Abstract

We propose a new algorithm to recover a geometrically correct image of an object or scene from a set of images distorted by the wave motion of a water surface. Under mild conditions where the wavy surface normals weakly satisfy a Gaussian distribution, we demonstrate that the geometric distortion can be removed and a corrected image can be recovered. Our method is based on higher-order spectra analysis—in particular, the bispectrum, similar to its use in astronomical speckle imaging. In adapting this technique to imaging through or over a moving water surface, special care must be taken, and specifically tailored techniques are discussed in this paper. Our algorithm has been tested under two different scenarios: the refraction of light through a water surface (the underwater case) and the reflection of light from a water surface (the reflection case). Results in both cases have been encouraging.

© 2010 Optical Society of America

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  1. M. Roggemann and B. Welch, Imaging Through Turbulence (CRC Press, 1996).
  2. M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging?” Opt. Laser Technol. 29, 51–55 (1997).
    [CrossRef]
  3. R. Shefer, M. Malhi, and A. Shenhar, “Waves distortion correction using cross correlation,” http://visl.technion.ac.il/projects/2000maor/ (2001).
  4. H. Murase, “Surface shape reconstruction of a nonrigid transparent object using refraction and motion,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1045–1052 (1992).
    [CrossRef]
  5. A. Efros, V. Isler, J. Shi, and M. Visontai, “Seeing through water,” in Neural Information Processing Systems (NIPS, 2004).
  6. A. Donate, G. Dahme, and E. Ribeiro, “Classification of textures distorted by water waves,” in Proceedings of the 18th International Conference on Pattern Recognition (IEEE, 2006), pp. 421–424.
  7. A. Donate and E. Ribeiro, “Improved reconstruction of images distorted by water waves,” in Proceedings of the First International Conference on Computer Vision Theory and Applications (INSTICC, 2006), pp. 228–235
  8. D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
    [CrossRef]
  9. Y. Tian and S. Narasimhan, “Seeing through water: image restoration using model-based tracking,” Proceedings of the IEEE International Conference of Computer Vision (ICCV) (IEEE, 2009), pp. 2303–2310.
  10. Z. Wen, D. Fraser, and A. Lambert, “Bicoherence: a new lucky region technique in anisoplanatic image restoration,” Appl. Opt. 48, 6111–6119 (2009).
    [CrossRef] [PubMed]
  11. C. Cox and W. Munk, “Slopes of the sea surface deduced from photographs of sun glitter,” Scripps Inst. Oceanogr. 5, 401–479(1956).
  12. D. Fried, “Probability of getting a lucky short-exposure image through turbulence,” J. Opt. Soc. Am. 68, 1651–1658(1978).
    [CrossRef]
  13. R. Tubbs, “Lucky exposures: diffraction limited astronomical imaging through the atmosphere,” Ph.D. dissertation (Cambridge University, 2003).
  14. S. Weddell and R. Webb, “Data preprocessing on sequential data for improved astronomical imaging,” in Proceedings of Image and Vision Computing (Academic, 2005), pp. 1–8.
  15. N. Law, C. Mackay, and J. Baldwin, “Lucky imaging: high angular resolution imaging in the visible from the ground,” Astron. Astrophys. 446, 739–745 (2006).
    [CrossRef]
  16. Z. Wen, D. Fraser, and A. Lambert, “Bicoherence used to predict lucky regions in turbulence affected surveillance,” in Proceedings of the IEEE International Conference on Video and Signal Based Surveillance (IEEE, 2006), p. 108.
    [CrossRef]
  17. C. J. Carrano, “Anisoplanatic performance of horizontal-path speckle imaging,” Proc. SPIE 5162, 14–26 (2003).
    [CrossRef]
  18. Z. Wang and A. Bovik, “A universal image quality index,” IEEE Signal Process Lett. 9, 81–84 (2002).
    [CrossRef]
  19. A. Lohmann and B. Wirnitzer, “Triple correlations,” in Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  20. C. Nikias and A. Petropulu, Higher-Order Spectra Analysis (PTR Prentice Hall, 1993).
  21. J. Fackrell and S. McLaughlin, “Quadratic phase coupling detection using higher order statistics,” in Proceedings of the IEE Colloquium on Higher Order Statistics (Academic, 1995), pp. 9–17.
    [CrossRef]
  22. J. Fackrell, S. McLaughlin, and P. White, “Practical issues concerning the use of the bicoherence for the detection of quadratic phase coupling,” in Proceedings of the IEEE Workshop on HOS (IEEE, 1995), pp. 1–5.
  23. M. Hinich and M. Wolinsky, “Normalizing bispectra,” J. Stat. Plan. Infer. 130, 405–411 (2005).
    [CrossRef]
  24. S. McLaughlin, A. Stogioglou, and J. Fackrell, “Introducing higher order statistics (HOS) for the detection of nonlinearities,” UK Nonlinear News (15 September 1995).
  25. W. Silva, T. Strganac, and M. Hajj, “Higher-order spectral analysis of a nonlinear pitch and plunge apparatus,” in Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Academic, 2005), pp. 1–20.
    [PubMed]
  26. T. D. de Wit, “Spectral and statistical analysis of plasma turbulence: beyond linear techniques,” in Space Plasma Simulation, J.Büchner, C.T.Dum, and M.Scholer, eds. (Springer, 2003).
  27. H. Farid and A. Popescu, “Blind removal of image nonlinearities,” in Proceedings of the IEEE Conference on International Conference of Computer Vision (IEEE, 2001), pp. 76–81.
  28. Z. Wen, D. Fraser, A. Lambert, and H. Li, “Reconstruction of underwater image by bispectrum,” in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. 3, pp. 545–548.
  29. D. Fraser, G. Thorpe, and A. Lambert, “Atmospheric turbulence visualization with wide-area motion-blur restoration,” J. Opt. Soc. Am. A 16, 1751–1758 (1999).
    [CrossRef]
  30. C. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913(1991).
    [CrossRef]

2009 (1)

2006 (2)

N. Law, C. Mackay, and J. Baldwin, “Lucky imaging: high angular resolution imaging in the visible from the ground,” Astron. Astrophys. 446, 739–745 (2006).
[CrossRef]

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

2005 (1)

M. Hinich and M. Wolinsky, “Normalizing bispectra,” J. Stat. Plan. Infer. 130, 405–411 (2005).
[CrossRef]

2003 (1)

C. J. Carrano, “Anisoplanatic performance of horizontal-path speckle imaging,” Proc. SPIE 5162, 14–26 (2003).
[CrossRef]

2002 (1)

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

1999 (1)

1997 (1)

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging?” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

1992 (1)

H. Murase, “Surface shape reconstruction of a nonrigid transparent object using refraction and motion,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1045–1052 (1992).
[CrossRef]

1991 (1)

1984 (1)

A. Lohmann and B. Wirnitzer, “Triple correlations,” in Proc. IEEE 72, 889–901 (1984).
[CrossRef]

1978 (1)

1956 (1)

C. Cox and W. Munk, “Slopes of the sea surface deduced from photographs of sun glitter,” Scripps Inst. Oceanogr. 5, 401–479(1956).

Baldwin, J.

N. Law, C. Mackay, and J. Baldwin, “Lucky imaging: high angular resolution imaging in the visible from the ground,” Astron. Astrophys. 446, 739–745 (2006).
[CrossRef]

Bovik, A.

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

Carrano, C. J.

C. J. Carrano, “Anisoplanatic performance of horizontal-path speckle imaging,” Proc. SPIE 5162, 14–26 (2003).
[CrossRef]

Carter, P. W.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Cox, C.

C. Cox and W. Munk, “Slopes of the sea surface deduced from photographs of sun glitter,” Scripps Inst. Oceanogr. 5, 401–479(1956).

Dahme, G.

A. Donate, G. Dahme, and E. Ribeiro, “Classification of textures distorted by water waves,” in Proceedings of the 18th International Conference on Pattern Recognition (IEEE, 2006), pp. 421–424.

Dainty, J. C.

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging?” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

de Wit, T. D.

T. D. de Wit, “Spectral and statistical analysis of plasma turbulence: beyond linear techniques,” in Space Plasma Simulation, J.Büchner, C.T.Dum, and M.Scholer, eds. (Springer, 2003).

Donate, A.

A. Donate and E. Ribeiro, “Improved reconstruction of images distorted by water waves,” in Proceedings of the First International Conference on Computer Vision Theory and Applications (INSTICC, 2006), pp. 228–235

A. Donate, G. Dahme, and E. Ribeiro, “Classification of textures distorted by water waves,” in Proceedings of the 18th International Conference on Pattern Recognition (IEEE, 2006), pp. 421–424.

Efros, A.

A. Efros, V. Isler, J. Shi, and M. Visontai, “Seeing through water,” in Neural Information Processing Systems (NIPS, 2004).

Fackrell, J.

S. McLaughlin, A. Stogioglou, and J. Fackrell, “Introducing higher order statistics (HOS) for the detection of nonlinearities,” UK Nonlinear News (15 September 1995).

J. Fackrell and S. McLaughlin, “Quadratic phase coupling detection using higher order statistics,” in Proceedings of the IEE Colloquium on Higher Order Statistics (Academic, 1995), pp. 9–17.
[CrossRef]

J. Fackrell, S. McLaughlin, and P. White, “Practical issues concerning the use of the bicoherence for the detection of quadratic phase coupling,” in Proceedings of the IEEE Workshop on HOS (IEEE, 1995), pp. 1–5.

Farid, H.

H. Farid and A. Popescu, “Blind removal of image nonlinearities,” in Proceedings of the IEEE Conference on International Conference of Computer Vision (IEEE, 2001), pp. 76–81.

Flacco, N. L.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Fraser, D.

Fraser, D.

D. Fraser, G. Thorpe, and A. Lambert, “Atmospheric turbulence visualization with wide-area motion-blur restoration,” J. Opt. Soc. Am. A 16, 1751–1758 (1999).
[CrossRef]

Z. Wen, D. Fraser, A. Lambert, and H. Li, “Reconstruction of underwater image by bispectrum,” in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. 3, pp. 545–548.

Z. Wen, D. Fraser, and A. Lambert, “Bicoherence used to predict lucky regions in turbulence affected surveillance,” in Proceedings of the IEEE International Conference on Video and Signal Based Surveillance (IEEE, 2006), p. 108.
[CrossRef]

Fried, D.

Hajj, M.

W. Silva, T. Strganac, and M. Hajj, “Higher-order spectral analysis of a nonlinear pitch and plunge apparatus,” in Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Academic, 2005), pp. 1–20.
[PubMed]

Hinich, M.

M. Hinich and M. Wolinsky, “Normalizing bispectra,” J. Stat. Plan. Infer. 130, 405–411 (2005).
[CrossRef]

Holohan, M. L.

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging?” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

Hubbard, B. E.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Isler, V.

A. Efros, V. Isler, J. Shi, and M. Visontai, “Seeing through water,” in Neural Information Processing Systems (NIPS, 2004).

Jones, N. M.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Lambert, A.

Lambert, A.

D. Fraser, G. Thorpe, and A. Lambert, “Atmospheric turbulence visualization with wide-area motion-blur restoration,” J. Opt. Soc. Am. A 16, 1751–1758 (1999).
[CrossRef]

Z. Wen, D. Fraser, A. Lambert, and H. Li, “Reconstruction of underwater image by bispectrum,” in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. 3, pp. 545–548.

Z. Wen, D. Fraser, and A. Lambert, “Bicoherence used to predict lucky regions in turbulence affected surveillance,” in Proceedings of the IEEE International Conference on Video and Signal Based Surveillance (IEEE, 2006), p. 108.
[CrossRef]

Law, N.

N. Law, C. Mackay, and J. Baldwin, “Lucky imaging: high angular resolution imaging in the visible from the ground,” Astron. Astrophys. 446, 739–745 (2006).
[CrossRef]

Li, H.

Z. Wen, D. Fraser, A. Lambert, and H. Li, “Reconstruction of underwater image by bispectrum,” in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. 3, pp. 545–548.

Lohmann, A.

A. Lohmann and B. Wirnitzer, “Triple correlations,” in Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Mackay, C.

N. Law, C. Mackay, and J. Baldwin, “Lucky imaging: high angular resolution imaging in the visible from the ground,” Astron. Astrophys. 446, 739–745 (2006).
[CrossRef]

Malhi, M.

R. Shefer, M. Malhi, and A. Shenhar, “Waves distortion correction using cross correlation,” http://visl.technion.ac.il/projects/2000maor/ (2001).

Matson, C.

McLaughlin, S.

S. McLaughlin, A. Stogioglou, and J. Fackrell, “Introducing higher order statistics (HOS) for the detection of nonlinearities,” UK Nonlinear News (15 September 1995).

J. Fackrell, S. McLaughlin, and P. White, “Practical issues concerning the use of the bicoherence for the detection of quadratic phase coupling,” in Proceedings of the IEEE Workshop on HOS (IEEE, 1995), pp. 1–5.

J. Fackrell and S. McLaughlin, “Quadratic phase coupling detection using higher order statistics,” in Proceedings of the IEE Colloquium on Higher Order Statistics (Academic, 1995), pp. 9–17.
[CrossRef]

Milder, D. M.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Munk, W.

C. Cox and W. Munk, “Slopes of the sea surface deduced from photographs of sun glitter,” Scripps Inst. Oceanogr. 5, 401–479(1956).

Murase, H.

H. Murase, “Surface shape reconstruction of a nonrigid transparent object using refraction and motion,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1045–1052 (1992).
[CrossRef]

Narasimhan, S.

Y. Tian and S. Narasimhan, “Seeing through water: image restoration using model-based tracking,” Proceedings of the IEEE International Conference of Computer Vision (ICCV) (IEEE, 2009), pp. 2303–2310.

Nikias, C.

C. Nikias and A. Petropulu, Higher-Order Spectra Analysis (PTR Prentice Hall, 1993).

Panici, K. R.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Petropulu, A.

C. Nikias and A. Petropulu, Higher-Order Spectra Analysis (PTR Prentice Hall, 1993).

Platt, B. D.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Popescu, A.

H. Farid and A. Popescu, “Blind removal of image nonlinearities,” in Proceedings of the IEEE Conference on International Conference of Computer Vision (IEEE, 2001), pp. 76–81.

Potter, R. E.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Ribeiro, E.

A. Donate and E. Ribeiro, “Improved reconstruction of images distorted by water waves,” in Proceedings of the First International Conference on Computer Vision Theory and Applications (INSTICC, 2006), pp. 228–235

A. Donate, G. Dahme, and E. Ribeiro, “Classification of textures distorted by water waves,” in Proceedings of the 18th International Conference on Pattern Recognition (IEEE, 2006), pp. 421–424.

Roggemann, M.

M. Roggemann and B. Welch, Imaging Through Turbulence (CRC Press, 1996).

Shefer, R.

R. Shefer, M. Malhi, and A. Shenhar, “Waves distortion correction using cross correlation,” http://visl.technion.ac.il/projects/2000maor/ (2001).

Shenhar, A.

R. Shefer, M. Malhi, and A. Shenhar, “Waves distortion correction using cross correlation,” http://visl.technion.ac.il/projects/2000maor/ (2001).

Shi, J.

A. Efros, V. Isler, J. Shi, and M. Visontai, “Seeing through water,” in Neural Information Processing Systems (NIPS, 2004).

Silva, W.

W. Silva, T. Strganac, and M. Hajj, “Higher-order spectral analysis of a nonlinear pitch and plunge apparatus,” in Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Academic, 2005), pp. 1–20.
[PubMed]

Stogioglou, A.

S. McLaughlin, A. Stogioglou, and J. Fackrell, “Introducing higher order statistics (HOS) for the detection of nonlinearities,” UK Nonlinear News (15 September 1995).

Strganac, T.

W. Silva, T. Strganac, and M. Hajj, “Higher-order spectral analysis of a nonlinear pitch and plunge apparatus,” in Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Academic, 2005), pp. 1–20.
[PubMed]

Thorpe, G.

Tian, Y.

Y. Tian and S. Narasimhan, “Seeing through water: image restoration using model-based tracking,” Proceedings of the IEEE International Conference of Computer Vision (ICCV) (IEEE, 2009), pp. 2303–2310.

Tong, K. W.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Tubbs, R.

R. Tubbs, “Lucky exposures: diffraction limited astronomical imaging through the atmosphere,” Ph.D. dissertation (Cambridge University, 2003).

Twisselmann, D. J.

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Visontai, M.

A. Efros, V. Isler, J. Shi, and M. Visontai, “Seeing through water,” in Neural Information Processing Systems (NIPS, 2004).

Wang, Z.

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

Webb, R.

S. Weddell and R. Webb, “Data preprocessing on sequential data for improved astronomical imaging,” in Proceedings of Image and Vision Computing (Academic, 2005), pp. 1–8.

Weddell, S.

S. Weddell and R. Webb, “Data preprocessing on sequential data for improved astronomical imaging,” in Proceedings of Image and Vision Computing (Academic, 2005), pp. 1–8.

Welch, B.

M. Roggemann and B. Welch, Imaging Through Turbulence (CRC Press, 1996).

Wen, Z.

Wen, Z.

Z. Wen, D. Fraser, and A. Lambert, “Bicoherence used to predict lucky regions in turbulence affected surveillance,” in Proceedings of the IEEE International Conference on Video and Signal Based Surveillance (IEEE, 2006), p. 108.
[CrossRef]

Z. Wen, D. Fraser, A. Lambert, and H. Li, “Reconstruction of underwater image by bispectrum,” in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. 3, pp. 545–548.

White, P.

J. Fackrell, S. McLaughlin, and P. White, “Practical issues concerning the use of the bicoherence for the detection of quadratic phase coupling,” in Proceedings of the IEEE Workshop on HOS (IEEE, 1995), pp. 1–5.

Wirnitzer, B.

A. Lohmann and B. Wirnitzer, “Triple correlations,” in Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Wolinsky, M.

M. Hinich and M. Wolinsky, “Normalizing bispectra,” J. Stat. Plan. Infer. 130, 405–411 (2005).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

N. Law, C. Mackay, and J. Baldwin, “Lucky imaging: high angular resolution imaging in the visible from the ground,” Astron. Astrophys. 446, 739–745 (2006).
[CrossRef]

IEEE Signal Process Lett. (1)

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

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

H. Murase, “Surface shape reconstruction of a nonrigid transparent object using refraction and motion,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1045–1052 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Stat. Plan. Infer. (1)

M. Hinich and M. Wolinsky, “Normalizing bispectra,” J. Stat. Plan. Infer. 130, 405–411 (2005).
[CrossRef]

Opt. Laser Technol. (1)

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging?” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

Proc. IEEE (1)

A. Lohmann and B. Wirnitzer, “Triple correlations,” in Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Proc. SPIE (1)

C. J. Carrano, “Anisoplanatic performance of horizontal-path speckle imaging,” Proc. SPIE 5162, 14–26 (2003).
[CrossRef]

Scripps Inst. Oceanogr. (1)

C. Cox and W. Munk, “Slopes of the sea surface deduced from photographs of sun glitter,” Scripps Inst. Oceanogr. 5, 401–479(1956).

Waves Random Complex Media (1)

D. M. Milder, P. W. Carter, N. L. Flacco, B. E. Hubbard, N. M. Jones, K. R. Panici, B. D. Platt, R. E. Potter, K. W. Tong, and D. J. Twisselmann, “Reconstruction of through-surface underwater imagery,” Waves Random Complex Media 16, 521–530(2006).
[CrossRef]

Other (17)

Y. Tian and S. Narasimhan, “Seeing through water: image restoration using model-based tracking,” Proceedings of the IEEE International Conference of Computer Vision (ICCV) (IEEE, 2009), pp. 2303–2310.

R. Shefer, M. Malhi, and A. Shenhar, “Waves distortion correction using cross correlation,” http://visl.technion.ac.il/projects/2000maor/ (2001).

A. Efros, V. Isler, J. Shi, and M. Visontai, “Seeing through water,” in Neural Information Processing Systems (NIPS, 2004).

A. Donate, G. Dahme, and E. Ribeiro, “Classification of textures distorted by water waves,” in Proceedings of the 18th International Conference on Pattern Recognition (IEEE, 2006), pp. 421–424.

A. Donate and E. Ribeiro, “Improved reconstruction of images distorted by water waves,” in Proceedings of the First International Conference on Computer Vision Theory and Applications (INSTICC, 2006), pp. 228–235

R. Tubbs, “Lucky exposures: diffraction limited astronomical imaging through the atmosphere,” Ph.D. dissertation (Cambridge University, 2003).

S. Weddell and R. Webb, “Data preprocessing on sequential data for improved astronomical imaging,” in Proceedings of Image and Vision Computing (Academic, 2005), pp. 1–8.

M. Roggemann and B. Welch, Imaging Through Turbulence (CRC Press, 1996).

Z. Wen, D. Fraser, and A. Lambert, “Bicoherence used to predict lucky regions in turbulence affected surveillance,” in Proceedings of the IEEE International Conference on Video and Signal Based Surveillance (IEEE, 2006), p. 108.
[CrossRef]

C. Nikias and A. Petropulu, Higher-Order Spectra Analysis (PTR Prentice Hall, 1993).

J. Fackrell and S. McLaughlin, “Quadratic phase coupling detection using higher order statistics,” in Proceedings of the IEE Colloquium on Higher Order Statistics (Academic, 1995), pp. 9–17.
[CrossRef]

J. Fackrell, S. McLaughlin, and P. White, “Practical issues concerning the use of the bicoherence for the detection of quadratic phase coupling,” in Proceedings of the IEEE Workshop on HOS (IEEE, 1995), pp. 1–5.

S. McLaughlin, A. Stogioglou, and J. Fackrell, “Introducing higher order statistics (HOS) for the detection of nonlinearities,” UK Nonlinear News (15 September 1995).

W. Silva, T. Strganac, and M. Hajj, “Higher-order spectral analysis of a nonlinear pitch and plunge apparatus,” in Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Academic, 2005), pp. 1–20.
[PubMed]

T. D. de Wit, “Spectral and statistical analysis of plasma turbulence: beyond linear techniques,” in Space Plasma Simulation, J.Büchner, C.T.Dum, and M.Scholer, eds. (Springer, 2003).

H. Farid and A. Popescu, “Blind removal of image nonlinearities,” in Proceedings of the IEEE Conference on International Conference of Computer Vision (IEEE, 2001), pp. 76–81.

Z. Wen, D. Fraser, A. Lambert, and H. Li, “Reconstruction of underwater image by bispectrum,” in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. 3, pp. 545–548.

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Figures (9)

Fig. 1
Fig. 1

Set of short-exposure sample images for the through-the-water imaging case (refraction).

Fig. 2
Fig. 2

Set of short-exposure sample images for the water-surface-reflection case (reflection).

Fig. 3
Fig. 3

Illustration of through-the-water imaging and the refraction law: when the water surface is flat and perpendicular to the line of sight, an observer sees an object in its true position b; when a surface wave exists, due to the effect of refraction, an observer sees the object in position a, which can be calculated by the Snell law.

Fig. 4
Fig. 4

Image reconstruction of water imaging reflection: (a) original object (b) geometry for water wave reflection; the temporal distribution of the normal of the surface N is Gaussian. A sample sequence of frames from a video stream is shown in Fig. 2. The images are upside down and distorted.

Fig. 5
Fig. 5

Comparison of the bicoherence method and the Wang method. The curves show the consistency between the two methods. While the dotted green curve represents the value of the bicoherence of 50 images, the red curve sketches the value calculated by the Wang method. Note that when the red curve goes up to a peak, the green dashed curve always goes down to a small value.

Fig. 6
Fig. 6

Simulation results. The result in (c) is produced by our algorithm using bicoherence for lucky region selection and bispectrum for image restoration.

Fig. 7
Fig. 7

Results of image reconstruction for through-the-water imaging.

Fig. 8
Fig. 8

Results of water-reflection image recovery: (a)–(c) under rough conditions and (d)–(e) under mild conditions.

Fig. 9
Fig. 9

Comparison of different results using different sized regions: (a) 32 × 32 and (b) 64 × 64 . From visual inspection, the result in (b) is better, clearer, and sharper.

Tables (1)

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Table 1 Comparison between Our Method and the Wang Method with Natural Data a

Equations (11)

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b ( u 1 , u 2 ) = | i = 0 N 1 I ( u 1 ) I ( u 2 ) I * ( u 1 + u 2 ) | i = 0 N 1 | I ( u 1 ) I ( u 2 ) | 2 i = 0 N 1 | I * ( u 1 + u 2 ) | 2 ,
b region = ( 1 N h k = 1 N h b k h ) 2 + ( 1 N v k = 1 N v b k v ) 2 ,
b k ( · ) = u 1 = 0 N u 1 1 u 2 = 0 N u 2 1 b line ( u 1 , u 2 ) N u 1 N u 2 ,
M = 4 σ i c i t i ¯ c i ¯ t ( σ i c 2 + σ i t 2 ) [ ( i ¯ c ) 2 + ( i ¯ t ) 2 ] ,
i ( x ) = o ( x ) * h ( x ) ,
I ( 3 ) ( u 1 , u 2 ) = I ( u 1 ) I ( u 2 ) I * ( u 1 + u 2 ) = O ( u 1 ) O ( u 2 ) O * ( u 1 + u 2 ) H ( u 1 ) H ( u 2 ) H * ( u 1 + u 2 ) = O ( 3 ) ( u 1 , u 2 ) H ( 3 ) ( u 1 , u 2 ) ,
I ( 3 ) ( u 1 , u 2 ) = O ( 3 ) ( u 1 , u 2 ) H ( 3 ) ( u 1 , u 2 ) .
ϕ O ( 3 ) = ϕ I ( 3 ) .
ϕ I ( 3 ) ( u 1 , u 2 ) = ϕ O ( u 1 ) + ϕ O ( u 2 ) ϕ O ( u 1 + u 2 ) .
{ d x = h p ( 1 1 / n ) + N d y = h q ( 1 1 / n ) + N ,
{ p = a i u i cos ( u i x + v i y w i t ) q = a i v i cos ( u i x + v i y w i t ) .

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