Abstract

Many robust phase unwrapping algorithms are computationally very time-consuming, making them impractical for handling large datasets or real-time applications. In this paper, we propose a generic framework using a novel wavelet transform that can be combined with many types of phase unwrapping algorithms. By inserting reversible modulo operators in the wavelet transform, the number of coefficients that need to be unwrapped is significantly reduced, which results in large computational gains. The algorithm is tested on various types of wrapped phase imagery, reporting speedup factors of up to 500. The source code of the algorithm is publicly available.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Efficient phase unwrapping algorithm based on cubature information particle filter applied to unwrap noisy continuous phase maps

Xianming Xie, Qingning Zeng, Kefei Liao, and Qinghua Liu
Opt. Express 27(7) 9906-9924 (2019)

Object recognition in subband transform-compressed images by use of correlation filters

Cindy Daniell, Abhijit Mahalanobis, and Rod Goodman
Appl. Opt. 42(32) 6474-6487 (2003)

Wavelet packet correlation methods in biometrics

Pablo Hennings, Jason Thornton, Jelena Kovačević, and B. V. K. Vijaya Kumar
Appl. Opt. 44(5) 637-646 (2005)

References

  • View by:
  • |
  • |
  • |

  1. A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to insar time series,” J. Opt. Soc. Am. A 24, 2737–2747 (2007).
    [Crossref]
  2. E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
    [Crossref] [PubMed]
  3. H. Zhang, M. J. Lalor, and D. R. Burton, “Spatiotemporal phase unwrapping for the measurement of discontinuous objects in dynamic fringe-projection phase-shifting profilometry,” Appl. Opt. 38, 3534–3541 (1999).
    [Crossref]
  4. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [Crossref] [PubMed]
  5. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
    [Crossref]
  6. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
    [Crossref] [PubMed]
  7. I. V. Lyuboshenko, H. Maître, and A. Maruani, “Least-mean-squares phase unwrapping by use of an incomplete set of residue branch cuts,” Appl. Opt. 41, 2129–2148 (2002).
    [Crossref] [PubMed]
  8. M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50, 6214–6224 (2011).
    [Crossref] [PubMed]
  9. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
    [Crossref]
  10. B. Friedlander and J. Francos, “Model based phase unwrapping of 2-d signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
    [Crossref]
  11. J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
    [Crossref] [PubMed]
  12. H. Y. H. Huang, L. Tian, Z. Zhang, Y. Liu, Z. Chen, and G. Barbastathis, “Path-independent phase unwrapping using phase gradient and total-variation (tv) denoising,” Opt. Express 20, 14075–14089 (2012).
    [Crossref] [PubMed]
  13. U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, D. Psaltis, and M. Unser, “Isotropic inverse-problem approach for two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 32, 1092–1100 (2015).
    [Crossref]
  14. D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  15. X. Xie and Y. Li, “Enhanced phase unwrapping algorithm based on unscented kalman filter, enhanced phase gradient estimator, and path-following strategy,” Appl. Opt. 53, 4049–4060 (2014).
    [Crossref] [PubMed]
  16. S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
    [Crossref]
  17. M. P. Hayes and P. T. Gough, “Synthetic aperture sonar: a review of current status,” IEEE J. Oceanic Eng. 34, 207–224 (2009).
    [Crossref]
  18. J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
    [Crossref]
  19. M. Zhao and Q. Kemao, “Quality-guided phase unwrapping implementation: an improved indexedinterwoven linked list,” Appl. Opt. 53, 3492–3500 (2014).
    [Crossref] [PubMed]
  20. W. Gao, N. T. T. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2d parallel windowed fourier transform for fringe pattern analysis using graphics processing unit,” Opt. Express 17, 23147–23152 (2009).
    [Crossref]
  21. H. Zhong, J. Tang, and S. Zhang, “Phase quality map based on local multi-unwrapped results for two-dimensional phase unwrapping,” Appl. Opt. 54, 739–745 (2015).
    [Crossref] [PubMed]
  22. O. Backoach, S. Kariv, P. Girshovitz, and N. T. Shaked, “Fast phase processing in off-axis holography by cuda including parallel phase unwrapping,” Opt. Express 24, 3177–3188 (2016).
    [Crossref] [PubMed]
  23. D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.
  24. P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
    [Crossref]
  25. W. Sweldens, “The lifting scheme: a custom-design construction of biorthogonal wavelets,” Appl. Computational Harmonic Anal. 3, 186–200 (1996).
    [Crossref]
  26. S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [Crossref]
  27. D. Blinder, “Matlab code: Efficient multiscale phase unwrapping methodology with modulo wavelet transform,” http://erc-interfere.eu/downloads.html (2016). [retrieved 17 June 2016].
  28. Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Machine Intelligence 26, 1124–1137 (2004).
    [Crossref]
  29. D. R. Karger, “Global min-cuts in rnc, and other ramifications of a simple min-cut algorithm,” in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (Society for Industrial and Applied Mathematics, 1993), pp. 21–30.
  30. T. McClure, “Automatic terrain generation,” http://www.mathworks.com/matlabcentral/fileexchange/39559-automatic-terrain-generation (2016). [retrieved 12 June 2016].

2016 (2)

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
[Crossref]

O. Backoach, S. Kariv, P. Girshovitz, and N. T. Shaked, “Fast phase processing in off-axis holography by cuda including parallel phase unwrapping,” Opt. Express 24, 3177–3188 (2016).
[Crossref] [PubMed]

2015 (2)

2014 (3)

2012 (1)

2011 (1)

2009 (2)

2007 (3)

2004 (1)

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Machine Intelligence 26, 1124–1137 (2004).
[Crossref]

2002 (2)

1999 (1)

1997 (1)

1996 (2)

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-d signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

W. Sweldens, “The lifting scheme: a custom-design construction of biorthogonal wavelets,” Appl. Computational Harmonic Anal. 3, 186–200 (1996).
[Crossref]

1989 (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

1982 (1)

An, D.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
[Crossref]

Asundi, A.

Backoach, O.

Barbastathis, G.

Barnhill, E.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
[Crossref] [PubMed]

Bioucas-Dias, J.

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

Blinder, D.

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

Boykov, Y.

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Machine Intelligence 26, 1124–1137 (2004).
[Crossref]

Bruylants, T.

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

Burton, D. R.

Chen, Z.

Dooms, A.

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

Ebrahimi, T.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Flynn, T. J.

Francos, J.

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-d signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

Friedlander, B.

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-d signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

Gao, W.

Gdeisat, M. A.

Ghiglia, D.

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

Girshovitz, P.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

Gough, P. T.

M. P. Hayes and P. T. Gough, “Synthetic aperture sonar: a review of current status,” IEEE J. Oceanic Eng. 34, 207–224 (2009).
[Crossref]

Hayes, M. P.

M. P. Hayes and P. T. Gough, “Synthetic aperture sonar: a review of current status,” IEEE J. Oceanic Eng. 34, 207–224 (2009).
[Crossref]

Herráez, M. A.

Hooper, A.

Huang, H. Y. H.

Huang, L.

Huang, X.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
[Crossref]

Huyen, N. T. T.

Itoh, K.

Johnson, C. L.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
[Crossref] [PubMed]

Kamilov, U. S.

Karger, D. R.

D. R. Karger, “Global min-cuts in rnc, and other ramifications of a simple min-cut algorithm,” in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (Society for Industrial and Applied Mathematics, 1993), pp. 21–30.

Kariv, S.

Kemao, Q.

Kennedy, P.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
[Crossref] [PubMed]

Kolmogorov, V.

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Machine Intelligence 26, 1124–1137 (2004).
[Crossref]

Lalor, M. J.

Li, X.

Li, Y.

Liu, Y.

Loi, H. S.

Lyuboshenko, I. V.

Mada, M.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
[Crossref] [PubMed]

Maître, H.

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

Maruani, A.

Munteanu, A.

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

Ottevaere, H.

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

Papadopoulos, I. N.

Pritt, M.

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

Psaltis, D.

Roberts, N.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
[Crossref] [PubMed]

Schelkens, P.

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Shaked, N. T.

Shoreh, M. H.

Skodras, A.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Su, X.

Sweldens, W.

W. Sweldens, “The lifting scheme: a custom-design construction of biorthogonal wavelets,” Appl. Computational Harmonic Anal. 3, 186–200 (1996).
[Crossref]

Tang, J.

Tian, L.

Unser, M.

Valadao, G.

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

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

Xie, X.

Xu, J.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
[Crossref]

Yau, S.-T.

Yi, P.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
[Crossref]

Zebker, H. A.

A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to insar time series,” J. Opt. Soc. Am. A 24, 2737–2747 (2007).
[Crossref]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

Zhang, H.

Zhang, Q.

Zhang, S.

Zhang, Z.

Zhao, M.

Zhong, H.

Appl. Computational Harmonic Anal. (1)

W. Sweldens, “The lifting scheme: a custom-design construction of biorthogonal wavelets,” Appl. Computational Harmonic Anal. 3, 186–200 (1996).
[Crossref]

Appl. Opt. (9)

M. Zhao and Q. Kemao, “Quality-guided phase unwrapping implementation: an improved indexedinterwoven linked list,” Appl. Opt. 53, 3492–3500 (2014).
[Crossref] [PubMed]

H. Zhong, J. Tang, and S. Zhang, “Phase quality map based on local multi-unwrapped results for two-dimensional phase unwrapping,” Appl. Opt. 54, 739–745 (2015).
[Crossref] [PubMed]

H. Zhang, M. J. Lalor, and D. R. Burton, “Spatiotemporal phase unwrapping for the measurement of discontinuous objects in dynamic fringe-projection phase-shifting profilometry,” Appl. Opt. 38, 3534–3541 (1999).
[Crossref]

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
[Crossref] [PubMed]

I. V. Lyuboshenko, H. Maître, and A. Maruani, “Least-mean-squares phase unwrapping by use of an incomplete set of residue branch cuts,” Appl. Opt. 41, 2129–2148 (2002).
[Crossref] [PubMed]

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50, 6214–6224 (2011).
[Crossref] [PubMed]

X. Xie and Y. Li, “Enhanced phase unwrapping algorithm based on unscented kalman filter, enhanced phase gradient estimator, and path-following strategy,” Appl. Opt. 53, 4049–4060 (2014).
[Crossref] [PubMed]

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
[Crossref]

IEEE Geosci. Remote Sens. Lett. (1)

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote Sens. Lett. 13, 666–670 (2016).
[Crossref]

IEEE J. Oceanic Eng. (1)

M. P. Hayes and P. T. Gough, “Synthetic aperture sonar: a review of current status,” IEEE J. Oceanic Eng. 34, 207–224 (2009).
[Crossref]

IEEE Trans. Image Process. (1)

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

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

IEEE Trans. Pattern Anal. Machine Intelligence (1)

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Machine Intelligence 26, 1124–1137 (2004).
[Crossref]

IEEE Trans. Signal Process. (1)

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-d signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

J. Opt. Soc. Am. A (3)

Magnetic Resonance in Medicine (1)

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4d phase unwrapping applied to magnetic resonance elastography,” Magnetic Resonance in Medicine 73, 2321–2331 (2014).
[Crossref] [PubMed]

Opt. Express (3)

Radio Science (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

Other (6)

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

D. Blinder, T. Bruylants, H. Ottevaere, A. Dooms, A. Munteanu, and P. Schelkens, “Modulo wavelets for interferometric phase data,” in Imaging and Applied Optics 2014, (Optical Society of America, 2014), paper JTu4A.24.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

D. Blinder, “Matlab code: Efficient multiscale phase unwrapping methodology with modulo wavelet transform,” http://erc-interfere.eu/downloads.html (2016). [retrieved 17 June 2016].

D. R. Karger, “Global min-cuts in rnc, and other ramifications of a simple min-cut algorithm,” in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (Society for Industrial and Applied Mathematics, 1993), pp. 21–30.

T. McClure, “Automatic terrain generation,” http://www.mathworks.com/matlabcentral/fileexchange/39559-automatic-terrain-generation (2016). [retrieved 12 June 2016].

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Forward modulo Haar wavelet transform lifting scheme. xi represent the ith level wavelet coefficients, si and di are the level low-pass and high-pass coefficients respectively.
Fig. 2
Fig. 2 Example of unwrapping a phase image. The wrapped phase image (a) is transformed with a modulo Haar transform (b), followed by a phase gradient estimation, which determines which coefficients can be unwrapped at what level (c). The color codes black, gray and white indicate whether the corresponding coefficients should be unwrapped at a previous, current, or a later wavelet level respectively.
Fig. 3
Fig. 3 Flowchart for the acceleration pipeline
Fig. 4
Fig. 4 Inverse lifting scheme for unwrapping. Depending on whether a coefficient of si is aliased, either the regular Haar transform or the modulo Haar transform is performed.
Fig. 5
Fig. 5 Computation times for the unwrapping every image on these stacked bar charts are shown for 0 up to 10 modulo wavelet decompositions. The top chart shows results for the accelerated “Fast 2D Phase Unwrapping”, the bottom chart displays the accelerated “PUMA” algorithm. The computation times for the modulo wavelet transform and the computation overhead for constructing the (sub)graphs for PUMA are added to the graph as well. The pyramid image was only decomposed up to 6 times, as its dimensions are not divisible by 16. Results are averaged over 10 runs.
Fig. 6
Fig. 6 Results for the unwrapping procedure. Each row represents a different dataset: “Synthetic”, “Face”, “Pyramids” and “MEMS” respectively. Column (a) shows the original wrapped phase images (black = −π up to white = +π). Columns (b) and (c) are the unwrapped figures for the “Fast 2D PU” algorithm, respectively without and with modulo wavelet acceleration. Analogously, columns (d) and (e) are the unwrapped figures for the “PUMA” algorithm, respectively unaccelerated and accelerated.
Fig. 7
Fig. 7 Four examples of the wrapped terrain maps generated by the “Automatic terrain generation” algorithm.
Fig. 8
Fig. 8 Visual comparison of the unwrapping errors of the algorithms w.r.t. the ground truth of one of the tested height maps. (a) is the reference height map and (f) is the wrapped version. (b), (c) and (d) are the error difference images of the accelerated PUMA algorithm using dilation with radii of 0, 1 and 2 pixels respectively; (g), (h) and (i) are the same for the accelerated Fast 2D Unwrap algorithm. (e) and (j) are the errors for the original PUMA and Fast 2D Unwrap respectively as a reference.
Fig. 9
Fig. 9 Images (a)–(d) represent four of the synthetic wrapped phase images respectively superimposed with Gaussian noise with standard deviations (σ) of 0, 0.1π, 0.2π and 0.4π. Images (e)–(h) are the corresponding unwrapped images using the accelerated PUMA algorithm.

Tables (4)

Tables Icon

Table 1 Computation times for the unwrapping algorithms for the different test images. The third and fourth column express the computation times in seconds, respectively for the default algorithm and the best result for the accelerated version. The relative speedup is shown in the last column. Results are averaged over 10 runs.

Tables Icon

Table 2 Reported MSE (in radians2) for various combinations of the accelerated PUMA and Fast 2D PU algorithms. The reference algorithm is displayed in the second column. The columns indicate the number of used modulo wavelet levels, the rows indicate the radius of the used dilation mask. Results are averaged over the 10 generated terrain images.

Tables Icon

Table 3 Relative speedup factors for the accelerated PUMA and Fast 2D PU algorithms, averaged over 10 instances.

Tables Icon

Table 4 Averaged runtimes over 10 instances per noise level for the accelerated PUMA algorithm using 5 wavelet decomposition levels. The runtime for the averaged runtime over all noise levels for the (unaccelerated) reference method is given in the last column, denoted “Ref”.

Equations (16)

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

Φ ( x ) = ( F ( x ) )
d j + 1 = P j ( s j , d j ) and s j + 1 = U j ( s j , d j + 1 )
d j = P j ( s j , d j + 1 ) and s j = U j ( s j + 1 , d j + 1 )
s i + 1 0 ( n ) = s i ( 2 n + 1 )
d i + 1 0 ( n ) = s i ( 2 n )
d i + 1 ( n ) = ( d i + 1 0 ( n ) s i + 1 0 ( n ) ) .
s i + 1 ( n ) = ( s i + 1 0 ( n ) + 1 2 d i + 1 ( n ) ) .
d H ( n ) = F ( 2 n + 1 ) F ( 2 n ) = ( F ( 2 n + 1 ) F ( 2 n ) ) = ( ( F ( 2 n + 1 ) ) ( F ( 2 n ) ) ) = ( Φ ( 2 n + 1 ) Φ ( 2 n ) ) = d M H ( n )
s M H ( n ) = ( Φ ( 2 n ) + 1 2 ( Φ ( 2 n + 1 ) Φ ( 2 n ) ) ) = ( Φ ( 2 n ) + 1 2 d M H ( n ) ) = ( F ( 2 n ) + 1 2 d H ( n ) ) = ( s H ( n ) )
s i + 1 ( n ) = ( s i ( 2 n ) + 1 2 ( s i ( 2 n + 1 ) s i ( 2 n ) ) ) = ( s i ( 2 n ) + δ n ) = ( s i ( 2 n + 1 ) δ n )
( s i ( 2 n + 1 ) 2 δ n ) = ( s i ( 2 n + 1 ) ( s i ( 2 n + 1 ) s i ( 2 n ) ) ) = ( s i ( 2 n + 1 ) ( s i ( 2 n + 1 ) s i ( 2 n ) ) ) = ( s i ( 2 n ) )
| ( s i + 1 ( n + 1 ) s i + 1 ( n ) ) | = | ( s i ( 2 n + 2 ) s i ( 2 n + 1 ) + δ n + 1 + δ n ) |
( L i + 1 , H i + 1 ) = W ( L i ) with W 1 ( U ( L i + 1 ) , H i + 1 ) = U ( L i )
M B = b B M b
s i + 1 0 = c ( s i + 1 1 2 d i + 1 , a i + 1 )
d i + 1 0 = c ( d i + 1 + s i + 1 0 , a i + 1 )

Metrics