Process. 64, 2103–2118 (2016).
[Crossref]

E. Wesfreid and M. V. Wickerhauser, “Adapted local trigonometric transforms and speech processing,” IEEE Trans. Signal Process. 41, 3596–3600 (1993).
[Crossref]

S. Ono and I. Yamada, “Signal recovery with certain involved convex data-fidelity constraints,” IEEE Trans. Signal Process. 63, 6149–6163 (2015).
[Crossref]

Int. J. Wavelets Multiresolut. Inf. Process. (1)

P. L. Søndergaard, B. Torrésani, and P. Balazs, “The linear time frequency analysis toolbox,” Int. J. Wavelets Multiresolut. Inf. Process. 10, 1250032 (2012).
[Crossref]

Inv. Probl. (1)

M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis in signal priors,” Inv. Probl. 23, 947–968 (2007).
[Crossref]

J. Am. Stat. Assoc. (1)

H. Zou, “The adaptive lasso and its oracle properties,” J. Am. Stat. Assoc. 101, 1418–1429 (2006).
[Crossref]

J. Math. Imaging Vis. (1)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20, 73–87 (2004).
[Crossref]

Opt. Commun. (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
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Opt. Eng. (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466–2472 (1997).
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K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrapping approach based on filtering by windowed Fourier transform,” Opt. Laser Technol. 37, 458–462 (2005).
[Crossref]

C. Quan, H. Niu, and C. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
[Crossref]

Opt. Lasers Eng. (4)

Q. Kemao, “Applications of windowed Fourier fringe analysis in optical measurement: a review,” Opt. Lasers Eng. 66, 67–73 (2015).
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Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
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Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193–1195 (2007).
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Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
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Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An accelerated linearized alternating direction method of multipliers,” SIAM J. Imaging Sci. 8, 644–681 (2015).
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Other (9)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

R. Chartrand, “Shrinkage mappings and their induced penalty functions,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2014), pp. 1026–1029.

O. Christensen, Frames and Bases: An Introductory Course (Springer, 2008).

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, 2011).

P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, eds. (Springer, 2011), pp. 185–212.

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing (ICASSP) (IEEE, 2015), pp. 3871–3875.

Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P. Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound, Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet, and S. Ystad, eds. (Springer, 2014), pp. 419–442.

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis” (2009) [Online]. Available: http://www.mathworks.com/matlabcentral/fileexchange/24852 .

J. Bioucas-Dias, “Code” (2012) [Online] Available: http://www.lx.it.pt/~bioucas/code.htm .

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Figures (6)

Fig. 1.
Fig. 1. Measured computational time of Gabor frame representation for an image of 64 × 64 pixels. It includes both forward and inverse transform as in Eq. (13). For reference, computational time of two WFT functions implemented by Kemao (wft2 [20] and wft2f [42]) are also shown as red dots.
Fig. 2.
Fig. 2. Comparison of denoising ability of hard- and soft-thresholding operators combined with Gabor frames as in Eq. (14). Noisy 100 × 100 image of the Gaussian function in Eq. (23) ( n o i s e l e v e l σ = 0.5 ) in (a) is used as test data. The channel number [how many frequency components are used for each direction (vertical and horizontal)] and shifting step (how many pixels are shifted to compute adjacent window) of a Gabor frame are adjusted to control redundancy shown in (b). Their computational times are depicted in (c). Second row is the result for hard-thresholding, while third row is for soft-thresholding. The results for lower left part in (b) are omitted because their redundancies are less than or equal to 1, which means they are not a frame [a forward and inverse transform pair cannot reconstruct a function as in Eq. (13)]. (d) and (g) are ISNR obtained by varying the threshold λ in Eqs. (5) and (9), where each line corresponds to each pixel in (b). The maximum values of each line in (d) and (g), which are the best achievable ISNR for (a), are illustrated in (e), (f), and (h), (i), respectively.
Fig. 3.
Fig. 3. Window functions used for the proposed method. The canonical tight windows g t 1 and g t 2 are illustrated in (a) and (b), respectively. For comparison, the Gaussian window used for calculating WFF is also depicted in (c); only part of it is shown in (c) for visual convenience since its support is not compact. These functions are symmetric (rotating the 25 × 25 pixel plane 90 deg does not change their appearance).
Fig. 4.
Fig. 4. ISNR for several noise levels. (a) is the results for WFF using soft-thresholding. (b) is for PEARLS, whose tuning parameter Γ is described in [14], and (c) is for the proposed method. Each line corresponds to testing data with different noise levels: σ = 0.25 (blue), 0.5 (red), 0.75 (yellow), and 1 (green). Some illustrative examples of these data are shown in Figs. 5 and 6.
Fig. 5.
Fig. 5. Some examples of the results for mildly contaminated data. The test data are obtained by replacing a quarter part of Eq. (23) with zero. Each column shows (from left to right) noisy data to be denoised, results for WFF, results for PEARLS, and results for the proposed method. Each row shows (from top to bottom) wrapped-phase maps, error of wrapped phase, and corresponding unwrapped phase obtained by PUMA algorithm [4] for noise levels σ = 0.25 , 0.5. The near side of 3D graphs of error and unwrapped phases corresponds to the upper left corner of the wrapped-phase maps. The parameters used for each method are listed in Table 1.
Fig. 6.
Fig. 6. Some examples of the results for severely contaminated data. The test data is obtained by replacing a quarter part of Eq. (23) with zero. Each column shows (from left to right) noisy data to be denoised, results for WFF, results for PEARLS, and results for the proposed method. Each row shows (from top to bottom) wrapped phase maps, error of wrapped phase, and corresponding unwrapped phase obtained by PUMA algorithm [4] for noise levels σ = 0.75 , 1. The near side of 3D graphs of error and unwrapped phases corresponds to the upper left corner of the wrapped-phase maps. The parameters used for each method are listed in Table 1.

Tables (3)

Tables Icon

Algorithm 1 Proposed algorithm (ALP-ADMM)

Tables Icon

Table 1. Parameters for Obtaining Figs. 5 and 6

Tables Icon

Table 2. Computational Time for Executing Each Method Once

Equations (24)

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( W g f ) ( y , k ) = R 2 f ( x ) g ( x y ) ¯ e 2 π i k , x d x ,
f ( x ) = R 2 R 2 ( W g f ) ( y , k ) g ( x y ) e 2 π i k , x d y d k ,
f = W inv g W g f .
f ˜ = W inv g T hard λ [ W g f ] ,
T hard λ [ z ] n = { z n ( | z n | λ ) 0 ( | z n | < λ )
φ ˜ = Arg [ W inv g T hard λ [ W g e i φ ] ] ,
T hard λ [ z ] arg min x [ x z 2 2 + λ 2 x 0 ] ,
x arg min x [ 1 2 x z 2 2 + λ x 1 ] ,
T soft λ [ z ] n = { ( 1 λ / | z n | ) z n ( | z n | λ ) 0 ( | z n | < λ ) .
{ g ( x a n ) e 2 π i b m , x } n , m Z 2 .
f ( x ) = n , m Z 2 ( F g f ) ( n , m ) g d ( x a n ) e 2 π i b m , x ,
( F g f ) ( n , m ) = R 2 f ( x ) g ( x a n ) ¯ e 2 π i b m , x d x .
f = F inv g d F g f .
f ˜ = F inv g d T hard λ [ F g f ] ,
x arg min x [ F d ( x ) + λ G ( L x ) ] ,
x arg min x [ 1 2 x d 2 2 + λ F g t x 1 ] ,
min x , y [ 1 2 x d 2 2 + λ { α F g t 1 ( x y ) 1 + ( 1 α ) F g t 2 y 1 } ] ,
x arg min x , y [ 1 2 x d 2 2 + λ { α F g t 1 ( x y ) 1 + ( 1 α ) F g t 2 D y 1 } ] .
x arg min x B C N [ F ( x ) + G ( L x ) ] ,
x ˜ ( 1 2 k + 1 ) x ^ + 2 k + 1 x x P B [ x k τ { L * ( ρ K k ( L x z ) + r ) + F ( x ˜ ) } ] z prox k ρ K G [ L x + k ρ K r ] r r + ρ k K ( L x z ) x ^ ( 1 2 k + 1 ) x ^ + 2 k + 1 x
prox λ G [ x ] = arg min y C N [ G ( y ) + 1 2 λ x y 2 2 ] ,
P B [ x ] n = { b n x n / | x n | ( | x n | b n ) x n ( | x n | < b n ) .
14 π e n 2 / 200 m 2 / 450 ,
ISNR = 10 log 10 [ e i φ e i φ true 2 2 e i φ ˜ e i φ true 2 2 ] ,

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