Abstract

We report on an algorithm for fast wavefront sensing that incorporates sparse representation for the first time in practice. The partial derivatives of optical wavefronts were sampled sparsely with a Shack–Hartmann wavefront sensor (SHWFS) by randomly subsampling the original SHWFS data to as little as 5%. Reconstruction was performed by a sparse representation algorithm that utilized the Zernike basis. We name this method sparse Zernike (SPARZER). Experiments on real and simulated data attest to the accuracy of the proposed techniques as compared to traditional sampling and reconstruction methods. We have made the corresponding dataset and software freely available online. Compressed wavefront sensing offers the potential to increase the speed of wavefront acquisition and to defray the cost of SHWFS devices.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Mackey, C. Dainty, Proc. SPIE 7108, 71080I (2008).
    [CrossRef]
  2. H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, J. Opt. Soc. Am. A 18, 497 (2001).
    [CrossRef]
  3. J. Z. Liang, D. R. Williams, D. T. Miller, J. Opt. Soc. Am. A 14, 2884 (1997).
    [CrossRef]
  4. A. Dubra, Y. Sulai, J. L. Norris, R. F. Cooper, A. M. Dubis, D. R. Williams, J. Carroll, Biomed. Opt. Express 2, 1864 (2011).
    [CrossRef]
  5. J. Primot, G. Rousset, J. C. Fontanella, J. Opt. Soc. Am. A 7, 1598 (1990).
    [CrossRef]
  6. J. Z. Liang, B. Grimm, S. Goelz, J. F. Bille, J. Opt. Soc. Am. A 11, 1949 (1994).
    [CrossRef]
  7. T. Nirmaier, G. Pudasaini, J. Bille, Opt. Express 11, 2704 (2003).
    [CrossRef]
  8. D. W. D. Monteiro, G. Vdovin, P. M. Sarro, Sens. Actuators A Phys. 109, 220 (2004).
  9. B. Pathak, A. Das, B. R. Boruah, Proc. SPIE 8557, 85570A (2012).
    [CrossRef]
  10. M. Elad, M. Aharon, IEEE Trans. Image Process. 15, 3736 (2006).
    [CrossRef]
  11. L. Y. Fang, S. T. Li, Q. Nie, J. A. Izatt, C. A. Toth, S. Farsiu, Biomed. Opt. Express 3, 927 (2012).
    [CrossRef]
  12. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
    [CrossRef]
  13. E. J. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
    [CrossRef]
  14. D. Xu, Y. Huang, J. U. Kang, Biomed. Opt. Express 4, 1519 (2013).
    [CrossRef]
  15. D. Baron, S. Sarvotham, R. G. Baraniuk, IEEE Trans. Signal Process. 58, 269 (2010).
    [CrossRef]
  16. E. J. Candes, T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
    [CrossRef]
  17. S. Farsiu, J. Christofferson, B. Eriksson, P. Milanfar, B. Friedlander, A. Shakouri, R. Nowak, Appl. Opt. 46, 5805 (2007).
    [CrossRef]
  18. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, S. Lim, Opt. Express 17, 13040 (2009).
    [CrossRef]
  19. Y. Rivenson, A. Stern, B. Javidi, Appl. Opt. 52, A423 (2013).
    [CrossRef]
  20. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
    [CrossRef]
  21. P. Clemente, V. Duran, E. Tajahuerce, P. Andres, V. Climent, J. Lancis, Opt. Lett. 38, 2524 (2013).
    [CrossRef]
  22. B. Hermann, E. J. Fernandez, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, P. Artal, Opt. Lett. 29, 2142 (2004).
    [CrossRef]
  23. R. J. Zawadzki, S. M. Jones, S. S. Olivier, M. T. Zhao, B. A. Bower, J. A. Izatt, S. Choi, S. Laut, J. S. Werner, Opt. Express 13, 8532 (2005).
    [CrossRef]
  24. J. M. Beckers, Annu. Rev. Astron. Astrophys. 31, 13 (1993).
    [CrossRef]
  25. O. Guyon, Astrophys J. 629, 592 (2005).
  26. M. Rostami, O. Michailovich, Z. Wang, IEEE Trans. Image Process. 21, 3139 (2012).
    [CrossRef]
  27. M. S. Hosseini, O. Michailovich, in 17th European Signal Processing Conference (2009), p. 115.
  28. J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).
  29. J. Porter, A. Guirao, I. G. Cox, D. R. Williams, J. Opt. Soc. Am. A 18, 1793 (2001).
    [CrossRef]
  30. A. Beck, M. Teboulle, SIAM J. Imaging Sci. 2, 183 (2009).
  31. B. Jaeken, P. Artal, Investig. Ophthamol. Vis. Sci. 53, 3405 (2012).
  32. L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
    [CrossRef]

2013 (4)

2012 (4)

L. Y. Fang, S. T. Li, Q. Nie, J. A. Izatt, C. A. Toth, S. Farsiu, Biomed. Opt. Express 3, 927 (2012).
[CrossRef]

B. Jaeken, P. Artal, Investig. Ophthamol. Vis. Sci. 53, 3405 (2012).

M. Rostami, O. Michailovich, Z. Wang, IEEE Trans. Image Process. 21, 3139 (2012).
[CrossRef]

B. Pathak, A. Das, B. R. Boruah, Proc. SPIE 8557, 85570A (2012).
[CrossRef]

2011 (1)

2010 (1)

D. Baron, S. Sarvotham, R. G. Baraniuk, IEEE Trans. Signal Process. 58, 269 (2010).
[CrossRef]

2009 (2)

2008 (2)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

R. Mackey, C. Dainty, Proc. SPIE 7108, 71080I (2008).
[CrossRef]

2007 (1)

2006 (4)

E. J. Candes, T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

M. Elad, M. Aharon, IEEE Trans. Image Process. 15, 3736 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. J. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

2005 (2)

2004 (2)

2003 (1)

2001 (2)

1997 (1)

1994 (1)

1993 (1)

J. M. Beckers, Annu. Rev. Astron. Astrophys. 31, 13 (1993).
[CrossRef]

1990 (1)

Aharon, M.

M. Elad, M. Aharon, IEEE Trans. Image Process. 15, 3736 (2006).
[CrossRef]

Andres, P.

Aragon, J. L.

Artal, P.

Awwal, A. A.

J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).

Baraniuk, R. G.

D. Baron, S. Sarvotham, R. G. Baraniuk, IEEE Trans. Signal Process. 58, 269 (2010).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Baron, D.

D. Baron, S. Sarvotham, R. G. Baraniuk, IEEE Trans. Signal Process. 58, 269 (2010).
[CrossRef]

Beck, A.

A. Beck, M. Teboulle, SIAM J. Imaging Sci. 2, 183 (2009).

Beckers, J. M.

J. M. Beckers, Annu. Rev. Astron. Astrophys. 31, 13 (1993).
[CrossRef]

Bille, J.

Bille, J. F.

Boruah, B. R.

B. Pathak, A. Das, B. R. Boruah, Proc. SPIE 8557, 85570A (2012).
[CrossRef]

Bower, B. A.

Brady, D. J.

Candes, E. J.

E. J. Candes, T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

E. J. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Carroll, J.

Choi, K.

Choi, S.

Christofferson, J.

Clemente, P.

Climent, V.

Cooper, R. F.

Cox, I. G.

Dainty, C.

R. Mackey, C. Dainty, Proc. SPIE 7108, 71080I (2008).
[CrossRef]

Das, A.

B. Pathak, A. Das, B. R. Boruah, Proc. SPIE 8557, 85570A (2012).
[CrossRef]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Drexler, W.

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Dubis, A. M.

Dubra, A.

Duran, V.

Elad, M.

M. Elad, M. Aharon, IEEE Trans. Image Process. 15, 3736 (2006).
[CrossRef]

Eriksson, B.

Fang, L. Y.

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

L. Y. Fang, S. T. Li, Q. Nie, J. A. Izatt, C. A. Toth, S. Farsiu, Biomed. Opt. Express 3, 927 (2012).
[CrossRef]

Farsiu, S.

Fercher, A. F.

Fernandez, E. J.

Fontanella, J. C.

Friedlander, B.

Goelz, S.

Grimm, B.

Guirao, A.

Guyon, O.

O. Guyon, Astrophys J. 629, 592 (2005).

Hermann, B.

Hofer, H.

Horisaki, R.

Hosseini, M. S.

M. S. Hosseini, O. Michailovich, in 17th European Signal Processing Conference (2009), p. 115.

Huang, Y.

Izatt, J. A.

Jaeken, B.

B. Jaeken, P. Artal, Investig. Ophthamol. Vis. Sci. 53, 3405 (2012).

Javidi, B.

Jones, S. M.

Kang, J. U.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Kuo, A. N.

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

Lancis, J.

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Laut, S.

Li, S. T.

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

L. Y. Fang, S. T. Li, Q. Nie, J. A. Izatt, C. A. Toth, S. Farsiu, Biomed. Opt. Express 3, 927 (2012).
[CrossRef]

Liang, J. Z.

Lim, S.

Lin, J.

J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).

Mackey, R.

R. Mackey, C. Dainty, Proc. SPIE 7108, 71080I (2008).
[CrossRef]

Marks, D. L.

McNabb, R. P.

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

Michailovich, O.

M. Rostami, O. Michailovich, Z. Wang, IEEE Trans. Image Process. 21, 3139 (2012).
[CrossRef]

M. S. Hosseini, O. Michailovich, in 17th European Signal Processing Conference (2009), p. 115.

Milanfar, P.

Miller, D. T.

Monteiro, D. W. D.

D. W. D. Monteiro, G. Vdovin, P. M. Sarro, Sens. Actuators A Phys. 109, 220 (2004).

Nie, Q.

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

L. Y. Fang, S. T. Li, Q. Nie, J. A. Izatt, C. A. Toth, S. Farsiu, Biomed. Opt. Express 3, 927 (2012).
[CrossRef]

Nirmaier, T.

Norris, J. L.

Nowak, R.

Olivier, S. S.

Pathak, B.

B. Pathak, A. Das, B. R. Boruah, Proc. SPIE 8557, 85570A (2012).
[CrossRef]

Porter, J.

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, J. Opt. Soc. Am. A 18, 1793 (2001).
[CrossRef]

J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).

Prieto, P. M.

Primot, J.

Pudasaini, G.

Queener, H.

J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).

Rivenson, Y.

Romberg, J.

E. J. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Rostami, M.

M. Rostami, O. Michailovich, Z. Wang, IEEE Trans. Image Process. 21, 3139 (2012).
[CrossRef]

Rousset, G.

Sarro, P. M.

D. W. D. Monteiro, G. Vdovin, P. M. Sarro, Sens. Actuators A Phys. 109, 220 (2004).

Sarvotham, S.

D. Baron, S. Sarvotham, R. G. Baraniuk, IEEE Trans. Signal Process. 58, 269 (2010).
[CrossRef]

Sattmann, H.

Shakouri, A.

Singer, B.

Stern, A.

Sulai, Y.

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Tajahuerce, E.

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

Tao, T.

E. J. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. J. Candes, T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

Teboulle, M.

A. Beck, M. Teboulle, SIAM J. Imaging Sci. 2, 183 (2009).

Thorn, K.

J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).

Toth, C. A.

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

L. Y. Fang, S. T. Li, Q. Nie, J. A. Izatt, C. A. Toth, S. Farsiu, Biomed. Opt. Express 3, 927 (2012).
[CrossRef]

Unterhuber, A.

Vdovin, G.

D. W. D. Monteiro, G. Vdovin, P. M. Sarro, Sens. Actuators A Phys. 109, 220 (2004).

Wang, Z.

M. Rostami, O. Michailovich, Z. Wang, IEEE Trans. Image Process. 21, 3139 (2012).
[CrossRef]

Werner, J. S.

Williams, D. R.

Xu, D.

Zawadzki, R. J.

Zhao, M. T.

Annu. Rev. Astron. Astrophys. (1)

J. M. Beckers, Annu. Rev. Astron. Astrophys. 31, 13 (1993).
[CrossRef]

Appl. Opt. (2)

Astrophys J. (1)

O. Guyon, Astrophys J. 629, 592 (2005).

Biomed. Opt. Express (3)

IEEE Signal Process. Mag. (1)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 25(2), 83 (2008).
[CrossRef]

IEEE Trans. Image Process. (2)

M. Rostami, O. Michailovich, Z. Wang, IEEE Trans. Image Process. 21, 3139 (2012).
[CrossRef]

M. Elad, M. Aharon, IEEE Trans. Image Process. 15, 3736 (2006).
[CrossRef]

IEEE Trans. Inf. Theory (3)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. J. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. J. Candes, T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

IEEE Trans. Med. Imaging (1)

L. Y. Fang, S. T. Li, R. P. McNabb, Q. Nie, A. N. Kuo, C. A. Toth, J. A. Izatt, S. Farsiu, IEEE Trans. Med. Imaging 32, 2034 (2013).
[CrossRef]

IEEE Trans. Signal Process. (1)

D. Baron, S. Sarvotham, R. G. Baraniuk, IEEE Trans. Signal Process. 58, 269 (2010).
[CrossRef]

Investig. Ophthamol. Vis. Sci. (1)

B. Jaeken, P. Artal, Investig. Ophthamol. Vis. Sci. 53, 3405 (2012).

J. Opt. Soc. Am. A (5)

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (2)

B. Pathak, A. Das, B. R. Boruah, Proc. SPIE 8557, 85570A (2012).
[CrossRef]

R. Mackey, C. Dainty, Proc. SPIE 7108, 71080I (2008).
[CrossRef]

Sens. Actuators A Phys. (1)

D. W. D. Monteiro, G. Vdovin, P. M. Sarro, Sens. Actuators A Phys. 109, 220 (2004).

SIAM J. Imaging Sci. (1)

A. Beck, M. Teboulle, SIAM J. Imaging Sci. 2, 183 (2009).

Other (2)

M. S. Hosseini, O. Michailovich, in 17th European Signal Processing Conference (2009), p. 115.

J. Porter, H. Queener, J. Lin, K. Thorn, A. A. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications (Wiley, 2006).

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 (6)

Fig. 1.
Fig. 1.

Schematic of optical setup used to acquire wavefront data. Red laser light was collimated and passed through linear polarizers (LP), a polarizing beam-splitter (PBS), a 5× beam expander, a quarter-wave plate, a deformable mirror (DM), and a relay telescope before being measured by a SHWFS. The symbols represent the polarization of the light while traveling from left to right in the system.

Fig. 2.
Fig. 2.

Flow diagram of experiments depicting (a) the denoised HLD, (b) the simulated noisy HLD and the sparse samples of a (c) compressed, (d) linearly interpolated, and (e) SPARZER reconstructed wavefront slope in the x-direction with simulated noise. The mean squared error of the wavefront slope was (d) 0.1451 and (e) 0.0254.

Fig. 3.
Fig. 3.

Reconstruction accuracy of the sparsely sampled HOAs of a typical human eye for various amounts of simulated noise at a 0.10 compression ratio. Error bars correspond to the 1000 realizations of additive noise (10 for cdcSR Wavelet).

Fig. 4.
Fig. 4.

Error in reconstructing the HOAs of a typical human eye sparsely sampled with various compression ratios. Error bars correspond to the 100 different sampling patterns at each compression rate (10 for cdcSR Wavelet).

Fig. 5.
Fig. 5.

Error in reconstructing wavefronts dominated by defocus aberration for various compression ratios (considered the worst-case scenario for our SPARZER method). Half-diopter (solid lines) and five diopters (dotted lines) of defocus are presented. Error bars correspond to the 100 different sampling patterns at each compression rate (10 for cdcSR Wavelet).

Fig. 6.
Fig. 6.

Error in reconstructing randomly generated high-order wavefronts sparsely sampled with various compression ratios. Error bars correspond to the five random HOA patterns at each compression rate.

Equations (2)

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

ϕ(x,y)x=fx=Zcxandϕ(x,y)y=fy=Zcy,
c^=argmin{12ψZcb22+λc1},

Metrics